Note: Descriptions are shown in the official language in which they were submitted.
I
FIELD OF THE INVENTION
This invention relates to the field of encoding data
related to a variable size character for access and display,
and particularly, to the encoding of display points
projected in the shape of a continuous smooth curve. The
field of this invention is the field of character and symbol
generation with continuous and smooth outline curves.
DESCRIPTION _ THE PRIOR ART
The prior art contains many examples of character
generating methods and systems. One such example is
US. Pa-tent 4,029,947 ('947) which generates characters from
a single encoded master but uses straight line interpolation
to approximate curves between a set of given points. Other
encoding systems are shown in US. Patent No. 4,298,945
('945) and US. Patent No. 4,199,815 ('815) which also show
a system of encoding straight lines using -the end points and
then interpolating points on the locus of straight lines
between the end points and about the outline of a character.
A further patent is US. Patent No. 4,338,673 ('673~, which
stores a character in a single size by encoding straight
line outlines of the character and then uses that encoding
to generate points along straight line approximations of the
outline at a desired size, as in the '945 patent and the
'815 patent.
However, these and other curve generation techniques do
not use a system whereby the coordinate points on the
outline of a character are encoded in a single master and
then, by utilizing a parametric cubic expression of a single
variable, a series of signals describing nodes on the locus
of a smooth continuous curve between those points, are
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generated for display of the character at any variable size.
Parametric cubic curves are known and shown in
"Fundamentals of Interactive Computer Graphics", JO Foley
and An dries Van Dam; Addison Wesley, Reading Mass, 19~2.
Shown therein are parametric cubic curves, as functions of a
single variable, used to represent curve surfaces.
Further, TEXT and METAFONT, Donald E. Knuth; American
Math Society, Digital Press, Bedford, Massachusetts, 1979,
shows the use of a parametric cubic curve wherein, given the
coordinate positions of a set of end points and the first
derivative (slope) of a curve at the end points, a
parametric cubic polynomial pa function of a single
parameter t), can be used to generate the locus of points on
a smooth continuous curve segment between those end points.
As shown by Knuth, the locus of each curve segment depends
on the location ox the -two end points of that segment, for
example, Al and Z2' and the angle of the curve at Al'
determined by the location of adjacent points Zoo Z2 and Z3.
In fitting the curve between two end points, the angles at
Al and Z2 are predetermined by Knuth. For example, where
the curve is from Z0 to 21 to Z2' Knuth assumes, as a rule,
the direction the curve takes through Al is the same as the
direction of the arc of a circle from Z0 to Al to Z2
However, not all curved outlines will satisfy this rule and,
in many cases, Knuth requires a further adaptation of that
foregoing process to produce the desired shape, namely of a
curve representing the actual smooth curved outline of a
character.
Nuthouse process starts using the rule described above
specifying a circle when fitting a curve between three given
points Al' Z2' Z3- Knuth does show that a parametric cubic
curve as a function of a single parameter t, as also shown
in "fundamentals of Interactive Computer Graphics", can be
modified, as shown on page 20 of Knuth, to include
"velocities" specified as "r" and "s" and which are
functions of the entrance angle and exit angle of the curve
segment at the given end points. The velocities' values
determine how the resulting curve will vary as a function of
t, either slowly describing a longer curve distance or more
quickly describing a shorter curve distance. The r and s
formulas are arbitrary and, in the case shown by Knuth, are
chosen to provide excellent approximations to circles and
ellipses when equals Q and + Q equals 90 degrees.
Additional properties chosen to be satisfied by Nuthouse
arbitrary formulas for r and s are further described in TEXT
and METAFONT.
In addition to Nuthouse internal requirement to specify
an entrance and exit angle at the curve end points, where no
previous angle or curve history is given, Knuth also must
use the circle approximation rule, specified above, for each
of the end points on the curve. This means that where the
actual curve to be generated is not accurately produced by
the Knuth circle approximation, an adjustment must be made.
Knuth makes this adjustment by manipulating adjoining
points. For example, where the curve is to be fit between
points Al Z2 and Z3. Knuth may manipulate the locations of
Z0 or Al or Z3 or Z4 to obtain an accurate fit. A further
problem in Knuth occurs when the sign of e is the same as
the sign of implying a) the curve entrance angle at point
Al is in the same quadrant as the curve exit angle at Z2' b)
a sine wave between Al and Z2 and c) an inflection point.
Knuth must resort to a manipulation of point locations to
obtain a desired smooth curve around the inflection point.
to
In summary, the processes described above for producing
outlines and showing the use of a parametric cubic curve
expressed as a polynomial capable of generating a series of
points along the locus of a curve between two given points
can only do so with the disadvantages noted above. In the
following summary of the invention, the method and system
for using a parametric cubic curve which accurately
reproduces a curve section between given points is shown and
described and which method and system avoids the
disadvantages discussed above.
,
SUMMARY OF THE INVENTION
This invention has as its object the generation,
encoding and display of a series of points (nodes) along the
locus of a curve segment between two given end points which
are defined as "knots". It has been specifically developed
to generate the locus of points (nodes) along the smooth
curved outline of characters or symbols between given knots
' aye Zap Zulu Zn-l~ Zen but it should be
understood that its use is not necessarily limited to that
purpose but may be used to generate and display smooth
continuous curves between any end points, regardless of the
application.
In the case of the applicant's preferred embodiment,
the knots are coordinate points along the outline of a
character, which may be an alphanumeric or any other
character or symbol and which coordinate points may be
representative of a master size encoded character at a
normalized size, on a dimensionless normalized encoding
grid. The coordinate points may be decoded for display at a
predetermined normalized display size or at an expanded or
reduced size. The knots may be along any outline or surface
whether two or three dimensional, although the applicant's
preferred embodiment is shown in a two-dimensional model for
use in smooth continuous curve generation. Using the method
and system of applicant, the nodes along a two-dimensional
curve or three-dimensional surface locus may be generated.
The system shown herein generates the nodes using the
encoded knot positions and the slopes of the curve at the
knots. The process generates signals representing the
nodes' coordinate values and is based upon the length (Z)
between the knots and the angular interrelationship (B) of
each of the knots with respect to a reference angle on the
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encoding grid.
In the preferred embodiment, the knots' coordinates are
encoded in a closed data loop and represent dimensionless
coordinates about the closed outline loop of the character
or symbol. The method described herein enters the closed
data loop at a set of encoded coordinates representing a
knot on the closed outline loop and initiates the node
generating process by using the angular interrelationships
of the knots about that entry knot. However, other methods
may be used to initiate the process without departing from
the inventive concept. In applicant's preferred embodiment,
once having entered the encoded closed data loop, the
analysis may proceed in a clockwise or counterclockwise
direction around the closed data loop.
It should be understood that the principles of this
invention are not limited to a closed outline loop or closed
data loop, but may be applied to open outline loops and to
open data loops.
The process may be used to generate either a smooth
continuous curve between the knots or some other form of
curve, forming a cusp at a knot for example, and which
although continuous, it is not smooth at all locations but
angular, as in the shape of a "K" or 'IT''. In the preferred
embodiment of the invention, the selection of a smooth
continuous curve or merely a continuous curve, having a cusp
for example, is made using stored rules which serve as
default command codes which use the inter knot angles formed
between the knots, and the distance between the knots to
produce a desired result. For example, where the angle
formed by two knots is in excess of a predetermined
threshold angle, a default command code may direct that a
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cusp be wormed. Other rules such as override control codes
may be similarly used to force a cusp or smooth curve,
regardless of the threshold conditions and which may be
encoded in the closed data loop to override the default
command codes as explained below.
Assuming, for the purpose of explaining the invention,
it is desired to connect all knots in the closed outline
loop with smooth continuous curves, the angles formed
between an entrance knot at which the representative encoded
data loop is entered and the first and second successive
knots thereto in a chosen progression of knots in the closed
outline loop are averaged to produce average angle values.
As the curve formed between knots is a continuation of a
curve which enters at the first of the knots (Zoo and which
exists at successive or at a second of the knots (Zulu the
tangent angles of the curve, which are the entrance angle at
the first knot and exit angle at the second knot can be
specified as average angles. The analysis continues by
proceeding around the closed outline loop in the established
progression of knots and the order thereof of the knots and
examining the angles formed between the knots as described
above.
As the knots form a master skeletal outline of the
character or symbol when juxtaposed on a dimensionless
encoding grid, arranged at a normalized size, the knots may
be rotated or scaled relative to such a normalized encoding
grid. Once the master encoded character or symbol (master
encoded character) is scaled and positioned as desired, then
the analysis described above, may continue wherein a) the
encoded closed data loop of knots in the closed outline loop
is entered at a first set of coordinates representing a
first knot, b) the angular and special relationships ox the
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knots on the normalized encoding grid are determined, using
the representative encoded data and c) for each set of knots
along the closed outline loop, assuming that each knot is to
be connected by a continuation of a smooth curve, the
average angles of the respective curve segments entering and
exiting the respective knots are determined. Where a smooth
continuous curve is desired, then the entrance angle of the
curve segment at a knot would normally be the same as the
exit angle of the curve from that same knot.
By using the foregoing method of analysis, namely,
determining the average angles made by the curve passing
through the knots, and using those average angles to
represent the slope of the tangents to the particular curve
segment at the respective knots, the disadvantages shown in
the prior art are overcome.
In particular, in this inventive concept, there is no
need to define a circle between the knots and then compute
the locus of nodes on the circle as Knuth does. A faster
process results using this invention which requires only the
inter knot angles of the knots relative to each other and to
a reference angle be known. The process as stated above,
uses a parametric cubic polynomial relationship of a single
variable t to generate signals or data, indicative of and
representing -the coordinates of nodes on the locus of a
smooth continuous curve segment between sets of respective
knots. Where a greater resolution and greater number of
nodes is needed to describe the locus, then the incremental
value of the parameter t, may be decreased providing a
greater number of discrete cumulative values of t and node
coordinates. Where less resolution and fewer nodes are
needed to define the locus, then the incremental value of t
can be enlarged, producing fewer discrete cumulative values
of -t and nodes.
In the preferred embodiment, the dimensionless
normalized master encoding grid represents an EM Square (M )
of 864 by 864 Dimensionless Resolution Units drowsy) in each
of the X, Y axes. The My is a measure used in the
typesetting trade wherein a character is set within an My
and is shown herein in a manner consistent with the
application of the inventive principles in the preferred
embodiment. When the master encoded character is to be
displayed, the master encoded character, at a normalized
size on the normalized encoding grid is scaled, the knot
coordinate positions at the scaled size are encoded and
expressed in the appropriate display intercept units such as
Raster Resolution Units (Russ). These scaled encoded knots
in the preferred embodiment are expressed in terms of the
display Russ and used to determine the incremental value of
t. The incremental value of t is used to generate discrete
cumulative values of t which are then applied to the cubic
parametric polynomial to generate the node coordinates.
Then through the cubic parametric polynomial relationship,
signals indicative of the coordinates of the nodes are
generated and stored as data. As defined by a parametric
expression of a single parameter t, the resultant node's x,
y coordinate values (X or Y are the axial directions in the
preferred embodiment), vary separately as separate functions
of t [i.e. I, v]. At the scaled size, the
incremental value of t may be related to the reciprocal of
the distance between knots expressed in Russ (i.e. Ed
l Zn-zn-l I ) or any other suitable method may be used to
produce a value of t. Values representing the node
coordinates are then generated using these incremental
values of t, each value being added, to the previous
cumulative values (with the second incremental value of t
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being added to the first incremental value of t and the
third incremental value of t being added to the previous
cumulative value of t and so on). In the preferred
embodiment, the value of t is set to vary from O to 1.
The resulting series of signals, stored as encoded
data, represent knots and nodes which define the locus of
the smooth continuous curve between the knots and the
outline of the character or symbol is a machine part
ultimately used to control or modulate a display to form the
desired character or symbol at the desired size, in a visual
image. The resultant data may be run length data, which is
applied directly to a raster beam, to position the beam and
energize the beam accordingly, or may be used to control a
free running raster.
Interpolation, rounding or truncating of value may be
used to locate the nodes on the display intercepts
corresponding to the raster line locations, where exact
coincidence is lacking.
As previously stated, as characters or symbols are not
always smooth curves but may contain cusps, a threshold test
may be used, such as one based on the exterior angle formed
by lines between knots. Where that exterior angle at a knot
is greater than a predetermined threshold angle, for
example, a cusp may be assumed. It being understood,
however, -that if the exterior angle is less than the
threshold angle, the analysis previously described with
regard for producing a locus of nodes to define a smooth
continuous curves would be used.
In the preferred embodiment, the master encoding grid
is a Cartesian coordinate system. As the preferred
embodiment is used in typesetting, the encoding grid
relative to which the character is encoded is set within an
My, which in the preferred embodiment contains 864 by 864
Dimensionless Resolution Units (Drums). The master encoded
character, at its normal size is set over a portion of the
available normalized encoding grid area of the My. As is
known in the typesetting field, expansion areas in the My
are also provided for large size characters. The character
may be scaled, rotated or projected by ordinary known
techniques and new coordinates for the knots may be
determined accordingly, as is known in the art. In the
preferred embodiment, the scaling is done in increments of
1/1024 Drums. The new coordinate locations of the knots for
the character at its scaled, rotated or projected positions
are determined. As only integers are used in the preferred
embodiment, any fraction or equivalent thereof is discarded.
In the process, the percent of reduction or enlargement is
first calculated in relation to a desired character size in
units of typesetter's points. The precision of the scaling
is increased by an auto scaling linear interpolation
increasing the resolution of the linear interpolation, as
explained below. The result is a scaled coordinate point in
Russ without the need to utilize floating point arithmetic.
In the preferred embodiment and as stated above,
override control codes are accessed responsive to data
stored with the stored knot coordinates to reduce storage
and the processing time.
The code 0 is used to indicate the end of all the
loops.
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A code 1 is used to indicate the movement in a
relatively long direction on an axis, for example, the X
axis. In this case, an X value is replaced with a new X
coordinate value.
The code 2 indicates the same process as a code 1 for
another axis, for example, the Y axis direction, where the Y
value is replaced with a new Y coordinate value.
A code 3, as in codes 1 and 2, indicates X and Y are
both replaced with new coordinate values.
A code 4 indicates the finish of a previous encoded
loop and the start of a new loop.
Codes 5, 6 and 7 indicate that the ZOO or MY directions
are respectively altered.
Codes 8 to 11 are editing commands forcing
predetermined conditions for the curve at the respective
knots as will be described.
The knots may be encoded in the closed data loop on a
4-bit memory boundary (nibble) and in the preferred
embodiment, the first nibble value of a complete information
set of nibbles is used to specify the number of nibbles used
in the complete information set.
Additionally, the data is packed in a novel manner
which can be interpreted as special information or control
codes, as will be explained.
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, . .. .
In summary, the inventive concept is a process and
system for transforming a machine part, in the form of
signals, encoded as data and representing a pattern of knots
on the outline of a master size symbol, into a similar
pattern at a reduced or enlarged size or transposed in
space, by generating a series of encoded data signals
representing nodes which more definitely define the said
pattern in the shape of smooth continuous curves or cusps
and which data signals may be directly used to control a
display process to visually display the pattern.
Accordingly, what is disclosed is a method and system
for generating a series of signals representing nodes on a
locus of a curve partially defined by a set of related
knots, encoded as data, with said knots defining the end
points of respective segments of said curve locus and with
said knots being in a successive order in relation to said
locus, and for encoding said node signals as data for use
when representing said curve segments in a separate
additional process responsive to the shape of said curve
locus, as represented by said encoded node signals. The
method and system involve defining the locations and the
successive order of said knots on said curve locus and
encoding as data, signals indicative of said knots, then for
a first knot, (Zoo representing a first end point of a
first curve segment, deriving a first angle, indicative of
the average of the inter knot angles between said first knot
(Zap)' and selected related knots and encoding as data,
signals indicative of said first angle, at a second of said
knots (Zb)' representing a second end point of said first
curve segment, establishing a second angle for said first
curve segment, and encoding as data, signals indicative of
said second angle, establishing a compiler for compiling
data according to a cubic parametric polynomial relationship
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between a parameter "t", said knots and angles at the said
end points of a said curve segment and the locus of a said
curve segment, establishing a range "R" of values for said
parameter 'it", applying said signals indicative of the said
locations of said first and second knots of said first curve
segment, to said compiler, applying said signals indicative
of the said first and second angles of said first curve
segment to said compiler, applying a signal indicative of a
distinct selected value of said parameter "t" within said
range "R", to said compiler to derive a signal indicative of
a respective node location on said first curve segment,
repealing the above by applying signals indicative of
additional distinct selected values of said parameter "t",
within said range "R", to derive a plurality of signals
indicative of respective node locations on said locus of
said first curve segment for respective distinct selected
values of said parameter "t", and encoding said signals
derived in step h) and i), in a data base to represent said
first curve segment.
Further disclosed is a method of encoding data and an
encoded data system representing knots on an outline defined
relative to a coordinate plane involving selecting sets of
coordinates on said outline, to represent the knots,
establishing a successive order of said knots, encoding said
knots in a data order indicative of said knot order, and
encoding by encoding a complete information set of data
providing a control code indicative of either i) the
coordinate locations of said knots or ii) a knot's direction
relative to others of said knots or iii) a predetermined
shape of said outline between a pair of said knots or iv)
data indicative of the shape of said outline at a knot or v)
encoding a complete information set providing the coordinate
distances between adjacent knots.
,
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Further disclosed is a method and system for encoding
data representing knots on an outline defined relative to a
coordinate plane and for generating a series of signals
representing nodes on the locus of a curve partially defined
by said set of knots, involving selecting sets of
coordinates on said outline, to represent said knots,
establishing a successive order of said knots, encoding said
knots in a data order indicative of said knot order, and
encoding a complete information set of data providing a code
indicative of a predetermined shape of said outline between
a pair of said knots or a complete information set providing
the coordinate distances between adjacent knots.
Further disclosed is a method and system for generating
a series of signals representing nodes on a locus of a curve
partially defined by a set of related knots, encoded as
data, with said knots defining the end points of respective
segments of said curve locus and with said knots being in a
successive order in relation to said locus, and for encoding
and decoding said node signals as data and for use of said
data when in an imaging process responsive to the shape for
said curve segments as represented by said encoded data,
involving defining the locations and the successive order of
said knots on said curve locus and encoding as data, signals
indicative of said knots, for a first knot, (Zap)'
representing a first end point of a first curve segment,
deriving a first angle, indicative of the average of the
inter knot angles between said first knot (Zap)' and selected
related knots and encoding as data, signals indicative of
said first angle, at a second of said knots (Zb)/
representing a second end point of said first curve segment,
establishing a second angle for said first curve segment,
and encoding as data, signals indicative of said second
angle, establishing a compiler for compiling data according
I
to a cubic parametric polynomial relationship between a
parameter "t", said knots and angles at the said end points
of a said curve segment and the locus of a said curve
segment, establishing a range "R" of values for said
parameter "t", applying said signals indicative of the said
locations of said first and second knots of said fist curve
segment, to said compiler, applying said signals indicative
of the said first and second angles of said first curve
segment to said compiler, applying a signal indicative of a
distinct selected value of said parameter "t" within said
range "R", to said compiler to derive a signal indicative of
a respective node location on said first curve segment,
repeating the above by applying signals indicative of
additional distinct selected values of said parameter "t",
within said range "R", to derive a plurality of signals
indicative of respective node locations on said locus of
said first curve segment for respective distinct selected
values of said parameter "t", encoding said signals derived
above in a data base to represent said first curve segment
accessing said data base signals, and controlling an imaging
means responsive to said accessed signals to reproduce said
curve.
Further disclosed is a method and system for linear
interpolation of coordinate points between first and second
end points, to produce coordinates on a straight line
outline and where said coordinates are located on a
coordinate system having a first coordinate direction and
second coordinate direction and encoded in machine readable
data words a radix "r", corresponding to the order of values
for designated positions in said data words, involving
encoding a first data word of Len" positions corresponding to
the distance between the said first and second end points in
the said first coordinate direction and placing said first
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data word into a first machine location, encoding a second
data word of "M" bits corresponding to the distance between
said first and second end points in said second coordinate
direction and placing said second data word into a second
machine location, determining the number of available
positions, between the most significant position of said
first data word and the most significant position of said
first machine location, available for shifting said first
data word in a first direction of the most significant
positions of said first machine location, shifting said
first data word by a maximum number of positions, equal to
the said number of available positions in said first
direction and the number of positions corresponding to the
number of significant positions used to encode said second
data word, and increasing the scale of said first data word
by a scale factor related to the number of said positions
shifted, deriving a third data word indicative of said
second data word in said second machine location divided
into said first data word shifted according to step d),
encoding data words indicative of the coordinate of said
straight line in said second coordinate direction, for
respective ones of said data words encoded according to step
f) encoding a multiple of said third data word, which are
related to a respective coordinate in said first coordinate
direction, on said straight line, reducing the scale of said
multiples of said third data words above, to the scale of
the first data word established prior to the above said
shifting and encoding said third data words produced above
with respective coordinates in said second coordinate
direction to produce said coordinates on said straight line.
Further disclosed is encoding data representing knots
on an outline defined relative to a coordinate plane and for
decoding said encoding data for use in a display process to
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produce images of said outlines represented by said encoded
data by selecting sets of coordinates on said outline, to
represent said knots, establishing a successive order of
said knots, encoding said knots in a data order indicative
of said knot order, the encoding including encoding a
complete information set of data providing a control code
indicative of either i) the coordinate locations of said
knots or ii) a knot's direction relative to others of said
knots or iii) a predetermined shape of said outline between
a pair of said knots or iv) data indicative of the shape of
said outline at a knot, or v) providing data indicative of
the coordinate distances between adjacent knots, decoding
said complete information sets in a decoding order related
to said data order responsive to said complete information
set being indicative of the coordinate distances between
adjacent knots, producing an image of a smooth continuous
curved outline or a straight line between said adjacent
knots or, responsive to said complete information set being
indicative of a said control code, as set forth in i), ii),
iii), or iv), producing an image of a smooth continuous
outline or a straight line according to the said coordinate
locations of said knots relative to adjacent knots in said
successive knot order or producing an image of said outline
being smooth at respective knots or being sharp and forming
cusps at respective knots.
Further disclosed is encoding data representing knots
on an outline loop defined relative to a coordinate plane,
for producing a display image of said outline and decoding
responsive to the interrelationship of said knots on said
outline loop, and imaging said outline loop responsive to
said decoded data involving selecting sets of coordinates on
said outline loop, to represent said cots establishing a
successive order of said knots, encoding said knots in a
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data order indicative of said knot order, encoding a
complete information set of i) data indicative of the
coordinate distances and inter knot angles between adjacent
knots, comparing the relative positions of successive knots
to at least a first inter knot criterion, responsive to said
step do of comparing, i) producing a first indication that a
set of said successive knots is within said criterion, or
ii) producing a second indication that a set of said
successive knots is outside said criterion, and i)
responsive to said first indication imaging said outline
loop in the form of a smooth continuous curve, or ii)
responsive to said second indication, imaging said outline
loop in the form of a straight line, between said set of
successive knots.
Further disclosed is encoding and accessing a single
data set of solution values functionally related to and
representing the solution set to at least two domains of a
variable, involving defining a single data set ox solution
values functionally related to a first domain of a variable
and to a second domain of a variable, arranging said data
set of solution values in an order related to said first
domain and said second domain, accessing said data set of
solution values relative to respective values in said first
domain to derive at least a part of said solution set to
said first domain, and accessing said data set of solution
values relative to respective values of said second domain
to derive at least a part of said solution set to at east
said second domain.
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BRIEF DFJSCRIPTION OF THE DRAWINGS
FIG. lo shows a set of knots it Z Us' Zulu Zoo
. . .) forming the skeletal outline of a master encoded
character.
FIG. lb shows the closed outline loops with reference
to a character.
FIG. to shows curve segments Zulu' Zap Zap Zulu' and
curve Zulu' Zoo and the relationships of the deviance
angles e and I, at knot Zap and Zulu
FIG. id and FIG. to shows in greater detail the
deviance angles of the curve segment Zap Zulu' at the
respective entrance and exit knots, as functions of the
inter knot angle (B) and the entrance and exit angles, and .
FIG. 2 shows the relationships of the knots, tangent
angles and angles and to a curve, for the purpose of
explaining the Hermit interpolator.
FIG. 3 shows a master encoding grid as may be used in
the preferred embodiment, at a normalized size, for encoding
a character or symbol such as the character G shown in
FIG. 3, as a master encoded character, at a normalized size.
FIG. shows the manner a look-up table may be arranged
so access in one direction would provide a value equal to
To, as explained herein, and in the opposite direction, a
value of To, as explained herein.
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Lo
FIG. 5 shows the angular relationships between a set of
knots Zap and Zulu when a sharp knot or cusp is to be
produced.
FIG. pa shows the angular relationships at a set of
knots when a knot between a straight line and a curve is
desired to be smooth.
FIG. 6b shows the angular relationships at a smooth
knot, and between a curve line and a straight line.
FIG. pa shows the angular relationships at a knot
between a curve line and straight line, forced by a control
code to override a default command code, which otherwise
would produce the results shown in FIG. 7b.
FIG. 8 shows the angular relationships about a set of
knots Zap and Zulu when it is desired that a smooth
continuous curve pass between the knots.
FIG. 9 shows a character as may be produced according
to the principles of the invention.
FIG 10 illustrates in block diagram a general type of
system with which the method of this invention con be
practiced.
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., .;, , . ,
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DESC~IPTIOM _ TOE PREFERRED EMBODIMENT
As stated in the summary, the object of this invention
is -to generate a series of display signals, representing
nodes describing the locus of a smooth continuous curve at a
desired size, from a normalized curve encoded as a set of
encoded knots. Normalized is used in its general and
ordinary meaning to denote a norm or standard size.
However, as would be apparent to those skilled in the art,
the displayed curve, described in terms of display intercept
coordinates (RUSS) is dependent upon the resolution of the
display, and the rate or relationship between the number of
raster resolution units (RUSS) at a given display
resolution to the dimensionless resolution units (DRUMS) in
which the curve is encoded at its normalized size and on the
normalized grid. The normalized encoded curve may also be
thought of as a master which is encoded at a master size
relative to the master encoding grid which may have any
suitable coordinate system and which after scaling, may be
used to generate the encoded data representing display
intercepts in raster units, a-t a given raster resolution for
any desired display size character. As the inventive
concepts disclosed herein are used in the typesetting
industry to produce typeset composition containing
characters having curved outlines in conformance to the
highest graphic standards, scaling starts with a
determination of the desired size of the character in any
chosen system of measurement. In the preferred embodiment,
character size is expressed in printer's points (351.282
micrometers/point or 0.01383 inches point It being
understood, however, that the invention can be used in
connection with other units of measurements and with
applications outside the printing or typesetting industry,
without deviating from or changing the inventive concepts
- 23 -
,,
I. ED Jo
shown herein. The invention, as described herein is with
reference to the printing industry, where the master
encoding grid is made synonymous with a dimensionless
encoding grid in the form of a typesetting My. This is the
application of the preferred embodiment of the invention and
discloses the best mode of using the invention.
The problem solved by this invention may be best
considered by viewing FIG. lo which shows a series of data
encoded knots (Zulu' Zap Zulu Zoo ) on a
dimensionless encoding grid having coordinates in the X and
Y axial directions, with the X direction coinciding with a
zero degree (0) reference angle. It should ye understood,
however, that any system of coordinates can be used with any
reference angle chosen without changing the manner in which
the inventive concepts are used.
As shown in FIG. lay a number of knots Zulu' Zap Za-tl'
Zoo' Zoo inclusive to Zion represent an outline loop which
may be smooth and continuous over a series of such knots or
continuous without being smooth over a series of such knots
or a combination of the foregoing. As stated, the knots
represent the skeletal outline of a predetermined normalized
or master size character or symbol (hereinafter referred to
as master size character), on a juxtaposed dimensionless
encoding grid where the coordinates have dimensions of
Dimensionless Resolution Units (Drums). In the case of the
encoded character, the master size is with reference to the
normalized encoding grid and the area within the grid. The
inter knot angles between the knots, with reference to the
reference angle, are shown as generally denoted by "B" and
particularly as Bawl, Bay Bawl and so on for the inter knot
angles at respective knots Zulu' Zap Zulu
FIG. lo shows a skeletal outline as may be typically used
- 24 -
Jo
,, .
for symbol wherein all the knots Zulu through Zion on the
symbol outline are arranged on a closed outline loop. This
relationship in its simplest form could be shown by the
knots for the outline of an O or D, encoded with two such
closed outline loops, for the exterior closed outline loops
11 and 15 and one for the interior closed outline loops 13
and 17 respectively, as shown in FIG. lb.
AS shown in FIG. lay a direction of progression of the
curve outline is chosen with reference to an defined by the
order of the knots around the outline. That progression of
the curve locus and order of knots is shown by numeral 19 in
FIG. Lao The chosen order of knots defining the said
progression, (i.e. a-l' Zap Zoo --- Zion) establishes
an outline loop of knots. That outline loop may be a closed
outline loop which ends upon itself, as shown in FIG. lb, by
closed outline loops 11, 13, 15 and 17. As will be
explained below, a compiler functioning according to a
parametric cubic polynomial is used to generate signals
indicative of nodes which are the coordinates of locations
on the locus of a smooth continuous curve between the knots.
The order of the knots, defines an outline loop and the
respective order of the nodes by their locations relative to
the knots and to each other on the outline loop, as would be
apparent to one skilled in the art. As explained below, the
data representative or indicative of these knots and nodes
are encoded in a data order indicative of the order of the
knots and nodes on the outline loop. This data order
establishes a data loop. As further explained below, the
data loop may be designed to close upon itself so that the
ending data location for the data loop is the starting data
location where data was accessed therefrom in the encoding
process and forming a closed data loop corresponding to the
closed outline loop.
- 25 -
lid f~3 I 6
As stated above, the knots Zap Zulu' etc., may be
encoded in the Cartesian coordinate system as X-Y points,
using as a reference the normalized encoding grid, or may be
encoded in any other system of coordinate points. The
outline between the knots is not encoded initially as it
will be represented by a generated series of nodes and which
will represent the smooth continuous curve locus ox the
outline, according to the principle of the invention. The
display intercept values for the nodes on the curve locus,
at a predetermined display size, are related to the encoded
knots on the dimensionless encoding grid shown in FIG. lo by
a parametric cubic polynomial relationship. Since the
invention is used in a two-dimensional system, the
parametric representation represents the curve locus of such
nodes Z(X,Y~ independently as a third order polynomial
function ox a parameter "t" which is independent of the
encoding grid coordinates. In the preferred embodiment, the
parametric cubic polynomial is shown in a Hermit form, it
being understood that those skilled in the art can use other
forms for defining the polynomial such as the Belier form,
defined in "Fundamentals of Interactive Computer Graphics",
referred to in the foregoing, and to which the improvements
of this invention are applicable.
The parametric representation of a curve is one for
which X and Y are represented as a third order (cubic)
polynomial relationship of a parameter to where:
1.1 I = accept + bxt2 + cut + d
1.2 v = await + byte + cyst + d
- 26 -
or
I.
ox
The Hermlte representation of the parametric cubic
polynomial uses the coordinate positions, of the knots, and
the tangent angles at the knots, such as Zap Zulu' etc. The
Belier representation uses the positions of the curve's end
points and two other points to define, indirectly, the
tangents at the curve's end points. The improvements of
this invention shown herein are applicable to any
representation by the parametric cubic polynomial which uses
the position of the curve's end points and the tangent
angles of the curve at the end pollinates, directly or
indirectly. For sake of explanation, only the Hermit form
will be discussed.
As shown in FIG. 2, given end points Pi, Pi and the
respective tangent vectors R1, R4 at the two end points Pi,
Pi and along the smooth continuous curve segment 18, a cubic
parametric polynomial relationship between a parameter "t"
and the locus of a curve segment 18, between a pair of end
points, Pi, and Pi, may be represented as the following
relationships below.
2.1 I= P1x+~3t2-2t3)(P4x-Plx)+t(t-1)2Rlx+t2(t-l)R4x
2.2 v= P1y+(3t2~2t3)(P4y~P1y)+t(t-1)2R1y+t2(t-l)R4y
where (Pox' Ply) POX Pry) are the coordinate values at Pi
and Pi respectively, and (REX, Ray) and Rex Ray) are the
tangent values at Pi and Pi respectively, with respect to a
straight line between Pi and Pi thereafter called entrance
and exit angles, respectively.
Cubic curves as a minimum are used, as no lower order
representation of curve segments can provide continuity of
position and slope at the end points where the curve
27 -
,. . .
I
segments meet, and at the same time can assure that the ends
of the curve segment pass through specific points.
The derivation of the Hermit parametric cubic
polynomials are shown in "Fundamentals of Interactive
Computer Graphics and the manner of using such parametric
cubic polynomials to define the points along a curve are
further discussed in TEXT and METAFO~T referred to in the
foregoing.
The Hermit form of the parametric cubic polynomial is
as stated is a series of curves as shown by Knuth in Chapter
2 in TEXT and METAFONT and which is written in Euler notation
as
3.1 I = Al (3t2-2t3)(Z2-Z1) + rt(1-t)2$1 - stout
3-2 $1 = essays); $2 = e issues); o <= t <= 1
and, where r and s are positive real numbers.
Equations 3.1, 3.2 define a curve having directions
represented by the deviance angles e and at Al' Z2
respectively.
The relationship shown in 3.1~ 3.2 may be encoded into
a compiler designed to process input data related to the
knot locations, the angles of the curve segment at its end
points, and then for separate selected values of a parameter
"t" to produce signals indicative of node locations on the
locus of the curve segment. The compiler is shown at the
end of the specification.
- 28 -
..
I;,
I
As stated above, the Hermit form of the parametric
cubic polynomial is used in the preferred embodiment of the
compiler but the principles of the invention can be used
with any other cubic polynomial using the directions of the
curve at end points of the curve and the end point
locations.
Assuming a curve direction for a given knot order from
knot to knot, and in particular, from knot Al to knot Z2'
with knot Al being the entrance knot for curve segment Al'
Z2 (hereafter curve segments will be defined by the
respective segment and knots such as "curve Al Z2") and
knot Z2 being the exit knot for curve Al Z2~ then is the
angular direction of the curve Al' Z2 at entrance knot Z2'
and O is the direction of curve Al' Z2 at exit knot Z2 In
the preferred embodiment, the inter knot angle (B) maple by
the straight line from knot Al and Z2~ is defined in terms
of a reference angle given for the normalized encoding grid.
Also, the angles and of the curve are related to that
same reference angle. Therefore, the inter knot angle
expressed below as B and the angles of the curve at the
entrance knot, defined below as and the angle of the curve
at the exit knot, defined below as I, are all defined with
regard to a reference angle.
was further explained below, according to the inventive
principles, deviance angles and I, shown used in equation
3.1 and 3.2, are the curve entrance and exit angles
respectively, defined with regard to the inter knot angles B
and applied to the compiler in -the form of the parametric
cubic polynomial as shown in equation 3.1.
- I -
In explaining the invention, and are used to
identify the entrance angle of a curve segment at a first
knot end point and the exit angle made by that same curve
segment at a successive second knot end point, in the knot
order. The first and second knots define the entrance and
exit knots of the curve segment, with regard to the order of
knots and the outline loop, as explained below. However,
and are defined relative to a reference angle "q" on the
master encoding coordinate grid. e and are the entrance
angle and exit angle in the cubic parametric polynomial of
3.1 and 3.2, defined relative to an inter knot angle B
between the entrance and exit knots (i.e. in the preferred
embodiment, e = - B ; = B - ). The preferred
embodiment uses a process of defining the entrance angle
and exit angle in terms of the inter knot angles on the
outline loop, and with respect to a reference angle and uses
the definition of the entrance and exit angles shown as
and when applying the angles and I, at the respective
entrance and exit knots, to the compiler, as based on and as
required by the derivation of the cubic parametric
polynomial, described herein. However, the angles e and
could be derived directly from the inter knot angles B or
recomputed from the relationship of the outline loop knots
and accessed directly without first deriving and and
without deviating from the principles of the invention.
The parameter "t" of 3.1 is allowed to vary over a
defined range as further explained below, and for each
discrete selected value of t, a discrete node on the locus
of the curve between the entrance and exit knots and defined
by the parametric cubic polynomial is generated by the
compiler. The nodes would be the coordinate location of
points on the locus of the curve described by equation 3.1
for each selected value of "t" and where knots Al and Z2
- 30 -
and the angle of the curve at those knots, given as and I,
derived above, was specified. The derivation of I, BY I, e,
and of "t" and their relationship according to the
principles of the invention are further explained and shown
below
It should be noted, however, that the for of -the
Hermit interpolator used in the preferred embodiment
imposes an opposite sign for than that shown in TEXT and
METAFONT shown in Knuth.
It should also be noted that the deviance angles and
representing the divergence between the straight line
angle inter knot angle By as used in the cubic parametric
polynomial of Al 3.2.
"r" and "s" affect the curve velocities, as described
below and the length of the curve between its respective end
points (i.e., curve Al' Z2 between Al' Z2) ''r" and "s" are
velocities at Al' Z2 respectively, a large velocity value
meaning the curve direction changes slowly while small
values indicate the curve undergoing more pronounced
directional changes [small values of r and s will then have
less influence on the value of I or v]. The
velocities, r and so in TEXT and METAFONT are represented as
Al _ 1 Sweeney
r -
I (l+lcos~l)sin~ I
4.2 s = 1 Sweeney
I (l+lcosl~l)sin
4.3 = (a + O / 2
- 31 -
: ,. , . .. .
:
~3~7~
Considering the effect of r and s on the curve locus
defined by the nodes, produced through the cubic polynomial
relationship, the values of r and s may be limited to
control the direction of the curve from the entrance knot to
the exit knot. In the preferred embodiment, r and s are
limited to the values of 0.1 to 4Ø However, these values
of r and s could be changed without departing from the
principles of the invention.
As discussed in the foregoing, the system shown therein
for using the relationship of the points along the smooth
curve in TEXT and METAFONT have disadvantages which are
eliminated by this invention which will become apparent by
reading the following explanation.
As explained above, in using the parametric cubic
polynomial relationship to define the nodes along a smooth
curve, TAO and METAFONT start with a circle approximation
rule and then must make adjustments to produce a smooth
continuous curve between given points. In this invention,
those disadvantages are eliminated by the invention
described in the following.
In the preferred embodiment, the sign of is reversed
relative to the angle rotation used in TEXT and METAFONT.
The principles of this invention as applied to the
preferred embodiment may now be particularly seen with
reference to FIG. lay As stated above, a skeletal outline
is described by a progression of a series of knots in the
don of Zap Zulu Zen Zulu which defines a closed
outline loop about the outline of the character or symbol.
The knots are encoded as coordinate points in a closed data
loop representing the closed outline loop. As stated, the
- 32 -
i
:,
object of the invention is to produce a series of nodes
representing display coordinates on the locus of a curve
between each of the knots, the curve being smooth and
continuous. However, as will be shown, the principles of
the invention can be modified so a series of points
describing straight lines between knots can be generated and
smooth continuous curves can be generated between knots
which are interspersed with straight lines. In addition,
Cusp can be formed between smooth continuous curve and/or
straight lines.
The smooth continuous curve is produced through the
parametric cubic polynomial relationship described above.
In using the invention, the following principles are
applied. The data loop may be entered at the data values
representing a knot, Zap for example, and the angular
relationship of the knot (Zap) referenced to a proceeding
knot (Zulu) and to a succeeding knot in the loop analyzed.
The analysis may then proceed in a clockwise direction
around the loop which may be considered in the forward
direction. However, as stated, the analysis can proceed in
a counter- clockwise or backward direction with the
direction being used to denote or indicate other values,
such as color, or other characteristics as may be needed.
As will be understood, the directions used herein are chosen
for explanation and do not limit the inventive principles.
FIG. lo represents a series of knots Zap Zulu' Zoo ~~~
Zion' Zulu arranged in a closed outline loop, encoded
relative to a master encoding grid and partially defining,
in skeletal form, the points on a closed outline loop, such
as outline 11, 13, 15, or 17, shown in FIG. lb. The
inter knot angles, the angles from knot to knot (i.e. from
knot Zulu to knot Zap are denoted generally by the letter
- 33 -
,,
~3~7~
"3" with a subscript reference indicating that the angle is
formed by a straight line from a respective knot, Byway for
example, to a successive knot Be in the outline loop. The
terms "preceding" and "successive" can be used with
reference to the defined order of knots in the outline loop,
i.e. either clockwise or counterclockwise, as the case may
be. In FIG. lay the knot angles B are shown as Bay Byway,
Byway, and so on for respective knots.
The manner of applying the cubic parametric polynomial
compiler in the preferred embodiment to define nodes between
the knots along the locus of a smooth curve utilizes the
deviance angle at the curve entrance knot and the deviance
angle at the curve exit knot. The deviance angles and Q
may also be referred to as entrance and exit angles.
According to the principles of the invention, and
assuming a smooth and continuous curve segment is desired to
be developed between a first knot Zap and a second knot Zap
a' Zulu' shown as numeral 21, in FIG to
then the inventive principles may be used to generate a
series of signals representing the display coordinates for
the nodes along the locus of the smooth continuous curve 21
as follows and similarly for the curves between others of
respective pairs of knots. The data encoding of the knots
representing each of the closed outline loops may also be
thought of as a loop, or closed data loop, indicative of and
representing the physical arrangement of the knots in the
closed outline loop about the character or symbol outline.
As stated, the outline loop and the data loop therefore, may
be thought of as an order of knots in the loop path and
related to a predetermined loop direction for such order
clockwise or counter- clockwise, for example). Within that
progression, and for the respective outline loop direction,
- 34 -
,
it can be easily seen that each knot (i.e. Zap represents
an exit point for the curve segment of the outline loop in
the knot order shown by the direction of arrow 20, from a
preceding loop knot (lye. Zap l) and at the same time, the
entrance point for the curve segment from that knot (i.e.
Zap to a successive knot (i.e. Zulu) The nodes, generated
as explained below are given an order in the outline loop
related to the order of knots on the outline loop and the
encoding for the nodes is similarly arranged on the data
loop in the order of the knots.
In explaining the invention, the convention used for
identifying a curve part of the outline loop between knots
is to refer to it by its knot end points, (i.e. curve part
21 is curve Zap Zulu) Similarly, a convention is used to
identify a curve entrance and exit angle as explained below.
The entrance angle for curve Zap Zulu (between knots
Zap and Zulu)' according to our convention would be the
angle represented by the first derivative of the curve Zap
Zulu at knot Zap and the exit angle would be the angle
represented by the first derivative of the curve segment at
the next successive knot Zulu
As would be understood, where a smooth continuous curve
is desired to pass through a knot, (i.e. Zap then the
entrance angle I, at a knot would be the same as the exit
angle at that same knot for the preceding curve segment
Zulu' Zap
In explaining the principles of the invention, the
entrance angles and exit angles for curve segments
starting and ending at knots are defined for the respective
knots according to the following convention. The curve
- 35 -
it:
, ,
segments may be thought of as having an entrance angle at
the knot where the outline loop, in the defined progressive
order of knots, enters a respective curve segment it
curve Zap Zulu) forming an entrance angle pa at that
entering knot, and an exit angle at the next successive
knot in the loop where the loop exits curve segment Zap Zulu
foxing exit angle pa at knot Zulu In the convention
chosen, the entrance and exit angles (pa' pa) for a curve
segment Sue' Zulu) of the loop are referenced to the
entrance knot (Zap) The entrance angle, pa' and the exit
angle, pa' therefore have a subscript reference to the
entrance knot Zap but refer to the curve segment of the loop
and the angle that curve segment makes at a first knot Zap
where the loop enters the curve Zap Zulu and the angle at a
successive knot (Zulu) relative to the outline loop
direction where it exits the curve Zap Zulu The entrance
angle, pa' is then the angle the loop makes as it passes
through knot Zap enters curve Zap Zulu' and continues on to
knot Zulu The exit angle, pa' is the angle the loop makes
as it passes through knot Zulu' exits curve Zap Zulu' enters
the next successive curve Zulu' Zoo and continues on to the
next subsequent exit knot, Zoo In summary, the entrance
angle , I, and exit angle, I, according to the convention
chosen to explain the principles of this invention, refer to
a curve segment of a loop outline between two knots such as
Zap and Zulu' the entrance angle pa being the tangent angle
or first derivative the outline loop makes as it enters
curve Zap Zulu at preceding knot end point Zap in the outline
loop direction, and the exit angle, pa' the tangent or first
derivative of the outline loop it makes as it exits that
curve Zap Zulu at knot end point Zulu and enters the next
successive curve Zulu Zoo' in the outline loop direction.
As can be seen, the exit angle, aye for the curve Zap Zulu
is the same angle as the entrance angle aye' for the next
- 36 -
.
e Zulu Zoo in the outline loop direction
Similarly, the entrance angle, pa is the exit angle, aye
for the proceeding curve Zulu Zap It will be understood by
those skilled in the art that these conventions can be
changed without changing the principles of the invention.
Whereas Knuth uses the rule that a smooth continuous
curve locus between points, such as from aye to Zap to Zulu'
must take the direction of a circle, the invention herein
avoids that rule and the problems created thereby by using
an average of the inter knot angles (B) relative to a knot to
define the respective knot entrance and exit angles, and
respectively, and thereby e and respectively.
For example, in FIG. to, for the curve Zap Zulu' shown
as numeral 21, the entrance angle pa at knot Zap of the curve
Zap Zulu' for example, is related to the average of the
inter knot angles Byway (from the proceeding knot Zulu to knot
Zap and Be (from Zap to the succeeding knot Zulu) That
average angle of the outline loop at any knot Z is also
expressed as when referring to the entrance angle at a
knot and into a curve segment and when referring to the
exit angle from a knot and out of a curve segment. In the
case of knot Zap the average entrance angle would be pa for
the curve Zap Zulu shown by numeral 21), proceeding from Zap
to Zulu in the order shown by arrow 20 and aye for the
average exit angle for the curve Zulu' Zap (shown by numeral
23), proceeding from Zap 1 to Zap in the order shown by arrow
20.
The angles or are always referred to as entrance
and exit angles respectively herein, but may or may not be
average angles or not depending on the application of each,
as explained herein.
- 37 -
.,
I
The invention is explained with reference to the
example of curve Zap Zulu' wherein Zap is the entrance knot
and Zulu is the exit knot, it being understood that the same
analysis would apply Jo other curve segments either
adjoining or further remote from curve Zap Zulu and having
respective entrance and exit knots.
In using the compiler to generate the nodes along a
smooth continuous curve, the encoded data loop representing
the knots along the closed outline loop may be entered at
any set of data representing any knot Z, or example,
designated to be the entrance knot and the average angle
determined by proceeding in the chosen loop direction using
the inter angular relationships of a knot to its related
knots in the loop, to determine the average angles and
at such related knots. In the preferred embodiment, these
related knots are adjacent knots to the next successive knot
to the entrance knot.
In the preferred embodiment, the inventive principles
of generating the nodes on the smooth continuous curve
locus, starts with a first average angle, determined for the
knot coordinate successive to the data loop entrance knot,
and which is represented by the data residing at the
location in the closed data loop where that data loop was
entered. The inter knot angle for the entrance knot is then
saved and used at the completion of the loop analysis to
determine the average entrance and exit angles at that
entrance knot when the outline loop analysis reaches that
entrance knot in the closed outline loop knot. Where the
data loop entrance knot is Zulu' its average angle would be
determined using the preceding inter knot angles Byway and
Byway for related knots Zulu and Zap
- 38 -
'
-
Following the above, the average angle pa and exit
angle aye at knot Zap is the average of Be 1 and Be and
expressed as
pa = ( Bay Be ) / 2 aye
The average exit angle pa and entrance angle aye at
knot Zulu is the average of Be and Byway and expressed as
= ( Byway + Be ) / 2 aye
As shown, the average entrance angle for any curve
segment is the average ox the inter knot angles at the
respective curve segment knot and at related knots, (i.e.
the preceding knot in the outline loop) and the average exit
angle is the average of the inter knot angles at the
respective curve segment exit knot and at related knots
(i.e. the next successive knot in the outline loop), as
shown above.
The parametric cubic polynomial compiler may then be
employed to relate the entrance and exit angles I, for the
respective entrance and exit knots to a series of nodes
describing the locus of a smooth curve between the
respective entrance and exit knots (i.e. Zap Zulu)
For the example above, the knots are Zap Zulu and the
respective entrance and exit angles are pa and pa for curve
Zap Zulu
The curve velocities r and s are related to the
deviance angles e and I, as set forth above and as used in
equation 4.1, 4.2, 4.3. As stated in the foregoing, e is
the deviance between the average entrance angle, for
- 39 -
example, pa for curve Zap Zulu' at an entrance knot Zap and
the inter knot angle spa at that entrance knot Zap Similarly,
is the deviance between the average exit angle, for
example, at the exit knot, Zulu' for the curve Zap Zulu and
the inter knot angle Be at that entrance knot Zap
The relationships between pa' Be' pa' eta a pa
more clearly seen in FIGS. id and to for knots Zap and Zap
respectively.
s a reminder, it should be remembered that the angles,
9 and are the tangents to the locus of the curve
represented by the nodes generated using the cubic
parametric polynomial compiler with respect to the
respective inter knot angles.
The coordinate values of Zap Zulu used through the
parametric cubic polynomial compiler to generate the node
values are the scaled intercept values given in Russ,
derived from the master encoded values of the knots, in
Drum. For example, assuming an encoding grid shown as an My
in FIG. 3 and defined as 864 X 864 Drums between master
encoding grid (zoo) coordinates 592, 736; 592, 1600; 1456,
1600 and 1456, 448 wherein zoo points 592, 736 are defined
as relative 0,0, relative to an x, y offset of 592, 736
respectively) and wherein the encoding grid is defined as
2047 by 2047 Drums.
In the preferred embodiment, the universe of the master
encoding grid comprising the My is 32768 X 32768 Drums and
the offset which positions the origin of the My can be
positioned anywhere in that universe. In the preferred
embodiment, the x and y offsets and shown in FIG. 3.
- 40 -
I",
The zoo coordinates of the intercept locations for
knots Zap at the displayed size in Russ may be derived prior
to generating the nodes by scaling, using the following
conversion factor (OF), for the preferred embodiment.
Where:
"P" in the desired display size in points per My (Pt/M2),
"Ryes" is the ratio of micrometers per point (uM/Pt) and
serves to convert -the display size from points to metric
units;
ROME" is the display resolution in Raster Resolution
Units per Micrometer;
DRY is the inverse of the encoding grid resolution, in
Drum per My;
and
Pressroom DRY) = (CF)*(RRU/DRU)
Err Master Coordinate Zn(Xn, Yin
(On Drum) (CFx Residuary) = On Russ, and
(Yin Drum) (Cry Residuary) = Yin Russ
The value of the parameter t may be derived from the
respective curve segment inter knot distance Ed = l Zn+l
Zen I given after scaling in Ruses described below. The
generated coordinates for the nodes are expressed in Russ
permitting run length encoding of the display outline
coordinates. Where Ed is a non-integer, the fractional
value may be discarded or rounded with redundant values
eliminated.
In the preferred embodiment of this invention, the
incremental value of t is related to the inverted value of
the absolute difference Ed between the entrance and the exit
knots,
i.e. Tenneco = 1/Zd = l/lZn~1-Znl
: - 41 -
..~ , . ., , .
Zen may be easily derived from the axial difference in
the direction, [i.e. Ed = [(Xn+1-Xn)/Cosine(Bn)], or the
axial difference in the Y direction,
[Ed = [(Yn+l-yn)/sine(Bn)]; Where On Yn+1; Xn+1' Yn+1 are
successive knots defining end points for a curve segment of
the closed outline loop and By is the inter knot angle there
between).
In practice, it is better to use the larger axial value
in the x or Y direction, to minimize error. In implementing
the invention, the preferred embodiment limits increments of
t to 1/1024. Discrete values of To = it - it ,
To = t (1-t) and To = tot (see equation 3.1, 3.2) may be
stored in a look-up table for 1024 values of t in the range
1/1024 t 1024/1024.
In practice t is stored in the range of 1023/1024
because the knot end point coordinate corresponding to t =
1024/1024 or 1 is saved, avoiding the need to calculate that
knot end point and providing a more reliable result.
Further, because of the relationship of To, To and To,
a look-up table as shown in FIG. 4, having values of To
starting with to and ending with a value of To for to may
be accessed in reverse order for values of To, as shown in
FIX&. 4.
The values, shown stored in a look-up table in FIG. 4,
are for discrete values of t in discrete steps of 1/1024
(i.e. 0/1024 to 1023/1024). However, t is shown being
inclusive from 0 to 1 for the purpose of explanation, it
being understood that in the preferred embodiment, varies
between 0 and 1023/1024 as the end point for To 1024/1024 is
Known.
- 42 -
.,
::
. , .. . . .
'7~6
Because the functions domain of the variable t for the
solution sets To = t(l-t)2 and To = t2(1-t) overlap, a table
of values may be accessed in one data direction with respect
to a first domain of variables for which the value
represents a solution set. Similarly, that solution set may
be accessed in an opposite data direction for a second
domain ox a variable
Also, because the function I 1 + Cost I Sin is
symmetric about 45 degrees, it is only necessary to store
values thereof for O to 45 degrees in half angle steps.
The solution set of values for the functions To and To
ma be accessed in a first data direction to provide the
functional solution set values for the domain t = O to t = 1
and in reverse order for the domain t = 1 to t = O. Values
of Sine may be stored in half angle steps between O and 90
degrees. Further, any derived nodes may be stored and used
again where the respective curve or corresponding symbol
outline loop is to be duplicated.
By forming a cumulative value of t as a multiple of the
incremental value of t and applying cumulative discrete
selected values of t within the parametric cubic polynomial
compiler, according to equation 3.1, display intercepts
represented by the generated nodes are produced which may
then be used as the display coordinates for the closed loop
outline at the desired display size.
s stated above, the value of 9 and used in the
compiler are derived, according to the principles of the
invention from the average entrance and exit angles, and
at a respective set of entrance and exit knots. The nodes
which are generated using the incremental value of t to
- 43 -
;.... .
increment the cumulative value of t and by applying discrete
selected cumulative incremented t values to the cubic
parametric polynomial compiler to produce respective node
coordinates for such discrete selected values of t.
The nodes, represent locations on the curve, which, as
stated above, form an angle at the entrance knot of the
curve as given by I, and forms an ankle at the exit knot
given by As stated, an order is chosen for the knots and
the order of knots defines an outline loop of knots, the
nodes being locations on the curve, between the knots also
have an order defined by that knot order and by the defined
outline loop. As would be understood by those skilled in
the art, that order is the succession of knots and nodes
encountered as one progresses along the outline loop as
specified. The data for the knots and nodes, as stated
above, is encoded in a data loop in that same order ox knots
and nodes. The encoded knots and nodes are placed in the
data loop, in that order corresponding to the outline loop.
Then, as one progresses in the defined order along the data
loop, corresponding to the outline loop, one would encounter
the encoding for the knots and nodes in the data loop in an
order, corresponding to the order one would encounter the
respective knots and nodes on the outline loop represented
by that said encoding.
In summary, given the knot locations, and the angles of
the curve at an entrance and exit knot as defined above, and
given an incremental value of t, then each discrete selected
cumulative value of to as described above, applied to the
cubic polynomial parametric compiler would produce
respective nodes, indicative of locations along the smooth
continuous curve described by the cubic parametric
polynomial. The encoding for the knots and nodes would be
- 44 -
Lo
in the same order along a data loop as one would encounter
the corresponding knots and nodes along the outline loop,
when proceeding in a chosen order along the outline loop.
The outline loop, when closing upon itself as shown in
FIG. lb, for example, is called a closed outline loop. The
data when encoded in a data loop which closes upon itself
corresponding to the closed outline loop is called a closed
data loop. In this way, accessing the closed data loop, in
a direction corresponding to the chosen direction along the
closed outline loop will return one to the initial or
starting data loop entrance location. As described below,
the data loop decoding is not complete until a determination
is made that the accessed location for the closed data loop
is starting location point and that accessing of all the
encoded data in the closed data loop has been completed.
Of course, as one skilled in the art would realize,
variations of the above closed outline loop and closed data
loop can be made without departing from the principles of
the invention.
These node coordinate values given in Raster Units may
then be used in run length or other suitable data forms -to
modulate a display to produce the curve at the desired size,
as is well known
The compiler according to the foregoing is shown below.
- 45 -
I
ENCODE IN
The novel manner of encoding the data pack, according
to the principles of the invention, is shown in the Table I
below, wherein p is the value of do, q is the value of dye
and 0, 1, 2, . . . D, E, E', are bit positions in the
encoded data pack. In the preferred embodiment, the bit
positions are valued according to the binary number system.
The Control Codes, explained below, are shown in Table II.
The encoding, shown for a maximum 12 binary words of 3
nibbles, in the preferred embodiment, is used to define a
control code or coordinate position for the master encoding
grid. It should be understood that the size of the word and
the number of nibbles used, as described below would
increase to accommodate a larger universe, commensurate with
a larger size master encoding grid.
The binary words are encoded in a series of nibble
length data words, arranged in the order of the data loop.
In this way, as will be shown below, the values of selected
data words with a nibble series may be used to define the
number of nibbles in a complete information set and the
initial nibble or bit positions in the next successive
complete information set.
A complete information set (IS) would be the set of
data words necessary to completely define the next
coordinate or a control code, as shown below.
- 45 -
.
o
Isle 1111
I
I) N
00 I) I
EYE
Jo
O o '--
Isle 11 11 11
(I) Q H
Us H H
id H H 1--1 H H
V
Al o o o or
o .
if' I
¦
I
Jo owe o o Jo I
em
h owe o o o
~:11 I
U) I .-1 .-1 -1 .-1
I
owe o
I' I
C51 I I I
I I
ml
U I
at
ED
d
.,~
a) 1- m
I O I
a)
Jo or co U
O I
Jo
I Lo
LO O
o o
o
Q,
h O Q,
no o h O
O
o
.,1 TV
Q,
o o o
Q, O O
o ) h I Jo ~11 id
a) o o o o o I I
us I o Al -,1: a) a Q, Q, O
o Hi assay h h h Q, Q,
Q o o O h h
h us Hi 'd I I) a) O
(d
to I h X X ,1 Al I:
4 1 0 I ) h h
O N I o u) h h h
h h h h
I
Us
h
I I 0 11 11 11
O I-
-' 'I
pa a) + +
5 a Jo
o I
En O Z;
o
5J h a
h o g o
O o n
O V I-
I X Q, I
o o I) h O
I O a) Ed O
Jo TV ) 0 A I
I 0
I I ,1 o Q,
-,1 Al to) h U) O O h O
So h h to so O
I: O -I I Jo
Of a) C I X
'
I: V) Q Q Q Q Q Q Q Q
V H H H H I 1 H H H H H H
': :
--
The data pack forming the closed data loop is used for
encoding the axial distances in Drum pox q=dy) between
knots, and for the double purpose of identifying and
accessing override control codes. These override control
codes are used to speed the decoding process and reduce the
requirement for encoded data describing coordinate points
while offering the further option of edit commands.
The data pack override control codes may be used to
override machine default command codes which would otherwise
be responsive to parameters expressed within the data pack.
As can be seen, the data pack offers the option of following
default command codes, responsive to predetermined
parameters, such as angles between knots or inter knot
distances, or override control codes for generating the
desired series of nodes and curve locus between knots.
In the preferred embodiment, the default command codes
are selected responsive to the parameters of the closed
outline loop described by the inter knot distances expressed
within the data pack. The process of the invention
described above for generating a series of nodes forming a
smooth continuous curve between a set of end points would
normally be used except where the distance between those end
points was greater than a predetermined distance, such as
128 units for the preferred embodiment. In that case, where
the inter knot coordinate distances were greater than 128
units, the default command code responsive thereto would
direct straight line interpolation and the formation of a
series of nodes to find the locus of that straight line
between the knots defining that respective coordinate
distance.
- 49 -
Similarly, where the exterior angle formed by a line
from the entrance knot to the exit knot and by a line formed
from the exit knot to a successive knot is greater than a
predetermined angle, such as 40 in the preferred
embodiment, then a default command code responsive thereto
would direct that a cusp be formed at the exit knot.
Straight line interpolation is well-known in the art.
However, within the inventive scheme herein is provided an
auto-scaling interpolation which provides an innovative and
simpler method of linear interpolation and at the same time
increases its accuracy and the precision of the closed
outline loop as defined by the series of generated nodes.
The auto-scaling linear interpolator is described below.
The format of the data pack shown in Table 1 is based 4
bit boundaries defining data words of 4 bits each. Data from
the data pack, representing discrete complete information
sets is accessed in a series of data words in nibbles of 4
bits at a time with the significance of that complete
information set (IS) and the number of data words therein
indicated by the value of the first nibble. The
correspondence between the first data word value of a series
of data words forming a IS and the Case indicated by that
first data word value is shown below.
First data word value Case Purpose
:
1 - 3 It Control Code
; O It Control Code
4 - 7 II Coordinate Distance
8 - B III Coordinate Distance
C - F IV Coordinate Distance
- 50 -
Jo ` .
I
series of data words may be a control code as in the
Cases It and It or the coordinate distances between knots,
represented by the zoo incremental values between such knots
in the preferred embodiment, in Cases II, III and IV.
Where the data words are nibbles, in the preferred
embodiment the first nibble value of a nibble series then
indicates -the Case, and directs the access of a
predetermined number of successive nibbles or data words
within the closed data loop from the IS and to complete the
override control code, as in Case It or It or to assemble
the number of data words necessary to complete the values
corresponding to the coordinate distance between the next
knot in the closed outline loop, as in Cases II, III and IV.
The number of nibbles or data words accessed responsive
to the first nibble or data word value are the nibble or
data word series necessary to form the IS for the
particular Case, after the first nibble or data word, as
follows:
In Case II, one nibble or a total of 2 nibbles are
required. In Case III, two more nibbles after the first
nibble, or a total of 3 nibbles are required, and in Case
IV, three more nibbles after the first nibble, or a total of
4 nibbles are required to provide the value of the inter knot
coordinate distance.
In the Cases It and IBM the number of successive
nibbles which must be accessed, responsive to the value of
the first nibble, is ordered responsively to the Case
indicated and as explained in detail below. For example,
Case I may require the accessing of 3 or 6 nibbles to
complete the Case It control code.
Jo .
- 51
A-t the end of the data word series of a IS, accessed
responsive to the first data word value, the next successive
nibble in the closed data loop would then be considered the
first data word value of the next data word series and
corresponding IS. That new first data word value of the
next data word series would then in a similar manner
indicate the number of successive data words to be accessed
to complete the completed information set for the override
control code or the value of the inter knot coordinate
distance, as the case may be. By encoding the closed data
loop on 4 bit boundaries, the data with respect to control
codes may be packed successively within the closed data loop
with the data respecting the inter knot coordinate distances
and, as such, the closed data loop may serve the double
function of providing override control codes as well as
inter knot distance data.
Table II provides a definition of the control codes,
control code value, the number of nibbles including the
first nibble to complete the control code instruction and
the purpose of the control code.
Further, as shown in Table I, and as stated above, a
first nibble value of 1 to 3 is indicative of Case It and
with that first nibble value indicating the particular
control code value for case Ian either 1, 2, or 3, and the
number of subsequent nibbles to be accessed for the IS
forming that particular Case It control code.
As shown in Table II, 3 additional nibbles are required
for a total of 4 nibbles, including the first nibble, to
complete the IS responsible to a control code value 1, 3
additional nibbles for a total of 4 nibbles are needed to
complete the IS responsive to a control code value of 2 and
- 52 -
, .,
6 additional nibbles for a total of 7 nibbles are needed to
complete the IS responsive to a control code value of 3.
Where the first nibble value of a IS is zero, then
Case It is indicated, as shown in the Table I, and the
access of the next successive nibble directed for completion
of the Case It instruction. In Case IBM 4 is added to that
next nibble value to obtain the control code value for case
IBM The control code values for Case It are shown in table
II with their corresponding purposes.
In summary, a first nibble value of 0, or 1 to 3,
indicates Case It or Case It as shown above, and directs the
accessing of a predetermined number of nibbles in the closed
data loop successive to the first nibble, to form the IS
for that Case and which is then used to determine the
control code value which in turn directs the processor. The
next nibble subsequent in the order of the closed data loop
to the last nibble of the previous nibble series
corresponding to a IS then becomes a new first nibble value
and it used to indicate either an override control code or
inter knot coordinate distance. Further, and as shown below,
where the first nibble bit as shown above indicates a Case
II, the ordering of one more nibble for a total of 2 nibbles
is required to complete the IS and to provide the
incremental coordinate distance. For Case III, 2 more
nibbles for a total of 3 nibbles are required to complete
the IS for the incremental coordinate distance. For Case
IV, 3 more nibbles for a -total of 4 nibbles are required to
complete the IS for the incremental coordinate distance for
code 4. Then the next successive nibble in the closed data
loop after the IS would be the new first nibble value
indicative of case Ill IBM II, III, or It, as the case may
be, leading to an indication of the number of successive
nibbles needed to complete the IS to complete the control
code or incremental coordinate distance.
for the override control code Case Ian where the next
successive knot in the closed outline loop is to be defined
by Case Ian control code values 1, 2 or 3, as shown in Table
II, then, the first nibble value will have a value of 1, 2
or 3 (thereby indicating override control code Case It), and
with the control code value being indicted by the specified
first nibble value (i.e. 1, 2, or 3). The number of
successive nibbles to be accessed to form the IS for that
Case It control code are indicated by the value of the first
nibble value.
Where the value of that first nibble value is 1, then
as shown in Table II, the next three nibbles in the closed
data loop are accessed and used to denote hori20ntal or
motion in the X direction with the next knot X coordinate
given by the next three nibbles, completing the nibble
series necessary to form the Completed Information Set for
that control code.
Where the value of that sty nibble is 2, then as shown
in table II, a vertical motion in the Y direction is
indicated with the next three nibbles in the closed data
loop being accessed, to complete the nibble series necessary
to form the IS and indicating the Y coordinate value of the
next knot.
Where the value of that sty nibble is 3, then the next
six nibbles in the closed data loop are accessed to complete
the nibble series and necessary to Norm the IS and
indicating the next knot, diagonally related to the
immediate knot, and with the ZOO coordinates therefore given
in each of the next 3 nibbles.
- 54 -
In the preferred embodiment, the least significant bit
of the control codes for Case Ian namely control codes 1, 2,
and 3 is used to provide a direction instruction for new x,
Y or ZOO coordinate values relative to the preceding knot in
the closed outline loop. That relative direction between
the new knot position and the preceding knot is then
followed when locating the positions of successive knots
corresponding to the information in a Case II, III or III
instruction indicating the incremental coordinate distance.
As will be seen below, the direction may also be changed by
a Case It control code 5, 6 or 7 which would negate the
established x, Y or MY direction.
In summary, in the closed outline loop new knot
coordinates as indicated by Case Ian control code values 1,
2 or 3, shown in table II, would be located in a direction
consistent with a previous X and Y direction instruction,
unless the least significant bit of the first nibble value
for the IS indicates a change in direction, as in Case IBM
control code values 5, 6 and 7.
In summary, the first data word value is the value of
the first data word of a series of data words forming the
Completed Information Set (IS) of the closed data loop,
indicating a override control, code as in case Ian for first
nibble values 1, 2 and 3, or as in Case IBM for value 0, or
the incremental coordinate distances as for first nibble
values 4-F, for cases II, III or IV.
Then, as stated above, if the sty data word value, in a
nibble size data word series is a zero, then control code
case It is denoted. Case IBM directs the access of another
nibble of 4 bits in the closed data loop and, if that nibble
is not equal to 0, its value is added to value 3 to obtain
- 55 -
Jo 6
I
the proper control code, as shown in Table II. In this way,
a total value of control codes O and 4 to 18 may be defined.
The meanings of each of the control codes having values 4-11
is shown in Table II.
If the control code value for the additional 4 bits is
zero, the end of the last loop is indicated (i.e. bits 4-7
are O value).
Code 4 indicates the start of a loop.
Codes 5, 6 and 7 provide direction information for the
next knot in the closed outline loop relative to the
preceding knot.
Codes 8, 9, 10 and 11 are editing override control
codes, which override the default command codes.
The editing override control codes are used to override
the default command codes, which would normally be
responsive to and result from a machine interpretation of
the data pack values for cases II, III and IV.
Code 8 directs Linear Interpolation Sharp Knot. As
shown in FIG. 5, it may be used to force the exit angle at a
knot (i.e. pa at knot Zulu) for curve Zap Zulu to be en
to the inter knot angle Be at the entrance knot, pa of that
same curve Zap Zulu As shown in FIG. 5, Code 8 forces the
curve Zap Zulu' at knot Zulu to have the same exit angle at
the exit knot (Zulu) as the inter knot angle Be at its
entrance knot Zap and produces a sharp cusp at Zulu To
complete the cusp at Zulu' the entrance angle aye for curve
Zulu Zoo would be forced equal to the inter knot angle Be+
at Zulu
- 56 -
Code 9 denotes linear Interpolation - Smooth Knot and
may be used, for example, to force a smooth knot located at
the end of a curve Zap Zulu which then becomes a straight
fine, or at the end of a straight line, which then becomes a
smooth continuous curve.
The use of override control code 9 is shown in FIG. pa
where a knot joins a straight line curve Zap Zulu to a
smooth continuous curve, Zulu Zoo In t aye
set equal to Be.
Where a knot joins a curve section Zap Zulu to a
straight line curve, Zulu' Zoo' as shown in FIG. 6b, then
pa' the exit angle for curve Zap Zulu is made equal to Byway,
the inter knot angle between Zulu' Zoo
It should be understood, however, that if a knot joins
two straight lines, this rule shown with respect to FIG. 5
is used in the preferred embodiment.
Control code 10 directs a curve interpolation at a
sharp knot and is used to form a cusp at the entrance knot
or the exit knot of a smooth continuous curve segment joined
by that knot to a straight line. Shown in FIG. pa, are two
examples, i.e. the curve Zap Zulu is formed with a sharp
knot at the entrance knot Zap and the curve Zulu' Zoo is
formed with a sharp knot at the exit knot Zoo This
control code is useful in overriding a default command code
which would otherwise require that the sharp knot at Zap for
example have an entrance angle for curve Zap Zulu equal to
Be which would introduce a distortion in curve Zap Zulu at
area 31, approximate the entrance knot Zap for curve Zap
Zulu as shown in FIG. 7b. Similarly, a distortion would be
introduced in a smooth continuous curve terminating in a
- 57 -
:
~3~'~6
sharp knot. For example, in curve Zulu Zoo as shown by
numeral 33 in FIG. 7b, approximate the exit knot Zoo' where
aye would be forced to equal Byway to form a cusp. code 10
overrides the default command code and forces e (i.e. pa) to
be equal to (i.e. pa).
Where a cusp is to be formed at knot Zap the default
command code would specify that the exit angle, pa at knot
Zap would be equal to the inter knot angle Byway at -the
Zulu' for curve Zulu' Zap and the entrance
angle pa for curve Zap Zulu would be equal to the inter knot
angle Be between the entrance knot Zap for curve Zap Zulu and
its exit knot Zulu As stated above, this would produce the
result shown in FIG. 7b, and a distortion of the smooth
continuous curve Zap Zulu shown in FIG. pa. With the result
of FIG. 7b, the deviance angle eta between Be and pa would be
zero as stated above. In this case, to avoid the distortion
shown by numerals 31, 33 in FIG. 7b and to produce the
smooth continuous curve as shown in FIG. pa, pa is forced
equal to pa the deviance angle, at exit knot Zulu' as stated
above. In the case shown in FIG. pa, arranged, for the sake
of explanation pa is equal to zero pa is equal to Be.
A similar result is forced by using a code 10 to
control the shape of curve Zulu Zoo at exit knot Zoo As
stated above, the default command code responsive to a cusp
at Zoo for example, would direct the curve Zulu' Zoo' to
the shape shown in FIG. 7b and particularly shown by numeral
33 approximate Zoo by directing that the exit angle aye at
knot Zoo is equal to the inter knot angle Byway at the
entrance knot Zulu for that same curve. However, by using
code 11 to direct what aye is equal to eye, then at exit
knot Z 2 for curve Zulu Zoo aye is q aye
aye' aye' eye Byway )
- 58 -
" .
For the sake of explanation, aye is equal to 0, for
curve Zulu Zoo
The effect of code 10 is to produce a smooth continuous
curve from or to a cusp such as at entrance knot Zap or at
exit knot Zoo and symmetrical about the respective curve
midpoints, as shown by numeral 35 and numeral 37,
respectively for curve Zap Zulu and for curve Zulu Zoo
control code 10 is to override a default command code
which would otherwise specify a cusp, such as where the
exterior angle as described above was above a threshold such
as 40 as shown in the preferred embodiment, or the
inter knot distance was greater than a predetermined
distance, such as 120 units in the preferred embodiment. In
this case, a smooth knot would be formed as described above,
by taking the average of the inter knot angles between a
preceding knot and the subject knot and the subject knot and
a successive knot in the outline loop direction of the
closed outline loop. However, as explained above, if the
absolute value of the inter knot angles between the preceding
knot and the subject knot and the subject knot and the
successive knot is greater than 180, then the supplemental
average is used by supplementing the average angle by 180.
This is to orient the angle in the correct direction where
the relationship of the average angle described above would
result in a resultant angle 180 out of phase with its
correct direction.
The application of the inventive principles will force
the angular relationships of and at the respective
knots to conform to the rules described above, as necessary
to produce the desired curve, straight line, cusp, or smooth
knot whether directed by the default command codes or by
- 59 -
:
. . Jo ... .
the override control codes. It should be noted, however,
-that where an override control code is used, it it used to
force a result contrary to what would ordinarily be produced
using the default command codes. For example if the
default command code would produce a cusp and a smooth knot
was desired followed by a smooth continuous curve or
preceded by a smooth continuous curve, then a code 10 would
be used. If the default command code would have produced a
smooth continuous curve, and a straight line joining a sharp
knot is desired, then a code 8 would be selected. If a
smooth knot is desired joined to a straight line, then a
code 9 may be selected to override the default command
control. All of the foregoing is shown in connection with
FIGS. 5, 6, pa, 7, pa and 8 and in the text accompanying
these figures.
A typical encoding for an A as shown in FIG. is shown
in the appendix. The compiler for decoding the packed
encoded information is also shown below, and is used with a
Motorola 68000 processor. As the desired output is a series
of intercepts at the intersection of the locus of the
character outline, and the display raster lines, these
intercepts can be converted into modulating information for
imaging the character on an imaging surface by any suitable
well-known imaging device
For the sake of explanation and in the way of an
example, a closed data loop for the encoded A shown in
FIG. 9, is set forth below and described.
CLOSED DATA LOOP FOR THE "A" OF FIG. 9
(Given in hexadecimal Notation)
83 98 3C 82 En YE 02 4C 01 5C 02 YE 16 49 00 5B
30 By 54 Do 75 96 83 78 81 A Do PA 37 lo 80 7C
- 60 -
.
. , , , .
I
57 PA 83 I Of 85 En 28 24 15 C0 20 49 77 96 88
73 OF 30 60 62 Fly ED OF 60 26 ED 28 20 C1 PA 88
06 50 10 06 95 99 7B 88 58 80 By 82 08 01 PA 70
OF OF 00 10 79 06 50 37 31 19 7B 73 53 81 51 79
As stated above, the closed data loop is encoded on
data words of 4 bit or nibble boundaries and the sequence of
the nibbles is as given below. In accordance with the
preferred embodiment, the first two bytes 5B of the closed
data loop indicate the total number of bytes in the data
loop for a closed outline loop. The first two bytes, 00 5b
indicates that there are 91 bytes total in the data loop for
the symbol A, describing outside closed outline loop 31 in
the direction of arrows 31 and inside closed outline loop 35
in the direction of arrows 35.
In accordance with the preferred embodiment, the
starting X and Y coordinates are given in three nibble pacts
of information for each respective X and Y location.
accordingly, the next three nibbles, 6 49 relates to the
starting X location. In accordance with the principles of
the invention and the preferred embodiment, the least
significant bit of the data for a new X or a new Y location
is a sign bit indicative of the direction. Accordingly, in
processing the data in the preferred embodiment, a shift of
one binary position is made to remove the sign bit, giving a
decimal data value of 804. As described above in the
preferred embodiment, since the origin of the My is offset
by 59Z units, 592 must be subtracted from 804 to provide the
correct X coordinate with reference to the origin of the My
of 212. The sign of the X direction, whether positive or
negative with regard to the origin of the My is given by the
sign bit and is positive if the sign bit is zero, in the
- 61 -
Jo j , .
preferred embodiment.
In accordance with the principles of the invention, the
starting Y position given as a new Y position is indicated
by the next nibble series of 3 nibbles or by EYE, which is
divided by 2 to remove the sign bit yielding the result of
750. In accordance with the offset of the My at 736 units,
736 is subtracted from 752 to yield a Y coordinate of 16.
The sign bit which is a 1, indicates a negative direction
for Y. Accordingly for the "A" of FIG. 9, the start point
x, y coordinates shown as numeral 39, is 212, 16 with the
new direction being x,-y.
As stated in the preferred embodiment, the data pack is
decoded by using the first two bytes to indicate the number
of bytes in the closed data loop corresponding to the
coordinate points around the closed outline loop, and by
three nibbles each comprising twelve bits corresponding to
the respective X and Y start locations. It should be noted
that wherever the X coordinates are defined by new
coordinate values, the least significant bit is used as a
sign bit to indicate the coordinate direction. The process
performs a divide by two which separates that bit to define
the aforesaid direction signed. Additionally, the X
position is referenced to the home or reference position in
the M which may be offset with regard to the origin of the
master encoding grid by subtracting the offset from a
coordinate position accordingly.
With this in mind, the following process is described
which produces the listing of coordinates shown below and in
FIG. 9.
As stated, the start position is at MY coordinates 2~2,
O. As the last bit of data accessed from the data pack, to
provide complete information for the preceding nibble series
- 62 -
. . .
was the tenth nibble corresponding to hexadecimal number 5.
The next nibble which follows a complete information series
of nibbles is a first nibble value.
As shown, the first nibble value 2 indicates a case It
and according to table II directs the access of the next
three nibbles, 5C0 indicative of the new X coordinate
position. In accordance with the procedure above, a
division by 2 removes the sign bit and a subtraction process
is performed relative to the My offset, to reference the new
Y position to the My home position or at zero. The first
nibble value "1", following the nibble series for the
completed information set of 5C02 then indicates a case It
and a new X coordinate. In accordance with the process
described above, a binary division of two is performed to
isolate the sign bit, indicate the direction and a
subtraction is performed to reference the new X position to
the home position. Accordingly, the new X position is 16.
The next coordinate position of 16,16 is given by the
complete information set EYE, with the nibble series first
nibble value indicating of a Case III. In accordance with
the format of the data pact as shown in Table I, P = X is
equal to 01110 and Q = y is equal to 00010. This translates
to 14 and 2 in decimal rotation respectively. In accordance
with the preferred embodiment, as it would be redundant to
use Cases II, III and IV, for a translation of 0, a 1 is
added to the results of p = x and q = y to provide the new X
and Y coordinate distances of 15 and 3 respectively. The p
value is added to previous X value of 16 to provide a new X
value of 31, and the q value is added to the previous Y
value of 16 to provide a new Y value of 19 accordingly.
- 63 -
As a reminder, it should be understood, that this
process of decoding the data pack being described is
designed to decode the encoded knot coordinates only. After
decoding the inventive principles described herein are
applied to the knots to either produce a series of nodes
describing a smooth continuous curve between the knots, as
between knots 16, 16, and knots 31, 19 or a straight line as
between knot 212, 0 and knot 16, O. In accordance with the
preferred embodiment, the direction of the knots indicated
by the completed information set EYE relative to the
previous knot is in accordance with the direction
established by the last previous sign bit accessed for the
respective X and Y directions as shown above, or by 567 as
described below.
The next nibble following the nibble series for the
completed information set above has the first nibble value
of 8 which is a Case III, directing the access of the next
two nibbles for providing the nibble series for complete
information set of 3C8 corresponding to a p = x value of 13
and a q = y value of 4 in accordance with the format shown
in Table II. In accordance with the preferred embodiment,
the direction of the knot denoted by this complete
information set follows the last previous direction
indication given and is accordingly added to the previous
coordinates of 31 and 19 to provide new coordinates of 44
and 23.
In accordance with the foregoing, the next nibble being
a first nibble value is 8 which accordingly directs the
access of the next two nibbles to provide a complete
information set of 398 producing new coordinate values of
54, 27, accordingly.
- 64 -
.~,
I
The next nibble corresponding to a first nibble value
of 8, as described above/ indicates a case 8 and directs the
access of the next two nibbles to form the nibble series
corresponding to the complete information set of hexadecimal
7C8 and according to the processing described above decoding
the new coordinates of 67, 35.
As can be seen, the smooth continuous curve between
knot end points 16,16 and passing through crypts 31, 19 and
44, 23 and 54, 27 and, 67, 35 are formed according to the
principles of the invention to form a smooth continuous
curve.
The next first nibble value of the next complete
information set is zero indicating Case It which directs the
access of another nibble to form a two nibble series
complete information set. As the next nibble was 8, for
Case IBM a value of 3 is added thereto forming a value of
11, indicative of control code 11 directing a smooth
continuous curve be formed between the last knot having
coordinate 67, 35 and the next knot which is to be indicated
: by the next complete information set accessed from the
closed data loop.
was the first nibble value of the next complete
information set is B indicating Case III, 3 nibbles are
accessed forming a 4 nibble series describing the next
complete information set and producing the new coordinates
of 85, 59.
The next first nibble value of the next nibble series
for the next complete information set is 3 indicating Case
Ian control code 3, shown in Table II, and which directs the
access of a pair of three nibbles each which are indicative
- 65 -
~,~,,.
of -the next X and Y positions and directions thereof
relative to the previous knot. In accordance with the
process described above, the new X and Y position is 349,
606 and as the distance between this new snot and the
previous knot is greater than 128 units, the default command
code directs linear interpolation. In accordance with the
above, the hexadecimal value AYE corresponds to the new X
position and AND corresponds to the new Y position. As
stated above, the least significant bit of each of the above
3 nibbles corresponds to the relative direction of the new X
and Y points.
The first nibble value of the next nibble series for
the next complete information set is 1 indicating a Case It
and directing the access of the next three nibbles of AYE to
replace the X coordinate with 372 and a new set of
coordinates 372, 606.
The next first nibble value for the next nibble series
is a 3 corresponding to a case Ian a control code 3 and the
access of a pair of 3 nibbles, i.e. 968, corresponding to
the new X position and 675, corresponding -to the new Y
positions and new X, Y coordinates of 612, 90 respectively.
The next first nibble value following the nibble series
for the complete information set above is D, corresponding
to Case IV which directs the access of the next three
nibbles to form the 4 nibble series 254, and which according
to the form shown in Table I provides a p = x value of 20
and a q = y value of 37. According to the process described
above, a 1 is added to the p = x value and the q = y value
to form decimal coordinate values of 21 and 38. Following
the most recent X, Y coordinate direction instructions
given, the X incremental value is added to the previous
- 66 -
.
coordinate 612 providing the new X coordinate of 633 and the
Y incremental value is subtracted from the previous Y of 90
to provide a new Y coordinate value of 52.
The next first nibble value of the next nibble series
for the next complete information set is a B corresponding
to Case III which directs the access of two more nibbles to
form a three nibble series, complete information set, and
producing the next coordinate values of 650, 32 for X and Y
respectively.
The next first nibble of the next nibble series for the
next complete information set is 8 corresponding to a Case
III, causing the access of the next two nibbles to produce a
three nibble information set and new coordinate values of
659, 25 for X and Y respectively.
The next first nibble value is a 9 corresponding to
case III and directing the access of the next two nibbles of
77 and corresponding to new X and Y coordinate 683, 17.
The next first nibble value is a 9 causing the access
of the next two nibbles 04 and corresponding to the new X, Y
coordinates with 704, 16.
The next new first value is 2 corresponding to Case It
and directing the access of the next three nibbles
corresponding to the 5C0 corresponding to a new Y value with
the direction indicated by the first significant bit therein
and producing new MY coordinates of 704, 0 respectively.
- 67 -
.
The next first nibble value of the next nibble series
for the next complete information set is 1 corresponding to
a Case It and directing the access of the next two nibbles
forming a three nibble series complete information set 824
and corresponding to the new X, Y location of 450, with
respect to the last previous direction given for the axial
directions X and Y.
The next first nibble value 4, the next complete
information set is 2 corresponding to case It and to
complete information sets LEO corresponding to new MY
coordinates 450, 16.
The next first nibble value is 8 corresponding to Case
III and as described above corresponding to complete
information set OWE and new MY coordinates 465, 17.
The next first nibble value is 9 corresponding to Case
III and complete information set 389 and new MY coordinates
490,21.
The next first nibble value is 8 corresponding to Case
III and complete information set AYE and new MY coordinates
501,25.
The next first nibble value is 7 corresponding to Case
II which directs the access of another nibble and according
to the format shown in Table II provides a p = x value of
binary 101 corresponding to a decimal value of 5 to which
one is added in accordance with the procedure above to
provide a decimal value of 6 for the new X coordinate
incremental distance. Similarly, the q = y value is binary
011 which corresponds to a decimal value of 3, to which one
is added in accordance with the process above to provide a
- 68 -
, . . .
I
new value of 4 corresponding to the incremental Y coordinate
distance. The new YO-YO coordinates are therefore 507,29
accordingly to the directions given in the last previous
direction instruction.
The next first nibble value is 8 corresponding to Case
III and the access of two more nibbles to form the complete
information set Aye and MY coordinates 516, 40.
The next first nibble value is 8 corresponding to Case
III, the access of the next two nibbles form a complete
information set C18 and new coordinates MY of 518, 53.
The next first nibble value is zero corresponding to
case It which directs the access of the next nibble to,
which 3 is added in accordance with procedures described
above to form the control code 5. us shown in Table II is
an X Negate which changes the X direction from its previous
direction. The next first nibble value is an 8
corresponding to Case III and in accordance with the
foregoing new X, Y coordinates 515, 67.
The next first nibble value is 8 corresponding to case
III and new coordinates of X, Y of 508, 86, respectively in
accordance with the above procedure.
The next first nibble value is a 0 corresponding to
case It which directs the access of the next nibble 6 to
which 3 is added according to the above procedure to provide
a control code of 9. 9 as shown in Table II is a line
smooth direction.
;
- 69 -
.'
-
The next first nibble value is F corresponding to Cohesive and in accordance with the procedure above causes the
access of three more nibbles forming the complete
information set 600F and new X, Y coordinates 459, 196 in
accordance with directions as given.
The next first nibble value is 1 corresponding to Case
It which directs the access of the next three nibbles, 62F
indicative of an X coordinate point in accordance with the
last previous directions given. Accordingly, the new X, Y
coordinates are 199, 196.
The next first nibble value is zero corresponding to
Case It and directs in the access of the next nibble having
a value of 6 -to which 3 is added in accordance with the
above procedure to produce a 9 corresponding to control code
9. A next first nibble value is zero once again
corresponding to Case It which directs the access of a next
nibble which is 3 to which 3 is added giving the control
code vow 6 which means negate Y.
The next first nibble value is F corresponding to Case
IV and in accordance with the above procedure causes the
access of the next three nibbles to complete the information
set 7D2F and producing new X, Y coordinates of 148, 70 which
is connected to the previous X, Y coordinates 199, 66,
according to the override control code 9 and in the new Y
direction.
The new first nibble value is zero corresponding to
case It and directing the access of the next nibble having a
value of 7 to which 3 is added to give it the control code
10 .
- 70 -
.
I
The next first nibble value is an A corresponding to
Case III and dixectlng the access of the next two nibbles to
give the complete information set AYE and new X and Y
coordinates 144, 52 respectively.
The next first nibble value is zero corresponding to
Case It which directs the access of another nibble having a
value 8 to which 3 is added in accordance with the above
procedure to provide the control code 11.
The next first nibble value is zero corresponding to
Case It directing the access of another nibble having the
value 2 to which 3 is added giving the control code 5 which
negates the previous X direction.
The next first nibble value is A corresponding to Case
III and the access of two additional nibbles giving the
complete information set B08 and new MY coordinates 145, 40.
The next first nibble value is zero corresponding to
case It ion accordance with the above procedure control code
11 .
The next first nibble value is 8 corresponding to Case
III~ complete information set 858 and new MY coordinates
151,31.
The next first nibble value is 8 corresponding to Case
III, complete information set 7B8 and new MY coordinates
163,23.
The next first nibble value is 9 corresponding to Case
III, complete information set 599 and new MY coordinates
189,17.
I,
I
The next first nibble value is 9 corresponding to
complete information set 069 and coordinates 212, 16.
The above my coordinates 212,16 bring the -traverse of
the closed data loop bringing it back to the starting point.
The next first nibble value is zero indicating Case It
with the next access nibble I to which 3 is added producing
the control code value 4 indicating the end of closed data
loop.
In accordance with the preferred embodiment, a routine
is added, not shown, which would be known to one skilled in
the art to ensure that the closed data loop decoding closes
upon the start location of the closed data loop and
completes the closed outline loop by ending at the start
point encoding of the closed data loop which represents the
start of the closed outline loop.
The second closed outline loop, shown by numeral 35 and
arrows 37, of the A now starts at the closed data loop coded
hexadecimal numbers 790, 650 corresponding to new X and Y
coordinate values 216, 232 and the X and Y directions,
derived according to the process given above with the regard
to the start points for the foregoing loop, 31.
The next first nibble value is 1 corresponding to Case
It and directing the access of the next three nibbles 815
and new MY coordinates 442, 232.
The next first nibble value is a 3 corresponding to a
Case Ian directing the access of a pair of three nibbles
735, and 97B corresponding to new X and Y coordinates of 330
and 477 in the negative X and negative Y directions
- 72 -
accordingly.
The next first nibble value is 1 corresponding to Case
It directing the access of the next three nibbles 731
corresponding to X coordinate 328, Y coordinate 477 in the
negative X and negative Y directions accordingly.
The next first nibble value is 3 corresponding to Case
It directing the access of a pair of three nibbles each
corresponding to hexadecimal value 690 and 790 and X and Y
positions 216, 232, respectively As 216, 232 are the start
points, the closed data loop representing the closed outline
loop 35 has been completed and the next first nibble value
is zero indicating Case It with the last nibble accessed
accordingly being zero indicating the end of loop. Once
again, the routine described above is used to insure that
the closed data loop ends at its start point.
The above is a representative encoding of a letter used
in the process to derive the coordinate values around the
outline of the character and any override control codes
which may be used to replace default command codes.
However, it should be understood that changes to the copings
could be used consistent with the principles of this
invention, as claimed herein, and that the invention should
not be restricted to the coding or decoding process shown
above, with respect to the preferred embodiment.
- 73 -
AUTO SCALING LINEAR INTERPOLATION
Linear interpolation is a well-known technique and is
not claimed as an invention in this application. The
auto scaling linear interpolation which is claimed and which
is described below is a method of increasing the precision
of a machine interpolation procedure which uses as a start
point, a first set of coordinates and as an ending point a
second set of coordinates. The coordinates are usually
expressed in respective coordinate directions such as x and
y, for example. In the preferred embodiment, the process of
linear interpolation is for the purpose of producing
coordinate points along a straight line between the first
and second end points, which coincide with a second
coordinate system such as the intercepts in a raster
display. Each of the coordinates are found by determining
the slope of the straight line between the two end points
which is equal to the incremental distance in a first
coordinate direction divided by the incremental distance in
a second coordinate direction (i.e. Y2-Y1/X2--X1). That
first coordinate incremental distance is expressed in an
encoded machine value as a first data word, (i.e. YO-YO),
in a first machine location. The second coordinate
incremental distance between the two end points in the
second coordinate direction (i.e. X2-X1), is also expressed
as a machine value and placed in a second machine location.
The machine location limits the precision by which a data
word may be expressed. For example, and as is well-known,
each machine is based upon radix, which is a number base.
The most common machine number base is the binary number
base. Each data word accordingly, has a number of bit
positions with each specified bit position being a specified
power of that radix (i e 24 23 22 21 2 2-1 2-2 )
Accordingly, a shift of a data word in a direction of the
I'
- 74 -
I;
,,
most significant bits corresponding to higher exponential
values or higher orders of the machine radix, results in an
increase in the scale or value of the data word. Further,
each shift in the direction of the most significant bits,
results in an effective multiplication of the data word
value by a scale factor of the machine radix raised to an
exponential power corresponding to the number of bits
shifted (i.e. 13 bit positions shifted is equal to a scale
factor of 213 in a binary machine or equivalent of 8196, in
decimal notation. Conversely, each shift of a data word,
towards the least significant bits, corresponding to lower
exponential values or orders of that machine radix,
corresponds to an effective division by the machine radix
value and conversely to a reducing scale factor. Further,
machine locations for storing data words are limited in the
number of bits available. Pus the precision of a data word
is a function of the data space and the number of bits
available for storing that data word, an increased precision
for expressing a data word are obtainable by extending the
number of bits or size of a machine location available for
specifying a data word. For example, as is well-recognized,
in a decimal system, the number 5.632498 is a more precise
value than 5.3624 which lacks the last three significant
places (i.e. .000098), of the former number. In binary,
the number 10111.101 is a more precise expression than that
number truncated or rounded to 10111.000, as the latter
number is missing the bits .101 and is therefore less
precise. However, the former binary expression requires a
larger machine location for specifying all the relevant bits
in that expression. The "point" in the above binary and
decimal values are used to indicate the positions, according
to the radix system used, to indicate position values equal
to or greater than l (decimal) and less than
(decimal) (i.e. integer and fractional values).
-- 75 --
The effect of shifting to increase the scale of the Y
increment is to eliminate the binary point in a third data
word representing the slope or Y incremental value and
thereby avoiding floating point arithmetic operations. The
point separates the bit positions in the radix system
selected, separating fractional from integer values in a
data word having values equal to or greater than 1 and less
than 1 (i. e. between the bit positions "2" and "2 1".
The binary point is equivalent to the decimal point in a
radix 10 system and equivalent to a "point" between those
machine positions having a value equal to or greater then
and less -then 1 in any system and as stated, separating the
fractional values from the integer values in the slope or Y
increment value. By shifting a machine representation of a
number in the direction of more significant bits, the scale
of the number is increased, thereby moving the "point"
effectively in the direction of the less significant bits.
If a sufficient number of bit positions are shifted, the
binary point is eliminated from the number. In this way,
floating point arithmetic is avoided.
According to the invention, linear interpolation
proceeds according to the known method by determining the
coordinate distance in a first coordinate direction and in a
second coordinate direction between a set of end points.
The method is used in the machine having a radix "r"
corresponding to the values of designated bit positions in
the encoded words used within the machine. The process
continues by encoding a first data word on "N" bits
corresponding to the distance between the first and second
end points in the first coordinate (i.e. YO-YO), and then
placing that first data word in a first machine location.
To complete the process of calculation the slope, according
to the interpolation method, a second data word of M bits
- 76 -
I
corresponding to the distance between the first and second
end points (i.e. X2-X1) in a second coorindate direction is
encoded and placed in a second location. The machine is now
ready to use the data value in the first location and a data
word value in the second location in a process of division
to produce a third data word corresponding to the slope of
the straight line between the first and second end points
(i.e. Y2-Y1/x2-X1). To increase the precision or resolution
of the third data word according to the inventive process,
the scale of the first data word (i.e. YO-YO), is increased
by determining the the number of available positions between
the most significant position of the first data word and the
most significant position of the first machine location.
The first data word is then shifted in a first direction
towards the most significant positions in that first machine
location, utilizing those more significant positions which
were unused, when expressing the first data word. The scale
of the first data word is again increased by shifting a
second time in the same first direction of the most
significant positions by a number of positions coorespondiny
to the number of bit positions used to encode the second
data word (i.e. X2-X1). In the preferred embodiment, the
first machine location is expanded to accommodate this
additional shift. It should be noted that the inventive
principle is not limited to the size of available data space
in any one machine or to any machine radix.
In the preferred embodiment, the second shifting
operation described above occurs in an extension to the
first machine location. The quotient corresponding to the
slope (i.e. Y2-Y1/X2-X1) produced as the third data word by
division of the first data word by the second data word will
reduce the first data word to a size coextensive with the
first machine location. As a result then, the data word may
=. ,
to
we
be raised by a scale factor corresponding to a shift in the
direction of the most significant positions corresponding to
the number of significant positions used to express the
denominator it X2-X1) or second data word, as the next
division step cancels that said shift of the first data
value by the said number of bit positions in the second data
word and corresponding reduces its position length to the
size of the first machine location.
The third encoded data word (i.e. the Y incremental
value for each X increment) is then stored and added to the
first data word (i.e. Ye). The second data word (i.e. X1)
is incremented a coordinate word value corresponding to the
first and second coordinate positions of a point along the
line are then stored. In Table III are given the coordinate
X, Y values; the value of Y incremented at its highest
precision and scale factor for the example shown.
In the iterative process, the third encoded data word
at the higher resolution, produced, corresponding to the
slope produced according to the above, is used iteratively
as a Y incremental value to derive a cumulative Y
incremental value related to a respective X coordinate value
in the first direction (i.e. Y) and stored as a cumulative
Y. As no change has been made to the precision (i.e. the
number of positions used to express the third data word),
the precision of the third data word is the same as the
first data word, produced by shifting the first data word
and increasing its scale, as described.
As interpolation is an iterative process, the data word
(i.e. the Y incremental value at the said higher
precision), is used to produce the cumulative Y, related to
the Y coordinate for a respective X coordinate.
- 78 -
---
As the purpose of producing data word values
corresponding to points along a straight line, is to produce
intercepts on a display coordinate system, the scale
cumulative Y words are reduced to the scale used to express
the end point coordinate values, by the step of truncating
or rounding. Truncating may be accomplished by discarding a
predetermined number of least significant bits or rounding
may be used by a shift in the direction of the least
significant bits, minus one, adding a bit corresponding to
the rounding value such as .5 (decimal) and discarding any
bits having values less than the least significant bit value
of interest. In the preferred embodiment, truncating or
rounding is to the position of the lowest integer value in
the radix 2 system (i.e. to bit position 2).
In the foregoing manner, the scale of cumulative Y is
then reduced to fit scale of the data words for the end
point coordinates. In the preferred embodiment, the
original precision is that of the coordinates on a display
which are the display intercepts, shown as X, Y in Table
III. The result of using this method is to avoid the
incremental error introduced into each cumulative Y value by
summing Y incremental at a lower scale.
The incremental value at the increased scale factor may
be used to derive a cumulative Y for each respective
coordinate X values by iteratively incrementing Y increment
by Y increment to produce a first cumulative Y and then
incrementing that first cumulative Y with the Y incremental
value and so on to produce a series of distinct cumulative Y
values at the higher scale factor for each X coordinate.
The discrete cumulate Y values may then be reduced by the
scale factor to the scale of the numerator (do) before
shifting and added to the initial Y coordinate value to
- 79 -
.
produce the correct Y coordinate for each respective X
coordinate. By producing the cumulative Y using Y increment
at the increased scale factor, cumulative errors in the Y
coordinate value due to an error in the cumulative Y value
due to the lower scale factor of Y increments are avoided.
An example is shown in Table III. In the preferred
embodiment, a radix 2 machine is used with locations ox 16
bits each used and with the most significant or Thea bit
being used for a sign bit. The Y register, therefore,
corresponding to a first data word of 120 would be 0000 0000
0111 1000. As the leftmost or most significant bit of the
register is not available for indicating the numerical
value, as it is a sign bit in the preferred embodiment, then
the first data word expressed in binary, may be shifted to
the left in the direction of the most significant bits, a
total of eight places corresponding to the eight zeros or
unused binary bit positions to the left of "111 1000." The
updated first data word, as shown in the register is now
01111000000000000 corresponding to a decimal value of
30,720.
The denominator corresponding to the distance between
the end points in a second coordinate direction and to the
second data word is the binary word 0000 0000 0011 0010
equal to a decimal value of 50. The quotient produced by
the division of the dividend or first data word by the
denominator or second data word produces a quotient having a
number of bits related to the number of bits in the dividend
reduced by the number of bits in the divisor or denominator.
In the divisor, the most significant bit is in the Thea bit
position corresponding to a decimal 25 or decimal of 32.
The numerator corresponding to the first data word and
expressed as the 16 bit binary word 0111 1000 0000 0000 may
- 80 -
.:
`'
, ,
be raised by an order of the radix, (i.e. 25) corresponding
to the order of the most significant bit position of the
denominator, with the assurance that the quotient will fit
within the register size containing the first data word
after shifting above and without loss owe any significant
bits in the quotient, such as, for example, bit
corresponding to the remainder.
To summarize the foregoing, the precision of data word
for the numerator value, used in deriving the slope can be
raised within the space allotted by a machine register by
shifting the significant bits of that data word to the left
or towards the most significant bit in the register equal to
the number of unused bits in the register or in the example
between the most significant bit position for the first data
word shown, or seventh bit position and the most significant
bit position available in the register. This causes the
binary word corresponding to the numerator to be stated at
its highest value thereby avoiding floating point
arithmetic. The precision of the slope used in linear
interpolation may be further increased by expanding the
register space available for storing that binary word and
shifting hat first data word to the left, effectively
raising it by a power of the radix used, corresponding to
the number of significant bits in the denominator or
divisor. As stated above, as the slope is the quotient
produced by the division of the dividend or numerator (first
data value), by the divisor or denominator (second data
value), a shift of the numerator by the number of
significant bits in the denominator, after division produces
a quotient having a number of significant bits no greater
than the dividend. In this way, the precision of the
quotient or incremental value in the first coordinate
direction is maintained equal to the precision of a
numerator.
To continue with the example above, as the divisor
contains 6 significant bits corresponding in binary notation
to it 25 or 32 then the numerator may be shifted to the left
for a total of 13 places by increasing the scale, raising
the value of the first data word by a scale factor of 213~
As 213 equals decimal 8192, the arithmetic accomplished in
binary form and expressed in decimal is
doe. 120 x 8192 = _0000 HEX where 120 = YO-YO;
32 50 X2-Xli
The first coordinate or Y incremental value is
expressed in decimal as 19660/8192 = 2.399902 or COO in
HEX. The actual incremental value defined by the coordinate
end points is 120/50 = 2.4. The difference between the Y
incremental value, expressed as a third data word and the
actual slope is 4xlO error. As shown in Table III, -the Y
incremental produced at the higher precision is used to
iteratively increment Y, which is then reduced in scale
accordingly and added to the first end point in the first
coordinate direction (Y direction) and that Y coordinate
value is stored. In reducing the scale, the Y value may be
truncated or rounded, to produce an intercept value, for
example. As shown in Table III below, the intercept value
is shown as produced by truncation and rounding.
Additionally, to increase the speed of the process, the
value o-f the first data word or numerator may be compared to
successive entries in an ordered set of values. By a
comparison of the value of the first data word it can
quickly be determined whether the coordinate difference
(i.e. YO-YO) is greater or lesser than a value in a
particular position of the set of values. Accordingly, the
- 82 -
I
'
set of values may be arranged in decreasing orders of the
machine radix, (i.e. 213, 212, . . . 2). By a successive
process of comparison in the decreasing order, it can easily
be determined where in this procedure, the first data word
in the first machine location is less than a value in the
set of values. Then that particular value in the set of
values could be referenced to an index value to indicate the
number of bits available in the numerator register for
shifting the first data word. It should be noted that
according to the principles of the invention, it is not
necessary where the denominator is a power of two, such as
two or four, to divide, saving additional time, as shifting
of the numerator accomplishes the same result.
Accordingly, the above process is equally valid for the
reverse process where the set of values is in increasing
order and the comparison is in the increasing order with the
particular value of interest in the set being the first
greater than the first data word.
In this way, it is possible to avoid floating point
arithmetic by increasing the value of the numerator and
corresponding to the number of unused bits available in the
register for shifting while further increasing the value of
the numerator corresponding to the number of significant
bits in the denominator to maintain the power of two of the
most significant bit of the quotient equal to the power of
two of the most significant bit of the dividend and thereby
increasing the precision of the quotient and the answer.
Additionally, it is possible to use a set of values such as
a look up table for example, to compare the value of the
numerator to values in the denominator, selecting as an
index number related to a value in said set of values to
indicating the number of positions available for shifting
- I -
the numerator by arranging the set of values in an order of
values corresponding Jo decreasing or increasing orders of
the radix and successively comparing the numerator value in
the order of said decreasing or increasing powers of the
radix until a particular value is found which is greater or
less, respectively than the numerator value. That
particular value can then correspond to an index value
corresponding to the power of two available in the register
for shifting the numerator and increasing its value by a
power of two.
As would be apparent to those skilled in the art, the
inventive principles can be applied to any radix system or
to any coordinate system.
- 84 -
I.
TABLE III
tart Point Delta_X=50
Xl~Y1 I Dwelt Y incremental = 19660 doe.;
End Point 4CCC Hex
ZOO = 50, 120 Scale Factor = 105134 doe.
(truncated) (rounded) Cumulative incremental
ZOO Y Y per X coordinate
1,2 [ 2] Sum= 19660 = 4CCChex
2,4 5] Sum= 39320 = 9998hex
3,7 [ 7] Sum= 58980 = E664hex
4,9 [ 10] Sum= 78640 = 13330hex
5,11 [ 12] Sum= 98300 = 17FFChex
6,14 [ 14] Sum= 117960 = lCCC8hex
7,16 [ 17] Sum= 137620 = 21994hex
8,19 19] Sum= 157280 = 26660hex
9,21 [ 22] Sum= 176940 = 2B32Chex
10,23 [ 24] Sum= 196600 = fox
11,26 [ 26] Sum= 216260 = 34CC4hex
12,28 [ 29] Sum= 235920 = 39990hex
13,31 [ 31] Sum= 255580 = 3E65Chex
14,33 [ 34] Sum= 275240 = 43328hex
15,35 [ 36] Sum= 294900 = fox
: 16,38 [ 38~ Sum= 314560 = 4CCCOhex
17,40 [ 41] Sum= 334220 = kooks
18,43 [ 43] Sum= 353880 = 56658hex
19,45 [ 46] Sum= 373540 = Buicks
20,47 [ 48] Sum= 393200 = fox
21,50 [ 50] Sum= 412860 = 64CBChex
22,52 [ 53] Sum= 432520 = 69988hex
23,55 [ 55] Sum= 452180 = 6E654hex
24,57 [ 58] Sum= 471840 = 73320hex
25,59 [ 60] Sum= 491500 = fishhooks
- 85 -
I
I ,,
, .
I
26,62 [ 62] Sum= 511160 = 7CCB8hex
27,64 [ 65] Sum= 530820 = 81984hex
28,67 [ 67] Sum= 550480 = 86650hex
29,69 [ 70] Sum= 570140 = 8B31Chex
30,71 [ 72] Sum= 589800 = fox
31,74 [ 74] Sum= 609460 = 94CB4hex
32,76 [ 77] Sum= 629120 = 99980hex
33,79 [ 79] Sum= 648780 = 9E64Chex
34,81 [ 82] Sum= 668440 = Axe
35,83 [ 84] Sum= 688100 = Affix
36,86 [ 86] Sum= 707760 = ACCBOhex
37,88 [ 89] Sum= 727420 = B197Chex
38,91 [ 91] Sum= 747080 = Buicks
39,93 [ 94] Sum= 766740 = Buicks
40,95 [ 96] Sum= 786400 = BFFEOhex
41,98 [ 98] Sum= 806060 = C4CAChex
42,100 [101] Sum= 825720 = Kooks
43,103 [103] Sum= 845380 = Sioux
44,105 [106] Sum= 865040 - Dixie
45,107 [108] Sum= 884700 = D7FDChex
46,110 [110] Sum= 904360 = DCCA8hex
47,112 [113] Sum= 924020 = E1974hex
48,115 [115] Sum= 943680 = E6640hex
49,117 [118] Sum= 963340 = EB30Chex
50,120 [120] Sum= 983000 = EFFD8hex
;
- 86 -
'
i
.
The effect of shifting to increase the scale of the Y
increment is to eliminate the binary point in a third data
word representing the slope and Y increment and thereby
avoiding floating point arithmetic operations. The binary
point separates the bit locations separating fractional from
integer values in a data word (i.e. having values equal to
or greater than 1 and less than 1 (i.e. between the bit
positions ~2 and "2 1". The binary point is equivalent to
the decimal point in a radix 10 system and equivalent to a
"ponytail between those machine positions having a value equal
to or greater than 1 and less than one in any system and
separating the fractional values from the integer values in
the slope or Y increment value.
By shifting in the direction of more significant bits,
the scale factor of Y increment is increased, thereby moving
the "point" effectively in the direction of the less
significant bits. If a sufficient number of bit positions
are shifted, the binary point is eliminated from Y
increment. In this way, floating point arithmetic is
avoided. The value of Y increment at the increased scale
factor may be used to derive a cumulative Y for each
respective coordinate X values by iteratively incrementing Y
increment by Y increment to produce a first cumulative Y and
then incrementing that first cumulative Y with Y increment
and so on to produce a series of distinct cumulative Y
values at the higher scale factor for each X coordinate.
The discrete cumulate Y values may then be reduced by the
scale factor to the scale of the numerator (do) before
shifting and added to the initial Y coordinate value to
produce the correct Y coordinate for each respective X
coordinate. By producing the cumulative Y using Y increment
at the increased scale factor, errors in the Y coordinate
value due to an error in the cumulative Y increment due to
- I -
Jo
I.
the lower scale factor of Y increment are avoided.
The compiler used in the preferred embodiment to
perform the method described herein is shown in the
following program listings written in Motorola 68000
assembler language and contained within six modules.
The first module is the main program utilizing the
control modules shown as modules 2-7. Its function is to
access the pack data and interpret the knot locations and
control codes therein.
Module 2 is a subset of module 1 and is used to
evaluate each complete information set, identifying the Case
and control code value thereof.
Module 3 is the compiler described earlier, which
functions according to equations 3.1 and 3.2. and which
compiles the knot locations, the angles of the curve at the
knots, and the related values of the parameter t to produce
the node locations on the curve segment locus.
Module 4 is the compiler which operates according to
the method of auto-scaling linear interpolation shown
herein.
Module 5 is a module which receives curve coordinate
data, corresponding to the display coordinate system and
sorts this data in the order of the raster lines on the
display, so that the display data is accessed in timed
relation to the generation of the raster lines, with data
for display on any one particular raster line accessed yin
time with the location of the imaging beam at that raster
line.
- 88 -
;
:.
'7
Module 6 is a general purpose memory allocation and
release mechanism for buffer and raster line data.
Module 7 is an apparatus for performing general
trigonometry.
As said, these modules are written in 68000 assembler
code as used in the preferred embodiment to perform the
invention as described. The language for the preferred
embodiment is further compiled into machine object language
for use on the Motorola 68000.
With respect to the system generally shown in Figure
10, there is illustrated a general type of apparatus with
which the invention can be practiced. More particularly, in
one aspect the CPU 1 is a microprocessor arrangement made up
of, for example, a Motorola 68000 processor. The processor
will be used to control the encoded information in data
store 5. The data store can reside in part, for example,
Magnetic Disk, CUD ROM, Magnetic Tape, ROM, RAM, Networked
Data Base Systems, other magnetic media or the like
encodable media. It is to be understood that the
above-identified specific elements are encompassed in said
data store 5. Control and initiation of the process can be
effected, for example, by a keyboard 3 through which
instructions can be given. The processor controls the
imaging apparatus 7.
In specific aspect, the arrangement described is
especially adapted for use as a phototypesetter of the type
using conventional laser raster scan technology.
alternatively, the system of Figure 10 can also correspond
to a desktop computer with CRT display.
- 89 -
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