JP2522109B2 - Curve fitting method - Google Patents

Description

The present invention relates to a method for approximating a curve in an outline font.

[Prior Art] As a means for expressing a curve such as a character or a figure, a font using an outline font has been rapidly spread in recent years.

Conventionally, in the case of actually displaying a curve using an outline font, the curve is sequentially divided until adjacent division points degenerate to the same point, and each division point is sequentially connected by a straight line to approximately represent the curve.

[Problems to be Solved] The above-described conventional method has a problem that a large amount of arithmetic processing is required each time curve data of an outline font is generated, and a large amount of time is required to generate the outline font.

An object of the present invention is to reduce the time required to generate an outline font.

[Means for Solving the Problem] The present invention represents a curve based on a curve formula parametrically expressed using two anchor points and at least one control point, and divides the curve into a plurality of parts. In the method of approximating a curve by connecting adjacent division points with a straight line, the values of the parameter functions of the curve equations that are determined for each of the n parameter values of the curve equation are stored in advance and the curve to be represented According to the above, only for m (m <n) of the n parameter values, the value of the curve expression is obtained based on the value of the parameter function,
A straight line connects the division points determined corresponding to the m parameter values.

Embodiments Embodiments of the present invention will be described below with reference to the accompanying drawings.

In the embodiments described below, unless otherwise specified, a cubic curve such as a cubic Bezier curve or a cubic spline curve is used as a curve expressing an outline font. Parametric representation of these curves is as follows.

Q = WA x PA + WB x PB + WC x PC + WD x PD (1) where WA, WB, WC and WD are functions expressed using the parameter "t", for example, in the case of cubic Bezier curve It looks like this:

WA = (1-t) ³ (2a) WB = 3t ・ (1-t) ² (2b) WC = 3t ²・ (1-t) (2c) WD = t ³ (2d) (0 ≤t≤1) PA, PB, PC and PD are coordinates on the XY plane that determine the shape of the curve, and are as follows.

PA (x, y): Anchor point PB (x, y): Control point PC (x, y): Control point PD (x, y): Anchor point Considering equation (1), the parameter function W
A, WB, WC, and WD are uniquely determined by the type of cubic curve as shown in equations (2a) to (2d), and PA, PB, PC, and PD depend on the shape of the curve. It is predetermined. Therefore, the parameter function WA,
For WB, WC and WD, the parameter “t” is subdivided as shown in FIG. 5, and WA, WB, and
The values of WC and WD can be calculated and the calculation results can be stored in a data table. PA, PB, PC and PD
As for, also, the coordinates (x, y) can be stored in advance in the data table. In this way, if each data is stored in the data table and the hardware calculation based on the equation (1) is performed using the multiplier and the adder, the curve data can be generated at high speed. That is, as shown in FIG. 5, t (0 ≦ t ≦ 1) is divided into 256, and the parameter functions WA, WB, WC, and WD are previously set for each t.
Then, if the hardware calculation is performed based on the equation (1), a total of 257 points on the curve can be obtained. A curve can be approximated by connecting these coordinates with a straight line.

By the way, in the above example, the curve is divided into 256 to be approximated, but for example, for a curve with a small curvature,
An approximate curve with sufficient accuracy can be expressed without increasing the number of divisions so much. This will be described below with reference to FIG. 3, FIG. 4 (A) and FIG. 4 (B). The curve shown in FIG. 3 is composed of a total of 14 small curves C1 to C14. That is, each of the small curves C1 to C14 is represented by the equation (1) by using different anchor points and control points. Fourth
Figures (A) and 4 (B) show the small curves shown in Figure 3.
C1 and C2 are shown respectively. The small curve C1 is divided into 16 as shown in FIG. 4 (A), and 17 points on the small curve C1
It is approximately expressed by Q0,0 to Q0,16. Therefore, in the example of FIG. 5, n = 0, 16, 32, ...
Is increased by 16 steps (256/16 = 16), and the calculation may be performed only for these n. The small curve C2 is divided into 32 as shown in FIG. 4 (B), and 33 points Q0,0 on the small curve C2
Approximately represented by ~ Q0,32. Therefore, in the example of FIG.
It is only necessary to increase n by 8 steps (256/32 = 8), such as 0, 8, 16, ... In this way, it is possible to further speed up the hardware calculation by previously obtaining an appropriate number of divisions according to the curve and performing only the calculation. A specific method for obtaining the number of divisions will be described later.

The method for approximating the curve shown in FIG. 3 will be described below with reference to the above description.

FIG. 1 is a block diagram showing the hardware used to approximate the curve shown in FIG.

CPU0 is a microprocessor that controls the entire hardware.

ROM0 is a read-only memory and stores programs for various processes performed by the microprocessor CPU0.

RAM0 is a random access memory, and stores each data described later in advance. RAM0 stores not only data related to cubic Bezier curves but also data related to various curves such as data related to cubic spline curves. These data are stored in RAM1 and RAM2, which will be described later, as necessary.
And sent to RAM3.

RAM1 is a random access memory, and stores the values of the parameter functions WA, WB, WC and WD shown in FIG. These data WA0-WA256, WB0-WB256,
WC0 to WC256 and WD0 to WD256 are sent from RAM0 and are stored in RAM1 as shown in FIG.

RAM2 is a random access memory, and anchor points PA and P are provided for each of the small curves C1 to C14 shown in FIG.
It stores each coordinate data of D, control point PB, and PC. Each of these coordinate data is sent from RAM0, and as shown in FIG.
Be stored in. In Figure 7, PAX0 to PDX0 are small curves
Each x coordinate of the anchor point and control point in C1, PAY0 to PDY0 is each y coordinate of these points, PAX1 to PDX1 is each x coordinate of these points in the minor curve C2, and PAY1 to PDY1 is each y of these points. Coordinates.

The RAM 3 is a random access memory and stores in advance data corresponding to the number of divisions of the small curves C1 to C14 shown in FIG. 3 for each of the small curves C1 to C14. The data corresponding to the number of divisions may be the number of divisions itself or the number of n steps shown in FIG. In the present embodiment, as shown in FIG. 8, the random access memory RAM3 has a step number n, that is, an increment Δn of n.
Is stored. In FIG. 8, Δn1, Δn2, ...
Is the number of steps stored corresponding to each of the small curves C1, C2, ... Shown in FIG. These data Δ
n1, .DELTA.n2, ... Are sent from RAM0 and are stored in RAM3 as shown in FIG.

The RAM 4 is a random access memory and stores the result of the hardware calculation based on the equation (1), in other words, the coordinate data Q (x, y) on each of the small curves C1 to C14. These coordinate data Q (x, y) are stored in the RAM 4 as shown in FIG. In FIG. 9, the data X0,0 to X0,16 and Y0,0 to Y0,16 are the x and y coordinates of the points Q0,0 to Q0,16 on the small curve C1 in FIG. 4 (A), respectively. Yes, the data X1,0 to X1,32 and Y1,0 to Y1,32 are the x and y coordinates of the points Q1,0 to Q1,32 on the minor curve C2 of FIG. 4 (B), respectively.

PNT1 is an address pointer of RAM1 and is composed of independent address pointers PNT1A, PNT1B, PNT1C and PNT1D. RA for PNT1A, PNT1B, PNT1C and PNT1D
Data stored in M1 WA0 to WA256, WB0 to WB256, WC
The addresses 0 to WC256 and WD0 to WD256 (see FIG. 6) are designated.

PNT2 is an address pointer of RAM2, and designates the address of each coordinate data (see FIG. 7) of anchor points and control points stored in RAM2.

PNT3 is an address pointer of RAM3, and the number of steps stored in RAM3 is Δn1, Δn2, ... (See Fig. 8).
The address of is specified.

PNT4 is an address pointer of RAM4, coordinate data X0,0 to X0,16 and Y0, obtained by hardware calculation.
The addresses 0 to Y0,16 etc. (see FIG. 9) are designated.

STP is a latch circuit for latching the step numbers Δn1, Δn2, ... (See FIG. 8) stored in the RAM3.

ADD is an adder circuit that adds the value of the currently selected address pointer (PNT1A, PNT1B, PNT1C or PNT1D) and the value of the latch circuit STP and sends the addition result to the currently selected address pointer Is. That is, it plays a role of increasing the value of the address pointer by the number of steps Δn1, Δn2, ... Latched by the latch circuit STP.

MLT is a multiplier circuit, and RAM1 shown in FIG.
The data stored in the RAM 2 and the data stored in the RAM 2 shown in FIG. 7 are multiplied. That is,
The multiplication (for example, WA × PA) portion in the equation (1) is calculated.

ALU is an arithmetic logic circuit that adds the multiplication value of the multiplier circuit MLT and the value of an accumulator ACC described later.

ACC is an accumulator, which holds the operation result of the arithmetic logic circuit ALU and stores this operation result in the arithmetic logic circuit.
It is sent to one input of the ALU. That is, the arithmetic logic circuit ALU and the accumulator ACC perform the operation of the addition part in the equation (1).

CONT is a control circuit, a microprocessor
Upon receiving an instruction from the CPU0, it controls each of the above arithmetic processing. The control circuit CONT includes a micro program for each of the above arithmetic processing.

Next, the operation of the circuit shown in FIG. 1 will be described with reference to the flowcharts shown in FIGS. 2 (A), (B) and (C).

First, initialization is performed based on the signal from the microprocessor CPU0.

(A): Data corresponding to t of each of the parameter functions WA, WB, WC, and WD shown in FIG. 5 is sent from RAM0 to RAM1, and these data are stored in RAM1 as shown in FIG. Memorized in. Coordinate data of anchor points and control points are sent from RAM0 to RAM2 for each of the small curves C1 to C14 shown in FIG. 3, and these data are stored in RAM2 as shown in FIG. To be done. RAM0 to RA
In M3, the step number data Δn for each of the small curves C1 to C14 corresponding to the number of divisions of each of the small curves C1 to C14 shown in FIG.
1, Δn2, ... Are sent and these data are stored in the RAM 3 as shown in FIG.

(B): The address pointers PNT2, PNT3, PNT4 are set to "0". As a result, the data PAX0 shown in FIG.
Is stored, the address where the data .DELTA.n1 shown in FIG. 8 is stored, and the address where the data X0,0 shown in FIG. 9 is stored are designated.

The initial setting is completed as described above, and then the arithmetic processing for the small curve C1 shown in FIGS. 3 and 4 (A) is performed.

(C): “0” is set in the address pointer PNT1A, “257” is set in the address pointer PNT1B, “514” is set in the address pointer PNT1C, and “771” is set in the address pointer PNT1D. 6, the addresses at which the data WA0, WB0, WC0 and WD0 are stored are designated respectively, and the latch circuit STP designates the address "0000" of the RAM3 designated by the address pointer PNT3. The data M [PNT3] stored in is latched, that is, the step number Δn1 shown in Fig. 8 (the step number "16" corresponding to the number of divisions of the small curve C1 shown in Fig. 4 (A)) is Will be latched.

(D): Next, the calculation routine shown in FIG. 2 (B) is performed as follows.

First, in order to obtain the x coordinate of the point Q0,0 shown in FIG. 4 (A), “WA × PA + WB × PB + PC × PC + WD × P” in the equation (1) is used.
The value of D "is calculated as follows.

(D1): Accumulator ACC is cleared and “0” is set.

(D2): The data M [PNT1A] stored in the address "0000" of the RAM1 designated by the address pointer PNT1A, that is, "WA0" (see FIG. 6) is the multiplier circuit WLT.
Sent to the "X" input of. Data M stored at address "0000" of RAM2 designated by address pointer PNT2
[PNT2] or "PAX0" (see FIG. 7) is sent to the "Y" input of the multiplier circuit MLT.

(D3): The sum of products calculation shown in FIG. 2 (C) is performed as follows.

(D101): Multiplier circuit MLT "X" input value "WA"
The product "WA0 x PAX0" of "0" and the value "PAX0" of the "Y" input is output to "Z" of the multiplier circuit MLT.
The calculation corresponding to the expression "WA x PA" is performed. The result of this calculation is sent to the "Y" input of the arithmetic logic circuit ALU, while the data "0" of the accumulator ACC is sent to the "X" input of the arithmetic logic circuit ALU.

(D102): The sum of the value of the "X" input of the arithmetic logic circuit ALU and the value of the "Y" input of the arithmetic logic circuit ALU is calculated, and the calculation result is sent to the accumulator ACC. As a result, the value of “WA0 × PAX0” is held in the accumulator ACC.

(D103): "1" is added to the value of the address pointer PNT2, and as a result, the pointer value becomes "0001".

(D4): The data M [PNT1B] stored in the address "0257" of the RAM1 designated by the address pointer PNT1B, that is, "WB0" (see FIG. 6) is the multiplier circuit MLT.
Sent to the "X" input of. Data M stored at address "0001" of RAM2 designated by address pointer PNT2
[PNT2], that is, "PBX0" (see FIG. 7) is sent to the "Y" input of the multiplier circuit MLT.

(D5): The sum of products calculation shown in FIG. 2 (C) is performed. The basic operation is the same as that described in (d3) above. In the multiplier circuit MLT, "WB0" shown in FIG.
Calculation corresponding to the product “WB0 × PBX0” with “PBX0” shown in the figure, that is, “WB × PB” in the equation (1) is performed. The arithmetic logic circuit ALU calculates the sum of this calculation result “WB0 × PBX0” and the value “WA0 × PAX0” held in the accumulator ACC. The calculation result “WA0 × PAX0 + WB0 × PBX0” is held in the accumulator ACC. That is, here, the calculation corresponding to “WA × PA + WB × PB” in the equation (1) is performed.

(D6): The data M [PNT1C] stored in the address “0514” of RAM1 designated by the address pointer PNT1C, that is, “WC0” (see FIG. 6) is the multiplier circuit MLT.
Sent to the "X" input of. Data M stored at address "0002" of RAM2 designated by address pointer PNT2
[PNT2] or "PCX0" (see FIG. 7) is sent to the "Y" input of the multiplier circuit MLT.

(D7): The sum of products calculation shown in FIG. 2 (C) is performed. The basic operation is the same as that described in (d3) above. In the multiplier circuit MLT, "WC0" shown in FIG.
Calculation corresponding to the product “WC0 × PCX0” with “PCX0” shown in the figure, that is, “WC × PC” in the equation (1) is performed. In the arithmetic logic circuit ALU, this calculation result “WC0 × PCX0” and the value “WA0 × PAX0 + WB0 × PBX” held in the accumulator ACC.
The sum is calculated with "0". The result of this calculation is "WA0 × PAX0 + WB0
“× PBX0 + WC0 × PCX0” is held in the accumulator ACC. That is, here, “WA × PA + WB × PB +” in the equation (1).
The calculation equivalent to WC x PC "is performed.

(D8): The data M [PNT1D] stored in the address "0771" of RAM1 designated by the address pointer PNT1D, that is, "WD0" (see FIG. 6) is the multiplier circuit MLT.
Sent to the "X" input of. Data M stored at address "0003" of RAM2 designated by address pointer PNT2
[PNT2], that is, "PDX0" (see FIG. 7) is sent to the "Y" input of the multiplier circuit MLT.

(D9): The sum of products calculation shown in FIG. 2 (C) is performed. The basic operation is the same as that described in (d3) above. In the multiplier circuit MLT, "WD0" shown in FIG.
Calculation corresponding to the product “WD0 × PDX0” with “PDX0” shown in the figure, that is, “WD × PD” in the equation (1) is performed. In the arithmetic logic circuit ALU, this calculation result “WD0 × PDX0” and the value held in the accumulator ACC “WA0 × PAX0 + WB0 × PBX”
The sum of 0 + WC0 x PCX0 "is calculated. The calculation result" WA0
“× PAX0 + WB0 × PBX0 + WC0 × PCX0 + WD0 × PDX0” is held in the accumulator ACC. That is, here, the calculation corresponding to “WA × PA + WB × PB + WC × PC + WD × PD” in the equation (1) is performed.

(D10): Point Q shown in FIG. 4 (A) as described above
The x coordinate of 0,0 is calculated. Accumulator A
The above calculation result stored in CC is the address pointer PN
It is stored in the address "0000" of RAM4 specified by T4.
This stored data M [PNT4] is "X0,0" in FIG.
It is expressed as

(D11): "1" is added to the value of the address pointer PNT4, and as a result, the pointer value becomes "0001".

(D12) to (d21): Next, point Q shown in FIG.
In order to obtain the y coordinate of 0,0, similar to the processing of (d1) to (d10), (d12) to (d2) shown in FIG. 2 (B).
The process of 1) is performed. Obtained calculation result “WA0 × PAY0 +
WB0 × PBY0 + WC0 × PCY0 + WD0 × PDY0 "is stored at the address" 0001 "of the RAM4 designated by the address pointer PNT4. This stored data M [PNT4] is" Y "in FIG.
It is expressed as 0,0 ".

(D22): "1" is added to the value of the address pointer PNT4, and as a result, the pointer value becomes "0002".

(D23): 8 is subtracted from the value of the address pointer PNT2, and as a result, the pointer value becomes "0000".

As described above, the calculation routine shown in FIG. 2 (A) ends, and the x coordinate “X0,0” of the point Q0,0 shown in FIG. 4 (A).
And the y-coordinate "Y0,0" is stored in the RAM 4 (see FIG. 9).

(E): Address pointers PNT1A, PNT1B, PNT1C and P
The number of steps Δn1 (the number of steps “16” corresponding to the number of divisions of the small curve C1 shown in FIG. 4A) latched by the latch circuit STP is added to the value of NT1D. Therefore, the pointer values of the address pointers PNT1A, PNT1B, PNT1C and PNT1D are "16", "273" (257 + 16) and "53", respectively.
It will be 0 "(514 + 16) and" 787 "(771 + 16).

(F): It is determined whether the value of the address pointer PNT1A is larger than "256".

Since the current value of the address pointer PNT1A is "0016", the processing shifts to the calculation routine (d), and the same processing as the processing described in (d) above is performed. As a result, the x coordinate “X0,1” and the y coordinate “Y0,1” of the point Q0,1 shown in FIG.
(See FIG. 9). The specific data is "X
0,1 "is" WA16 x PAX0 + WB16 x PBX0 + WC16 x PCX0 + WD16 x
PDX0 "and" Y0,1 "is" WA16 x PAY0 + WB16 x PBY0 + WC "
16 × PCY0 + WD16 × PDY0 ”(see FIGS. 6 and 7).

As described above, the value of the address pointer PNT1A is "2.
Similar processing is performed until it becomes larger than 56 ". As a result, the x coordinate" X0,0 "of points Q0,0 to Q0,16 shown in FIG.
~ "X0,16" and y coordinate "Y0,0" to "Y0,16" are sequentially RAM
It is stored in 4 (see FIG. 9).

(G): When the value of the address pointer PNT1A becomes larger than "256", the following processing is performed. 8 is added to the value of the address pointer PNT2, and as a result, the pointer value is "00.
08 "(see FIG. 7). 1 is added to the value of the address pointer PNT3, and as a result, the pointer value becomes" 0001 "(see FIG. 8).

As described above, the processing for the small curve C1 shown in FIGS. 3 and 4A is completed.

(H): It is judged whether or not the value of the address pointer PNT3 becomes "14". This value of "14" corresponds to the number of divisions of the curve shown in FIG. 3, and this value is appropriately selected for each curve.

Since the current value of the address pointer PNT3 is "0001", the processing shifts to the calculation routine (c), and the same processing as the processing described in (c) to (g) above is performed. As a result, the x-coordinates "X1,0" to "X1,32" of points Q1,0 to Q1,32 shown in FIG.
And the y-coordinates "Y1,0" to "Y1,32" are sequentially stored in the RAM 4 (see FIG. 9).

As described above, the value of the address pointer PNT3 is "1".
Similar processing is performed on the small curves C1 to C14 shown in FIG. 3 until the value becomes 4 ". The value of the address pointer PNT3 is" 14 ".
Then, it is determined that the processing for the small curves C1 to C14 shown in FIG. 3 has been completed.

As already mentioned, it is possible to select an appropriate number of divisions for each small curve according to its curvature, etc., and it is possible to speed up hardware calculation by selecting the optimal number of divisions for each small curve. Is. Generally, when the curvature is small,
An approximate curve can be represented accurately even with a small number of divisions.

Therefore, a method for obtaining the optimum number of divisions of the small curve will be described below.

First, the first method will be described with reference to FIG.

In FIG. 10, PA and PD are anchor points of the minor curve CV1, and PB and PC are control points of the minor curve CV1. Lab, Lbc, Lbc, Lcd, and Lda are the lengths of straight line segments connecting between PA and PB, between PB and PC, between PC and PD, and between PD and PA, respectively. Generally, the lengths of the line segments Lab and Lcd become longer as the curvature of the small curve CV1 is larger. Therefore, the optimum number of divisions “N” of the small curve CV1 can be obtained using the following evaluation formula.

ΔL = (Lab + Lbc + Lcd−Lda) +1 (3) N = 2 _{{log (a) ΔL}} (4) 2 ≦ a ≦ 8 (“a” is the bottom) The bottom “a” is the approximation accuracy of the curve. The number of divisions “N” shown in the equation (4) is the number of divisions in the data table (“256” in the example of FIG. 6).
Is chosen not to exceed.

FIG. 11 shows an example (when “a” = 3) of obtaining the number of divisions using the above evaluation formula. This is the control point PB
2 shows an example in which the y-coordinates of PC and PC are sequentially reduced, that is, the curvature of the small curve is sequentially reduced. The "step" shown in Fig. 11 is "25
6 / N ", which is the step number data Δn shown in FIG.
It is equivalent to 1, Δn2, ………. As shown in FIG. 11, when the above evaluation formula is used, it is understood that the smaller the curvature of the small curve, the smaller the number of divisions.

Although the values of the respective line segments are used as they are in the equation (3), a value obtained by squaring the lengths of these line segments may be used.

Next, the second method will be described with reference to FIGS. 12 and 13.

In FIG. 12, PA and PD are anchor points of the small curve CV2, and PB and PC are control points of the small curve CV2. S1 is an area of a quadrangle formed when the points PA, PB, PC and PD are sequentially connected by a straight line.
Generally, the area S1 of the quadrangle increases as the curvature of the small curve CV2 increases. Therefore, the optimum number of divisions “N” of the small curve CV2 can be obtained by the following evaluation formula.

N = 2 _{{log (a) S1}} (5) 2 ≦ a ≦ 8 (“a” is the bottom) The bottom “a” is appropriately selected according to the approximation accuracy of the curve, and ( The number of divisions “N” shown in equation 5) is the number of divisions in the data table (“256” in the example of FIG. 6)
Is chosen not to exceed.

In FIG. 13, PA and PD are anchor points of the small curve CV3, and PB and PC are control points of the small curve CV3. S2 is the area of a triangle formed when the points PA, PB and PD are sequentially connected by a straight line, and S3 is the area of a triangle formed when the points PA, PC and PD are sequentially connected by a straight line. is there.

Generally, the sum “S2 + S3” of the areas of the triangles S2 and S3 increases as the curvature of the small curve CV3 increases. Therefore, the optimum number of divisions “N” of the small curve CV3 can be obtained by the following evaluation formula.

N = 2 _{{log (a) (S2 + S3)}} (6) 2 ≦ a ≦ 8 (“a” is the bottom) The bottom “a” is appropriately selected according to the approximation accuracy of the curve, Further, the number of divisions “N” shown in the equation (6) is the number of divisions in the data table (“256” in the example of FIG. 6)
Is chosen not to exceed.

Although the values of the respective areas are used as they are in the expressions (5) and (6), a value obtained by squaring these areas may be used.

Next, the third method will be described with reference to FIG.

In FIG. 14, PA and PD are anchor points of the small curve CV0, and PB and PC are control points of the small curve CV0. D1 is the first division point on the small curve CV0,
Points other than anchor points PA and PD. Specifically, the values of WA to WD are calculated with t = 1/2 in the expressions (2a) to (2d), and these values are substituted into the expression (1). The small curve CV0 between the point PD and the first division point D1 is called the first division curve. D2 is the second dividing point on the small curve CV0 located between PD and D1, and PD
And points other than D1. Specifically, the values of WA to WD are calculated with t = 3/4 (= 1/2 + 1/4) in the equations (2a) to (2d), and these values are substituted into the equation (1). is there. The small curve CV0 between the point PD and the second division point D2 is called a second division curve. Similarly, t
The third on the small curve CV0 at = 7/8 (= 1/2 + 1/4 + 1/8)
The division point of D3 (not shown), t = 15/16 (= 1/2 + 1/4)
+ 1/8 + 1/16) D4 to the fourth dividing point on the small curve CV0
(Not shown), and so on, the division points Dn (not shown) on the small curve CV0 are sequentially determined. When the first to nth division points D1 to Dn are obtained in this way, it can be obtained at high speed by using the data table shown in FIG. L1 is the length of the line segment connecting point PA and point PD with a straight line,
L2 is the length of the line segment connecting the point PA and the first division point D1 with a straight line, L3 is the length of the line segment connecting the point PD and the first division point D1 with a straight line, and L4 is the first line segment. Is the length of a line segment connecting the dividing point D1 and the second dividing point D2 with a straight line, and L5 is the length of the line segment connecting the point PD and the second dividing point D2 with a straight line.

The operation for obtaining the number of divisions will be described below with reference to FIG.

After obtaining the first division point D1, the lengths of the line segments L1, L2, and L3 are obtained, and the following difference data is calculated.

ΔL1 = L2 + L3-L1 (7a) It is determined whether the first difference data ΔL1 satisfies the following conditions.

ΔL1 <ΔL0 (7b) where ΔL0 is a predetermined constant. When the first difference data ΔL1 satisfies the condition of the expression (7b), the number of divisions of the small curve CV0 in FIG. 14 becomes “1”. When the first difference data ΔL1 does not satisfy the condition of the expression (7b), after obtaining the second dividing point D2, the lengths of the line segments L4 and L5 are respectively obtained, and the following calculation is performed.

ΔL2 = L4 + L5-L3 (8a) It is determined whether the second difference data ΔL2 satisfies the following condition.

ΔL2 <ΔL0 (8b) When the second difference data ΔL2 satisfies the condition of expression (8b), the number of divisions of the small curve CV0 in FIG. 14 is “2”. When the second difference data ΔL2 does not satisfy the condition of the expression (8b), the division is performed in the same manner as the above method,
Sequential division points D3, D4, ..., Dn and difference data ΔL3, Δ
L4, ..., Calculate ΔLn. It is determined whether or not the nth difference data ΔLn satisfies the following conditions.

ΔLn <ΔL0 (9b) When the nth difference data ΔLn satisfies the condition of expression (9b), the number of divisions of the small curve CV0 in FIG. 14 is “2 ^n-1 ”.

In the fourth method described above, t = 1/2, t = 3/4 (= 1/2 + 1/4), t = 7/8 in the equations (2a) to (2d).
(= 1/2 + 1/4 + 1/8), t = 15/16 (= 1/2 + 1/4 + 1/8 + 1)
/ 16), ... We obtained each division point as, for example, t = 1
/ 2, t = 1/4 (= 1 / 2-1 / 4), t = 3/8 (= 1 / 2-1 / 4 + 1 /
8), t = 5/16 (= 1 / 2−1 / 4 + 1 / 8−1 / 16), ... In short, when obtaining the kth division point, subtraction is performed even if "1/2 ⁿ " is added to the value of "t" at the (k-1) th division point. Good.

[Effect] According to the present invention, the value of the parameter function of the curve formula is stored in advance, and the value of the curve formula is obtained using only the necessary number of parameter functions. Therefore, a large amount of data is generated each time the curve data is generated. The calculation processing is not required, and the time required to generate the outline font can be significantly reduced.

[Brief description of drawings]

1 to 14 are all for explaining an embodiment of the present invention, and FIG. 1 is a block diagram, FIG. 2 (A),
FIGS. 2 (B) and 2 (C) are flowcharts showing the operation of FIG. 1, FIGS. 3, 4 (A) and 4 (B) are explanatory diagrams of outline fonts, and FIG. FIG. 6 is a conceptual diagram of a data table, FIGS. 6 to 9 are address maps of each data stored in RAM, FIG. 10 is an explanatory diagram regarding the first method for obtaining the number of divisions of a small curve, and FIG. 11 is The figure which shows the specific example when the number of divisions is obtained by the first method, FIGS. 12 and 13 are explanatory diagrams regarding the second method of obtaining the number of divisions of the small curve, and FIG. 14 is the division of the small curve It is explanatory drawing regarding the 3rd method of calculating | requiring a number. PA, PD …… Anchor point PB, PC …… Control point

Claims (1)

(57) [Claims] 1. A curve is expressed based on a parametric expression using two anchor points and at least one control point, the curve is divided into a plurality of parts, and adjacent division points are straight lines. In the method of approximating the above curve, the values of the parameter functions of the above curve equations, which are determined respectively for the n parameter values of the above curve equation, are stored in advance, and the values are calculated according to the curve to be represented. For m (m <n) out of n parameter values, obtain the value of the above curve formula based on the value of the above parameter function, and connect a dividing point determined corresponding to the above m parameter values with a straight line. Curve approximation method characterized by.