JPH0271384A - Broken line approximating device for cubic bezier curve - Google Patents

Abstract

PURPOSE:To effectively and smoothly approximate a curve to a broken line without increasing the processing time and the quantity of data by changing the dividing frequency, i.e., the number of approximate curves in accordance with the curvature of the curve. CONSTITUTION:A cubic Bezier curve coordinate input means input a start point Q11, a 1st control point Q12, a 2nd control point Q13, and an end point Q14 of a cubic Bezier curve. An error arithmetic means calculates the errors of a middle point P of a segment Q11Q14 connecting both points Q11 and Q14 and a middle point R of a segment Q12Q13 connecting both points Q12 and Q13. An error discriminating means discriminates whether those calculated errors are larger than the prescribed value or not. If not, a broken line data generating means produces the segment Q11Q14 connecting the start and end points of the corresponding curve as the broken line data on a curve. While the curve is divided and the actions are repeated in case the errors are larger than the prescribed value.

Description

[Detailed Description of the Invention] [Summary] Regarding cubic heges and curve line approximation devices used, for example, in computer graphic display devices, figure printing devices, etc., curves can be approximated by broken lines without increasing processing time and data amount. A curve coordinate input means for inputting a starting point, a first control point, a second control point, and an end point of a cubic Bezier curve, and a line connecting the starting point and the ending point. error calculation means for calculating an error between the midpoint of the minute and the midpoint of the line segment connecting the first control point and the second control point; and determining whether the error is greater than or equal to a predetermined value. Do it! ′(
a difference determining means; a broken line data generating means for generating a line segment connecting the starting point and the ending point of the curve as a broken line of the curve when the error is less than a predetermined value; In the case of t, divide the cubic Hessoe curve, process each of the divided cubic Hessoe curves, and repeat the operations of the error calculation means and the error discrimination stage 1. and curve dividing means.

[Industrial application field]

The present invention is limited to two insulators7. - Taggraphic display device: This invention relates to a broken line approximation device for a cubic Hezier curve, which is used in graphic printing devices and the like.

[Conventional technology]

Generally, the cubic curve is, as shown in Figure 2,
Starting point QI +, ending point (,l, 4. Two control points Q,
2Q1. A curved line is represented by the four curved fixing seats 4g1. The point of the curve T of this cubic/G1-curve can be found at the node Ip by using these four curve definition coordinates (according to Tomio Yamaguchi's "21/Bib,-Tade Display"). Shape processing T science (2) pi (1-15
, Nikkan T-gyo, 1982). Zunai, CL, (starting point),
Q, 2 (control point), Ql:l (control point), Q,, (end point) Q) When four curve definition coordinates are given, divide the curve by the ratio ts: ]-13 using the following procedure.・You can find the points on curve 1.

(1) The point where Q, I, Ql2 is internally divided by the ratio ts+]-ts is defined as Q2□, and the point between Q, □, Q, , is defined as the ratio ts
: Let the point to be internally divided by lts be Q22, and Q, 1. The point that internally divides the interval between Q and , by the ratio tjLs:Its is defined as Q2. shall be.

(2) Furthermore, Q2. ．． 022 internally divided by the ratio Ls:14s is defined as Qll, and Q2. , Q2y is the ratio is
: ] Let the point internally divided by ts be Q, 2.

(3) Furthermore, -・degree, 0. , , ,0. :1
2 internally divided by the ratio L:, :1-1s is Q4. shall be.

At this time, Q4□ is Qz, CL2. Ql:1. Qx
6, it becomes a point on the curve that divides the given curve by the ratio ts:ILs.

At the same time, the original cubic Hezier curve expressed by the 4' curve defining coordinates of Qx, Q+□, Ql3, and Q14 is
4 of Q, ll, Q21, Q:ll, Q4+
a cubic Hezier curve represented by two curve-defining coordinates, and Q41. Q3□Q2. , Q, , is divided into two cubic curves represented by four curve-defining coordinates.

Conventionally, when approximating a cubic curve with a straight line, the above-mentioned division is repeated i-1 times a predetermined number of times, and a straight line is created by sequentially connecting the points on the obtained curve. It was.

[Problem to be solved by the invention] There is a relationship proportional to an exponential function between the number of divisions and the number of points of the music vAJ-1 to be found. Therefore, in order to express a curve with a large curvature with a broken line with high precision, needs to be increased by the same number as above, which results in an increase in processing time and data ψ, but also excessive accuracy for curves with small curvature, and vice versa There is a problem that the quality of the curve deteriorates.

Therefore, an object of the present invention is to approximate a cubic Hessier curve with a polygonal line with high precision without increasing processing time or data amount.

[Failure to solve the problem]

” - Alternative means for solving the above problem 4.4 are shown in FIG. That is, the cubic Hezier curve coordinate input means inputs the starting point Ql, the first control point Q, 2, the second control point Q19, and the end point 0, 4 of the cubic Hezier curve, and the error calculation means inputs the starting point Qll. Calculate the error between the midpoint P of the line segment Q, , Q14 that connects and the end point Q14, and the midpoint R of the line segment Q1□QIff that connects the first control point Q, 2 and the second control point QI3. The error determining means determines whether the error is within a predetermined value or not. As a result, if the error is less than a predetermined value, the line segment Q connecting the starting point and ending point of the curve,
Q, , is generated as curved line data. On the other hand, if the error is 1- or more than the predetermined value, the curve dividing means divides the cubic Hezi curve and divides the divided cubic Hezi y-+fb
In 'rr'R, the operations of the error calculating means and the error determining means are repeated.

C action] - By using the above-mentioned means, as shown in Figure 3, the error C is the difference between the line segment Q, , Q,, which was drawn near A, and the actual curve X in that section. Use the maximum distance of At this time, the distance between the curve × and the line segment Q + + (, l l 4 is iit:R
Since it is difficult to do so, we use the convex hull (the generated curve The midpoint P of the connecting line segment and the two control points Q1□,
The distance from the midpoint R of the line segment connecting Q13 is used as the error. The distance between the point P and the point 0 is longer than the maximum distance between the line segment QllQ14 and the curve X due to the convex closure of the cubic Hezi pentagonal curve. If the distance (error) between point P and point Q is large, 3
Next ＼Je curve is divided. Therefore, the number of divisions changes according to changes in the shape of the curve X. As a result, the number of divisions increases for a curve with a large curvature, but the number of divisions decreases for a curve with a small curvature.

〔Example〕

FIG. 4 shows a graphic display device to which the curved line approximation device according to the present invention is applied. In the fourth figure, 1 is a figure data input device, 2 is a coordinate transformation device, and shi is a curve generator 1.
4 is a drawing processing device, and 5 is a display control device that controls the CRT 6. The curved line approximation device according to the present invention is used as a curve generation device. Further, the present invention is also applied to a graphic printing device.

In this case, the display control device '#, 5 and CR of FIG.
T6 is a print control device and a printing device.

FIG. 5 is a block circuit diagram showing an embodiment of the curved line approximation device according to the present invention. In FIG. 5, a data input circuit 501 inputs and processes four curve definition coordinates 011 to Q14 representing a cubic Hessian curve.

The curve division circuit 502 divides the curve of the human coordinates from the data input circuit 501 or the curve data returned from the error determination circuit 504 by ts:]-ts. The error calculation circuit 503 calculates the error between the input coordinates from the data input circuit 501 or the divided curve and the line segment that approximates it. The error determination circuit 504 compares the error calculated by the error calculation circuit 503 with a preset error value, and if the error is within the error value, passes the data to the data output circuit 505. If it is not within the dark value, the data is returned to the curve dividing circuit 502 again.

The data output circuit 505 performs a process of outputting data of a curve approximated by a broken line.

Each circuit 502 and 503 will be described in detail below.

The curve dividing circuit 502 divides the curve at a ratio Ls:1-ts based on the properties of a cubic Hezier curve, for example.
How to find the point (Reference: Written by LllLJ Tomio °“
Shape processing engineering using computer display (2)
" FI Fl T', lA pQ~15, Showa 5
7) It is used to divide a curve. As shown in FIG. 6, the curve dividing circuit 502 includes registers 601 to 6.
04, coordinate division circuits 605-6Q, registers 611-6
It is composed of 18 G. That is, as shown in Figure 2, if Ql, ~Q+4 are given as coordinates representing a curve, these coordinates are stored in the register as 01~60.
It is stored in 4. Locus 4ffQ++ and Qla are stored as they are in registers 611 and 618. Coordinate division circuit (
)05 generates the coordinate Q21 obtained by dividing the coordinates Q, , , Q, 2 by the ratio Ls+I−Ls and stores it in the register 612, and the coordinate dividing circuit 606 generates the coordinate Q21, which is obtained by dividing the coordinates Q, . The ratio t between Q and 1
The coordinate division circuit 607 generates the coordinate Q22 divided by s:1-1s, and the coordinate division circuit 607 divides the coordinates Q and 34Q14 into the ratio ts:
Generates the coordinate Q21 divided by Lts and stores it in the register 617.
Store in. Further, the coordinate dividing circuit 608 has the coordinate Q2.
, the coordinate Q obtained by dividing the space between Q2□ by the ratio ts:ILs,
1 is generated and stored in the register 613, and the coordinate division circuit 6
09, the coordinates Q, 2 obtained by dividing the coordinates Q2□ and Cl23 by the ratio Ls: 14s are output 11 and stored in the register 616ζ. Furthermore, the coordinate division 111J i step 6Q is the coordinate Q
3+, (11□ divided by the ratio ts:l-1s) generates coordinates Q, 1 and stores them in registers 614 and 615.

At this time, as shown in FIG. 2, the coordinates 4. Ba, QQ+z
+ Q13+ Curve ratio ts・ given by Qla
The coordinates are on the curve divided by 1-ts, and at the same time,
At this coordinate, the curve is divided into two curves (l11.Q2Q
, ,Q, , and Q, ,Q3□, Q2:1 ,0.1
−).

Note that the registers 601 to 6 in the 6th IN'l 4
Switching of the data input circuit 501 or the error determination circuit 504 for the input coordinates Q, .about.Q of 04 is performed by a switch (2) not shown.

Each of the coordinate division circuits 605 to 6Q in FIG. 6 is constituted by, for example, the mounting devices 701, 702, IJII'lJ, and the device 703 as shown in FIG. 7G.

The midpoint R of the line segment Q12Q11 in FIG. 3 and the line segment Q,,Q,.

When calculating the error (distance) from the midpoint P, if the coordinates of It people in 2 are (XI, Yl), (X2, Yz), the calculation of 17 (i distance) between the two points is r( It is expressed as shown in (1).

Distance -1(X, -X,)24-ir, Y2)Z
I I/2 (1) Equation (1) requires multiplication and division, which is disadvantageous when high-speed processing is required. In order to process this using only addition and subtraction, the coordinate difference between the two points is used as shown in equation (2).

Distance-I XI-X2 1 t l Yl'-Y
z:...(2) By doing this, the calculation can be performed simply and at high speed, and the distance calculated by equation (2) is the same as the distance calculated by equation (1). is always longer than 1. Therefore, when comparing the obtained distance with the error and making a judgment, the accuracy of the approximated curve does not become low, so the approximated curve is smooth.

Details of the error calculation circuit 503 shown in FIG. 5, which is used to calculate the error shown in (2), are shown in FIG. In Figure 8, 801, 802, 803, 804, 808 +:+
Registers 805 and 806 are midpoint calculation circuits that calculate the midpoint of the two input coordinates, and 807 is a midpoint calculation circuit that calculates the midpoint of the two input coordinates, and 807 calculates the AI formula shown in equation (2) from the two manually entered midpoints R and P. This is a coordinate difference performance 'Q circuit that calculates the coordinate error determined by For example, as shown in FIG. 9, the midpoint calculation circuits 805 and 804i include an adder 901 and a divider
902.

First, the data input circuit 401 outputs 1'
Qz, Q, +2, Qll, required for 4-difference calculation
Q14 are stored in registers 801-804, respectively. Gogode, Q, 1 is the starting point, Q14 is the ending point, Q...
Q and l are control points (I2). As a result, the midpoint calculation circuit 805 has Q, □ and (
1, 3 are input, and the control points Q, 2 . The midpoint R of the line segment connecting Q and 3 is found.

Further, Qll and Q14 are manually input to the midpoint calculation circuit 806, and the midpoint P of the line segment connecting the starting point Ql+ and the ending point Q14 is obtained as an output of the midpoint calculation circuit 806.

Furthermore, the coordinate difference calculation circuit 807 includes a midpoint calculation circuit 8
Midpoint R found by 05 and midpoint calculation rgJ path 806
The midpoint P found by is input, and the difference between these two midpoints is determined as the output of the coordinate difference calculation circuit 807. This calculation result is stored in the register 8084 as the error e.
．． : Stored.

In addition, the manual coordinates Q1. of the registers 801 to 804 shown in FIG. Switching to the data input circuit 501 or the curve dividing circuit 502 in Q14 is performed by a switch not shown.

[The error determination circuit 504 in FIG. 5 compares the above error e with a threshold value, and if the error is smaller than the threshold value, the line segment Q,...
Q, , is output from the data output section 805.

Also, +i! ′i difference is larger than the dark value, the curve Q
llQ21 + Qtl + Q41 is again input to the curve dividing circuit 502 and divided. At this time, another curve, Q41. Q3□, Q23. Q...
The curve Q, is temporarily held inside the error determination circuit 504.

Q21, Q31. When Qa+(IJ division 'l>(P near), it is input to the curve division circuit 502 and processed.

Through this series of processing, the third-order Hezier curve is output as a polygonal line.

The above-described embodiment can also be implemented as a program using a general-purpose microcomputer. For example, as shown in FIG.
In step Q02, the distance (error) e is calculated based on the statement (2), and in step Q03, it is determined whether the distance 1%le is less than the darkness value. As a result, if the distance fd4e≦darkness value, the line segments Q, , Q, are output as broken line data in step Q06, and if the distance (・〉threshold value), the line segments Q04, Q0 are output.
Proceed to step 5.

In step Q04, the curve is divided by forming cubic Heger curve data, and in step Q05, it is divided into eight pieces Q1□, Q2. ,..., Q14 is stored in memory. Then, return to step 1o02c.

Then, the curve valve 1; +1 is repeated -4 degrees. Also, the curve C knobs Q02 to Q06 are repeated until the polygonal line approximation of the curve is completed in steps QF+7 and HIO2.

(Effect of the invention) As explained above, according to the present invention, the curvature of the curve is
Since the number of divisions, that is, the number of approximate broken lines changes with ,
It is possible to efficiently and smoothly approximate a curved line without increasing the processing time or the amount of data.

[Brief explanation of the drawing]

Figure 1 is a diagram illustrating the basic configuration of the present invention; Figure 2 is a diagram explaining the division of the following Heje curve; Figure 3 is a diagram explaining the details of the present invention; Figure 4 is a diagram explaining the invention Figure 5 shows an embodiment of the curved line approximation device according to the present invention in 21 x 4 blocks. 11. Figure 6 is a circuit diagram showing the curve division circuit in Figure 5. Figure 7 is a circuit diagram showing an example of the coordinate division circuit in Figure 6. Figure 8 is a circuit diagram showing an example of the coordinate division circuit in Figure 5. 1 is a circuit diagram showing the error calculation circuit, FIG. 9 is a circuit diagram showing the central calculation circuit of FIG. 8, and FIG. Q is a flowchart showing another embodiment. 501...Data input circuit, 502...Curve division circuit, 503...Error calculation circuit, 504...Error discrimination circuit, 505...Data output circuit.

Claims (1)

[Claims] 1. Cubic Bezier curve coordinate input for inputting the starting point (Q_1_1), first control point (Q_1_2), second control point (Q_1_3), and end point (Q_1_4) of the cubic Bezier curve. means, the starting point (Q_1_1) and the ending point (Q_1
The midpoint of the line segment (Q_1_1Q_1_4) connecting _4) with
P) and a line segment (Q_1_2Q_1) connecting the first control point (Q_1_2) and the second control point (Q_1_3).
_3) An error calculating means for calculating the error between the midpoint (R), an error determining means for determining whether the error is greater than or equal to a predetermined value, and an error determining means for determining whether the error is greater than or equal to a predetermined value; broken line data generating means for generating a line segment (Q_1_1Q_1_4) connecting the starting point and the ending point as broken line data of the curve; and dividing the cubic Bezier curve when the error is greater than a predetermined value, and dividing each of the divided three A broken line approximation device for a cubic Bezier curve, comprising curve dividing means for repeating the operations of the error calculating means and the error determining means for the next Bezier curve. 2. The broken line approximation device for a cubic Bezier curve according to claim 1, wherein the error calculation means calculates a distance between the two midpoints (P, R) as the error. 3. The broken line approximation device for a cubic Bezier curve according to claim 2, wherein the error calculation means calculates a coordinate difference as a distance between the two midpoints (P, R).