JPH0561982A - Curve approximation method - Google Patents

Abstract

PURPOSE:To shorten the processing time and to obtain a curve excellent in the approximate accuracy by stipulating the latter coordinates as a dot on the approximate curve when the interval of the XY coordinates of the dot obtained at each another process is separated by 1 dot. CONSTITUTION:By a point Pk (k=1-m) to show an anchor point and a control point and a parameter function Wk shown by using a parameter (t), an original curve shown by an expression is expressed approximately by using a matrix- shaped dot. The value of the parameter (t) used when the coordinates of the dot on the approximate curve are obtained is changed, the coordinates on the original curve are obtained and the coordinates of the matrix-shaped dot closest to the coordinates are obtained. The interval of the coordinates and the coordinates of the dot obtained first is separated by 1 dot in one axial direction and even in any axial direction, the interval is not separated by 2 dots or above, and then, the coordinates of the dot obtained third are stipulated as the dot on the approximate curve continuous immediately after the dot obtained first.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method of approximating a curve used for an outline font or the like.

[0002]

2. Description of the Related Art Conventionally, when a curve is displayed on a display medium using matrix-shaped dots such as a CRT or a printer, the curve is divided into a plurality of short vectors and then DDA (Digi
tal Differential Analyser (digital differential analysis).

[0003]

SUMMARY OF THE INVENTION In the above conventional method,
Since the curve is divided into short vectors, the processing time becomes long and the approximation accuracy becomes poor.

An object of the present invention is to provide a curve approximation method which has a short processing time and excellent approximation accuracy.

[0005]

A curve approximation method according to the present invention is a parameter represented by using a point "Pk" (k = 1, ..., M) representing an anchor point and a control point and a parameter "t". Function "Wk" (k =
1, ..., m) In the method of expressing the approximated curve of the original curve by approximating the original curve expressed by using the matrix dots, the parameter “t” is used to express the approximated curve on the approximated curve. The first step of obtaining the coordinates of one dot,
The second step of obtaining the coordinates on the original curve by changing the value of the parameter "t" used in the first step, and the matrix shape closest to the coordinates on the original curve obtained in the second step. In the third step of obtaining the coordinate of the dot, and the distance between the coordinate of the one dot obtained in the first step and the coordinate of the dot obtained in the third step is one dot apart in at least one axial direction and Also in the axis method (4), when the dots are not separated by 2 dots or more, the coordinate of the dot obtained in the third step is defined as a dot on the approximate curve immediately following the one dot, and the fourth step.

[0006]

EXAMPLE A cubic curve such as a cubic Bézier curve or a cubic spline curve is often used as a curve that normally expresses an outline font. Therefore, such a cubic curve will be described below as an example.

Parametrically expressing the cubic curve, it is expressed as follows: Q = WA × PA + WB × PB + WC × PC + WD × PD (1) Here, WA, WB, WC and WD are functions expressed using the parameter "t", for example, 3
In the case of the next Bezier curve, WA = (1-t) ³ (2a) WB = 3t · (1-t) ² (2b) WC = 3t ² · (1-t) (2c) WD = t ³ ( 2d) (0≤t≤1) (2e). PA, PB, PC and PD are coordinates on the XY plane that determine the shape of the curve, PA (x, y): anchor point PB (x, y): control point PC (x, y): Control point PD (x, y): Anchor point.

Here, considering the equation (1), the parameter functions WA, WB, WC and WD are, for example, (2
As shown in the equations a) to (2d), it is uniquely determined by the type of cubic curve, and PA, PB, PC and PD are predetermined according to the shape of the curve. Therefore, for the parameter functions WA, WB, WC and WD, the parameter “t” (0 ≦ t ≦ 1) is subdivided as shown in FIG. 5, and WA, It is possible to calculate the values of WB, WC and WD and store the calculation result in the data table. The coordinates of PA, PB, PC and PD can be stored in advance as data.

In this way, if each data is stored in advance and the hardware calculation based on the equation (1) is performed using the multiplier, the adder, etc., high speed processing becomes possible.

When the original curve expressed as described above is represented by dots at the intersections of the XY matrix (here, XY
The matrix means a virtual XY matrix before displaying on a display medium such as a CRT or a printer, but it may be a physical XY matrix when actually displaying on such a display medium. .. ), The coordinates of the intersections of the XY matrix can take only discrete values (for example, x
= 0, 1, 2, ..., Y = 0, 1, 2, ...), and the approximated curve represented by dots cannot faithfully reproduce the original curve. Therefore, among the intersections of the XY matrix, the intersections that are as close as possible to the original curve are sequentially selected to represent the approximate curve.

Based on the above, the outline of a method of obtaining the coordinates of the intersection of the XY matrix in the approximate curve will be described (see FIGS. 1, 2 and 5).
In the following description, unless otherwise specified, the anchor points PA, PD and the control point PB,
PC is PA (x, y) = (0,0) PB (x, y) = (10,20) PC (x, y) = (40,25) PD (x, y) = (50,0) ) Will be explained.

In FIGS. 1 and 2, Q0, Q1, Q
2, ... Are dots representing the approximate curve, and x and y are the X and Y coordinates of the approximate curve. The coordinates of the origin "Q0" of the approximate curve are "x = 0, y = 0".

First, assuming that the value of t is 1/2, 1/4, ..., And the values of the XY coordinates of the original curve are sequentially obtained based on the equation (1). Then, these values are rounded off and the values (x, y) of the X and Y coordinates of the intersections of the XY matrix are sequentially obtained. In this way, when the above-mentioned operation is repeated and the increment of the X coordinate is Δx and the increment of the Y coordinate is Δy, the following relation (Δx = 0 and Δy = 1) or (Δx = 1 and Δy = 0) or (Δx = 1 and Δy = 1) (3) The XY coordinate (x, y) when satisfying (3) is set as the first dot Q1 of the approximate curve (in the example of FIGS. 1 and 2, x = 0, y = 1 ”.) At this time, the value of n is set as the initial value of the increment Δn of n (“ Δn ”in the examples of FIGS.
= 2 ". ), The second and subsequent dots Q2, Q3,
... will be sought after.

To obtain the second dot Q2,
First, as “t = 4/128” (value obtained by adding the initial value “2” of Δn to the value “2/128” of t when the first dot D1 is obtained), based on the equation (1), Then, the X-Y coordinate value of the original curve is obtained. Then, those values are rounded off to obtain XY
The XY coordinates (x, y) of the intersection point of the matrix are obtained.

In the example of FIGS. 1 and 2, "x = 1, y =
2 ", that is," .DELTA.x = 1, .DELTA.y = 1 ", and the condition of the expression (3) is satisfied, so this value becomes the XY coordinate of the second dot Q2. In this way, the value of n is
(Δn = 2 in the example of FIGS. 1 and 2), and the XY coordinates (x, y) of each dot are sequentially obtained. Then, when Δx and Δy have the following relationship (Δx = 0 and Δy = 0) (4), the value of n is set to “x” until the XY coordinates (x, y) satisfy the condition of the expression (3). It is sequentially increased by 1 "(in the example of FIGS. 1 and 2, the 17th dot Q17 corresponds). Then, the value of Δn at that time is changed to a new Δn.
(“Δn = 3” in the examples of FIGS. 1 and 2). On the other hand, when Δx and Δy have the following relation (Δx> 1 or Δy> 1) (5), the value of n is set to “x” until the XY coordinates (x, y) satisfy the condition of the expression (3). It is sequentially decreased by 1 ″ (in the example of FIGS. 1 and 2, the 20th dot Q20 corresponds). Then, the value of Δn at that time is changed to a new Δn.
(“Δn = 2” in the examples of FIGS. 1 and 2).

The above is the outline of the method for obtaining the XY coordinates (x, y) of the intersection of the XY matrix in the approximate curve.

Next, a concrete embodiment of the method for obtaining the coordinates of the intersection of the XY matrix in the approximated curve will be described with reference to FIG.
~ It demonstrates with reference to FIG.

FIG. 3 is a block diagram showing the hardware configuration of the embodiment.

The CPU 0 is a microprocessor and controls the entire hardware. ROM0 is a read-only memory and is a microprocessor CPU0
The programs for various processes performed by the are stored. The RAM0 is a random access memory, which stores not only data on the cubic Bezier curve but also each data described later in advance. These data are sent to RAM1 and RAM2 described later as needed.

The RAM 1 is a random access memory, and has parameters corresponding to the data table shown in FIG.
It stores the respective values of the data functions WA, WB, WC and WD. These data WA0-WA128, WB0-
WB128, WC0 to WC128, WD0 to WD128 are RA
It is sent from M0 and is stored in the RAM1 as shown in FIG. PNT1 is an address pointer of RAM1 and is an independent address pointer PNT1.
It is composed of A, PNT1B, PNT1C and PNT1D. PNT1A, PNT1B, PNT1C
And PNT1D are data WA0 to WA128, WB0 to WB128, WC0 to WC12 stored in the RAM1.
8 and WD0 to WD128 (see FIG. 6) are respectively designated.

The RAM 2 is a random access memory and stores the coordinate data of the anchor points PA and PD, the control points PB and PC.

These coordinate data are sent from the RAM0 and are stored in the RAM2 as shown in FIG. In FIG. 7, PAx to PDx are X coordinates of anchor points and control points, P
Ay to PDy are the Y coordinates of each of these points. P
NT2 is an address pointer of RAM2,
The address of each coordinate data (see FIG. 7) of the anchor point and the control point stored in FIG.

The RAM 3 is a random access memory and stores the result of the hardware calculation based on the equation (1), in other words, the data of the XY coordinates (x, y) of the approximate curve. Is. These XY coordinates (x,
The data of y) is stored in the RAM 3 as shown in FIG. The data shown in FIG. 8 is shown corresponding to the data of the XY coordinates (x, y) shown in FIG. PNT3
Is an address pointer of RAM3.

MLT is a multiplier, and R in FIG.
The data stored in AM1 is multiplied by the data stored in RAM2 of FIG. That is,
The calculation is performed on the multiplication part (eg, WA × PA) of the equation (1). ALU is an arithmetic logic circuit for adding the multiplication value of the multiplier MLT and the value of the accumulator ACC described later. The ACC is an accumulator, which holds the operation result of the arithmetic logic circuit ALU and sends the operation result to one input of the arithmetic logic circuit ALU. That is, the arithmetic logic circuit ALU and the accumulator ACC perform the operation of the addition part of the equation (1).

MEM1 is a memory circuit, which temporarily stores the operation result Q (x, y) of the equation (1). S
UB1 is a subtraction circuit, which subtracts the value stored in the memory circuit MEM1 from the current calculation result and produces the subtraction result. CMP1 is a comparison circuit, and the increment Δx in the X-axis direction and the increment Δy in the Y-axis direction of the subtraction result are
It is to compare whether or not “Δx> 1 or Δy> 1” is satisfied.

MEM2 is a memory circuit, which temporarily stores the pointer value of the address pointer PNT1A. SUB2 is a subtraction circuit, which subtracts the pointer value stored in the memory circuit MEM2 from the current pointer value (pointer value of the address pointer PNT1A),
The result of the subtraction is generated.

The MEM3 is a memory circuit for storing "+1" or "-1".

MPX is a multiplexer, and passes either the output of the subtraction circuit SUB2 or the output of the memory circuit MEM3. ADD is an adder that adds the current pointer value (pointer value of the address pointer PNT1A) and the output value of the multiplexer MPX. CMP2 is a comparison circuit for comparing whether or not the pointer value of the address pointer PNT1A exceeds "128". In other words, it is determined whether or not the value of the parameter "t" exceeds "1".

CONT is a control circuit, which receives an instruction from the microprocessor CPU0 and controls the above-mentioned arithmetic processing. This control circuit C
The ONT includes a microprogram for each of the above arithmetic processing.

Next, the operation of this embodiment will be described using a flow chart (see FIGS. 1 to 8).

First, the following initialization is performed based on the signal from the microprocessor CPU0. RAM0 to RAM1 have data WA0 to WA128 and WB0 to W corresponding to the parameter functions WA, WB, WC and WD.
B128, WC0 to WC128, and WD0 to WD128 are sent and stored in the RAM 1 as shown in FIG. The pointer value of the address pointer PNT1 is "t = 1/2 (64
/ 128) ”compatible data WA64, WB64, WC6
4. Initially set to the address of RAM1 where WD64 is stored. That is, each address pointer PNT1
The pointer values of A, PNT1B, PNT1C and PNT1D are “64”, “193”, “322”,
It becomes "451". From RAM0 to RAM2, anchor points PA and PD, control point PB
And coordinate data of the PC are sent and stored in the RAM 2 as shown in FIG. The pointer value of the address pointer PNT2 is initialized to "0". The address “0” and the address “1” of the RAM 3 correspond to the coordinates (x, y) of the origin of the approximate curve, and therefore RA
The data "0" is stored in the address "0" of the M3, and the data "0" is stored in the address "1" (see FIG. 8).
reference). The pointer value of the address pointer PNT3 is
Initially set to "2". In the memory circuit MEM1, the coordinates (x, y) of the origin of the approximate curve, that is, “x = 0, y =
0 ”is initialized. “0” is stored in the memory circuit MEM2.
Is initialized. (F1).

Next, the first dot "Q" of the approximate curve
In order to obtain 1 ", the operation corresponding to the equation (1) is performed as follows. In the multiplier MLT, the value stored at the address (address" 64 ") designated by the address pointer PNT1A. ("WA64", Figure 6
Reference) and the value stored in the address (address “0”) designated by the address pointer PNT2 (“PAx
, (See FIG. 7) and the multiplication result “WA64
XPAx "is sent to the Y input of the arithmetic logic circuit ALU, and X
It is added to the input value (accumulator ACC value "0"). The addition result "WA64 * PAx" is temporarily held in the accumulator ACC. Subsequently, PNT1B is selected as the address pointer PNT1, and the pointer value of the address pointer PNT2 is incremented (+1). Then, in the multiplier MLT, the address designated by the address pointer PNT1B (address “1
93 ”) (“ WB64 ”, see FIG. 6)
And the value ("PBx", stored at the address (address "1") designated by the address pointer PNT2,
(See FIG. 7) and are multiplied. Multiplication result “WB64 × PB
x "is sent to the Y input of the arithmetic logic circuit ALU and added to the value of the X input (accumulator ACC value" WA64 x PAx "). The addition result" WA64 x PAx + WB64 x "
PBx "is temporarily stored in the accumulator ACC. Similar to the above operation, the product-sum result" WA64 * PAx + WB64 * PBx + WC64 * PCx "for the X coordinate is obtained.
+ WD64 × PDx ”is obtained, and the product-sum result is rounded off by the arithmetic logic circuit ALU. Similarly, the product-sum result“ WA64 × PAy + WB64 × PBy + W ”for the Y coordinate is obtained.
C64.times.PCy + WD64.times.PDy "is obtained, and the product-sum result is rounded off by the arithmetic logic circuit ALU. (F2, f
3).

In the subtraction circuit SUB1, the XY coordinates Q0 (0, 0, 0, 0, 0) of the origin of the approximated curve stored in the memory circuit MEM1 is calculated from the rounded values (x, y) of the XY coordinates.
The value of 0) is subtracted. Increment Δx in the X-axis direction of the subtraction result
And the increment Δy in the Y-axis direction are sent to the comparison circuit CMP1 and are compared to see if “Δx> 1 or Δy> 1” is satisfied. (F4).

When the subtraction result is "Δx> 1 or Δy>1","t = t / 2", that is, "t = 1/4 (32
/ 128) ”compatible data WA32, WB32, WC3
2. In order to read WD32, the pointer value of each address pointer PNT1A, PNT1B, PNT1C, PNT1D is set to the address value of RAM1 where these data are stored. (F5).

Until the subtraction result does not satisfy "Δx> 1 or Δy>1", the operations of f2 to f4 are repeated. That is, the above operations f2 to f4 are repeated until the subtraction result satisfies (Δx = 0 and Δy = 1) or (Δx = 1 and Δy = 0) or (Δx = 1 and Δy = 1) (3). Is. The value (x, y) of the XY coordinates when the condition of the expression (3) is satisfied is the first dot Q1 of the approximate curve.
The coordinates are (x, y) (“x = 0, y = 1” in FIGS. 1 and 2).

The value of the coordinate Q1 (x, y) is stored in the address ("0002", "0003", see FIG. 8) of the RAM 3 designated by the address pointer PNT3. The value of the coordinate Q1 (0,1) is stored in the memory circuit MEM1 instead of the coordinate Q0 (0,0) of the origin of the approximate curve. (F6).

Subsequently, the routine for obtaining the dots "Q2", Q3 ", ... After the second approximation curve is performed as follows: The subtraction circuit SUB2 causes the pointer of the address pointer PNT1A. The value "0" stored in the memory circuit MEM2 is subtracted from the value (value "2" of "n" in FIG. 1) to calculate the value of Δn shown in FIG. 1 (in the case of FIGS. 1 and 2). Is “Δn = 2”). continue,
The value of the address pointer PNT1A (“2” in the case of FIGS. 1 and 2) is stored in the memory circuit MEM2. (F
7).

Address pointers PNT1A, PNT1B, PNT1C and PNT1D are added by the adder circuit ADD.
And the value of Δn are respectively added, and the addition result is obtained as the address pointers PNT1A and PNT1.
It becomes a new pointer value of B, PNT1C and PNT1D. (F8).

Subsequently, the product-sum operation and rounding are performed in the same manner as (f2) and (f3). (F9, f1
0).

In the subtraction circuit SUB1, the value of the XY coordinate after being rounded off (in the case of FIGS. 1 and 2, "x =
1, y = 2 ") to the value of the XY coordinate Q1 (x, y) stored in the memory circuit MEM1 (" x = 0, y = 1 "in FIGS. 1 and 2). The value is subtracted. The subtraction results Δx and Δy are sent to the comparison circuit CMP1. "Δ
x> 1 or Δy> 1 ”, the address pointers PNT1A, PNT1B, PNT1C and PNT1D
Each pointer value of 1 and the value “−1” stored in the memory circuit MEM3 are added by the adder circuit ADD. as a result,
Address pointers PNT1A, PNT1B, PNT1C
And the pointer values of PNT1D are decremented (-
1) is done. (F11, f12). “Δx = 0 and Δy
= 0 ”, the address pointers PNT1A, PN
Each pointer value of T1B, PNT1C and PNT1D and the value “+1” stored in the memory circuit MEM3 are added by the adder circuit ADD. As a result, the address pointers PNT1A, PNT1B, PNT1C and PNT1D
Each pointer value of is incremented (+1). (F
13, f14). In the case of FIG. 1 and FIG. 2, since the subtraction result does not satisfy any of the conditions of f11 and f13, the value of the XY coordinate (x = 1, y = 2) after the product-sum calculation and rounding are performed. ) Directly becomes the coordinates of the second dot "Q2". (F15).

The subtraction circuit SUB2 subtracts the value "2" stored in the storage circuit MEM2 from the pointer value of the address pointer PNT1A ("4" in FIGS. 1 and 2) to calculate the value of Δn. (“Δn = 2” in FIGS. 1 and 2) (f17).

Address pointers PNT1A, PNT1B, PNT1C and PNT1D are added by the adder circuit ADD.
And the value of Δn are respectively added, and the addition result is obtained as the address pointers PNT1A and PNT1.
It becomes a new pointer value of B, PNT1C and PNT1D. (F18).

Thereafter, the routine of f9 to f18 is repeated in the same manner as described above until the value of n exceeds "128", in other words, while the condition of "0≤t≤1" is satisfied. Be done. (F16). In this way, the dots Qm (x, y) of the approximate curve are sequentially determined, and these values are sequentially stored in the RAM 3 as shown in FIG. In the example of FIGS. 1 and 2, the case where the 20th dot Q20 is obtained is f
This corresponds to the routine of 12 and the case of finding the 17th dot Q17 corresponds to the routine of f14.

As described above, X- in the approximate curve
The coordinates of the intersection points of the Y matrix are sequentially obtained.

In the above embodiment, when the result of performing the routine of f5 is "t <1/128", in other words, "n <1", the "1/2 subspan method" is used. Then, the following processing may be performed (see FIG. 9). First, the original curve is divided into two segments, and new anchor points and control points are defined for each segment. In the example of FIG. 9, the anchor points of one segment are PA and PE, the control points are PF and PG, the anchor points of the other segment are PE and PD, and the control points are PH and PI. Using these new anchor points and control points, each segment is processed in the same manner as the above method. Then, the processing results for each segment are finally combined to determine the coordinates of the intersection of the XY matrix in the approximate curve.

Although the data table is constructed by dividing "t" into 128 in the above embodiment, the data table may be constructed with the number of divisions of 256, 512, ... When handling large coordinate data. Good. When handling small coordinate data using a data table having a large number of divisions, a value larger than "± 1" (for example, "± 3") is stored in the memory circuit MEM3 shown in FIG. If this is done, convergence to a value that satisfies a predetermined condition is accelerated, and high-speed processing is possible.

The concept of the present invention can be applied not only to the two-dimensional curve described above, but also to the three-dimensional curve.

[0048]

By using the curve approximation method of the present invention, it is possible to perform curve approximation with a short processing time and excellent approximation accuracy.

[Brief description of drawings]

FIG. 1 is an explanatory diagram showing an embodiment of the present invention.

FIG. 2 is an explanatory diagram showing an example of the present invention.

FIG. 3 is a block diagram showing an embodiment of the present invention.

FIG. 4 is a flowchart showing the operation of the embodiment of the present invention.

FIG. 5 is an explanatory diagram showing a part of the embodiment.

FIG. 6 is an explanatory diagram showing a part of the embodiment.

FIG. 7 is an explanatory diagram showing a part of the embodiment.

FIG. 8 is an explanatory diagram showing a part of the embodiment.

FIG. 9 is an explanatory diagram when a part of the embodiment is changed.

Claims (4)

[Claims] 1. A parameter function "Wk" represented by using a point "Pk" (k = 1, ..., M) representing an anchor point and a control point and a parameter "t".
With (k = 1, ..., m), In the method of expressing the approximated curve of the original curve by approximating the original curve expressed by using the matrix dot, the above-mentioned approximated curve is expressed by using the parameter “t”. The first step for obtaining the coordinates of one dot, the second step for obtaining the coordinates on the original curve by changing the value of the parameter "t" used in the first step, and the second step The third step of obtaining the coordinates of the matrix-like dots closest to the coordinates on the original curve, and the distance between the coordinates of one dot obtained in the first step and the coordinates of the dots obtained in the third step are , At least one dot in at least one axial direction and not more than two dots apart in any of the axial methods, the dot coordinates obtained in the third step are set on the approximate curve immediately after the one dot. Fourth as a dot And a curve approximation method. 2. A parameter function "Wk" represented by using a point "Pk" (k = 1, ..., M) representing an anchor point and a control point and a parameter "t".
With (k = 1, ..., m), In the method of expressing the approximated curve of the original curve by approximating the original curve expressed by using the matrix dots, the change range of the parameter “t” is divided into a plurality of sections. The first step of dividing and setting the parameter value of the parameter "t" for each division point and obtaining the coordinate of one dot on the approximate curve using the parameter value , Parameters other than the above parameter values used in the first process
The second step of obtaining the coordinates on the original curve using the data values, the third step of obtaining the coordinates of the matrix-like dots closest to the coordinates on the original curve obtained in the second step, and the first step. When the distance between the coordinates of one dot obtained in the process and the coordinates of the dot obtained in the third process is 1 dot apart in at least one axis direction and not more than 2 dots apart in any axis method, And a fourth step of determining the coordinates of the dot obtained in the third step as a dot on the approximate curve immediately following the one dot. 3. A parameter function "Wk" represented by using a point "Pk" (k = 1, ..., M) representing an anchor point and a control point and a parameter "t".
With (k = 1, ..., m), In the method of expressing the approximated curve of the original curve by approximating the original curve expressed by using the matrix dot, the above-mentioned approximated curve is expressed by using the parameter “t”. A first step for obtaining the XY coordinates of one dot, a second step for obtaining the XY coordinates on the original curve by changing the value of the parameter "t" used in the first step, and the second step The third step of obtaining the XY coordinates of the matrix-like dots closest to the XY coordinates on the original curve obtained in step 3, and the XY coordinate of one dot obtained in the first step and the third step
The distance between the dot and the XY coordinates obtained in the process is “0 dot in the X-axis direction and 1 dot in the Y-axis direction” or “1 dot in the X-axis direction and 0 dot in the Y-axis direction” or “1 dot in the X-axis direction”.
A dot and one dot in the Y-axis direction ", a fourth step of defining the XY coordinates of the dot obtained in the third step as a dot on the approximate curve immediately after the one dot. Characteristic curve approximation method. 4. A parameter function "Wk" represented by using a point "Pk" (k = 1, ..., M) representing an anchor point and a control point and a parameter "t".
With (k = 1, ..., m), In the method of expressing the approximated curve of the original curve by approximating the original curve expressed by using the matrix dots, the change range of the parameter “t” is divided into a plurality of sections. The first step of dividing and setting the parameter value of the parameter "t" for each division point and obtaining the XY coordinate of one dot on the approximate curve using the parameter value And parameters other than the above parameter values used in the first process
The second step of obtaining the XY coordinates on the original curve using the data, and the third step of obtaining the XY coordinates of the matrix-like dots closest to the XY coordinates on the original curve obtained in the second step. The XY coordinates of one dot obtained in the first process and the third coordinate
The distance between the dot and the XY coordinates obtained in the process is “0 dot in the X-axis direction and 1 dot in the Y-axis direction” or “1 dot in the X-axis direction and 0 dot in the Y-axis direction” or “1 dot in the X-axis direction”.
A dot and one dot in the Y-axis direction ", a fourth step of defining the XY coordinates of the dot obtained in the third step as a dot on the approximate curve immediately after the one dot. Characteristic curve approximation method.