JP2677273B2 - A polygonal line approximation device for cubic Bezier curves - Google Patents

Description

The present invention relates to a polygonal line approximation device for a cubic Bezier curve used in, for example, a computer graphic display device, a graphic printing device, etc., and approximates a curve with a polygonal line without increasing processing time and data amount. For the purpose of: a cubic Bezier curve coordinate input means for inputting a start point, a first control point, a second control point, and an end point of a cubic Bezier curve, and a line segment connecting the start point and the end point A point, and a midpoint of a line segment connecting the first control point and the second control point,
Error calculating means for calculating the coordinate difference of the curve, an error determining means for determining whether the coordinate difference is a predetermined value or more, and a line segment connecting the start point and the end point of the curve when the coordinate difference is less than the predetermined value. And polygonal line data generating means for generating the curved line data of the curve, and dividing the cubic Bezier curve when the coordinate difference is a predetermined value or more, and calculating the error for each of the divided cubic Bezier curves. Means and curve dividing means for repeating the operation of the error determining means.

[Industrial applications]

The present invention relates to a polygonal line approximation device for a cubic Bezier curve used in, for example, a computer graphic display device or a graphic printing device.

[Conventional technology]

Generally, the 3-pair Bezier curve is as shown in FIG.
A curve is represented by four curve defining coordinates of a start point Q ₁₁ , an end point Q ₁₄ , and two control points Q ₁₂ , Q ₁₃ . The points on the curve of this cubic Bezier curve can be easily obtained by using these four curve definition coordinates (Fujio Yamaguchi "Shape processing engineering by computer display (2)" p10
~ 15, Nikkan Kogyo, Showa 57). That is, Q ₁₁ (starting point), Q
Given the four curve definition coordinates of ₁₂ (control point), Q ₁₃ (control point), and Q ₁₄ (end point), the points on the curve that divide the curve by the ratio ts: 1-ts are calculated by the following procedure. You can ask.

(1) Q ₂₁ is the point that internally divides between Q ₁₁ and Q ₁₂ with the ratio ts: 1−ts, and Q ₂₂ is the point that internally divides between Q ₁₂ Q and ₁₃ with the ratio ts: 1−ts.
The point where Q ₁₃ and Q ₁₄ are internally divided by the ratio ts: 1−ts is Q ₂₃ .

(2) Furthermore, the point that internally divides Q ₂₁ and Q ₂₂ with the ratio ts: 1-ts is Q _31, and the point that internally divides Q ₂₂ and Q ₂₃ with the ratio ts: 1-ts is Q.
₃₂ .

(3) Once again, Q ₄₁ is the point that internally divides between Q _{31 and} Q ₃₂ with the ratio ts: 1−ts.

At this time, Q ₄₁ is a point on the curve dividing the curve given by Q ₁₁ , Q ₁₂ , Q ₁₃ , and Q ₁₄ by the ratio ts: 1−ts.

At the same _{time, Q 11, Q 12, Q} 13, the original cubic Bezier curve represented by four curves define the coordinates of Q _{14 is, Q 11, Q 12, Q} 13, Q 14
It is divided into two curves, a cubic Bezier curve represented by the four curve defining coordinates of the above and a cubic Bezier curve represented by the four curve defining coordinates of Q ₄₁ , Q ₃₂ , Q ₂₃ , and Q ₁₄ .

Conventionally, when approximating a cubic Bezier curve with a straight line sequence,
The division by the above-described method is repeated a predetermined number of times to form a straight line string in which the points on the obtained curve are connected in order.

[Problems to be solved by the invention]

The number of divisions and the number of points on the curve to be obtained have a relationship proportional to the exponential function, and therefore, in order to represent a curve with a large curvature with a polygonal line with high accuracy, it is necessary to increase the number of times. As a result, the processing time and the amount of data increase, but the accuracy becomes excessive for a curve with a small curvature.
In the opposite case, there is a problem that the quality of the approximated curve of the curve having a large curvature deteriorates.

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

[Means for solving the problem]

The means for solving the above-mentioned problem is shown in FIG. That is, the cubic Bezier curve coordinate input means uses the cubic Bezier curve start point Q ₁₁ , the first control point Q ₁₂ , and the second control point Q.
₁₃ and the end point Q ₁₄ are input, and the error calculating means uses the middle point P of the line segment ▲ ▼ connecting the start point Q ₁₁ and the end point ₁₄ and the first point
Line segment connecting the control point Q ₁₂ and the second control point Q ₁₃ of ▲
The difference between the midpoint R and the coordinate difference is calculated, and the error determining means determines whether or not the difference between the coordinates is equal to or greater than a predetermined value. As a result, when the coordinate difference is less than the predetermined value, the polygonal line data generating means generates the line segment ▲ ▼ connecting the start point and the end point of the curve as the polygonal line data of the curve. On the other hand, when the coordinate difference is equal to or larger than the predetermined value, the curve dividing means divides the quadratic Bezier curve, and the operations of the error calculating means and the error determining means are repeated with respect to the divided cubic Bezier curves. Is.

(Operation)

According to the above-mentioned means, as shown in FIG. 3, the maximum distance between the approximated line segment () and the actual curve X in the section is used as the error. At this time, the curve X
Since it is difficult to calculate the distance between the line segment and the line ▲ ▼, use the convex hull (the generated curve X exists inside the four curve definition coordinates), which is the property of a cubic Bezier curve. Then, the coordinate difference between the midpoint P of the line segment connecting the start point Q ₁₁ and the end point Q ₁₄ of the Bezier curve and the midpoint R of the line segment connecting the two control points Q ₁₂ and Q ₁₃ is used. The distance between the point P and the point R is longer than the maximum distance between the line segment ▼ and the curve X due to the convex hull of the cubic Bezier curve. When the coordinate difference between the point P and the point R is large, the cubic Bezier curve is divided. Therefore, the number of divisions changes as the shape of the curve X changes. As a result, the number of divisions increases for a curve having a large curvature, but the number of divisions decreases for a curve having 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 FIG. 4, 1 is a graphic data input device, 2 is a coordinate conversion device, 3 is a curve generation device, 4
Is a drawing processing device, and 5 is a display control device for controlling the CRT 6. The curved line approximation apparatus according to the present invention is a curve generation apparatus 3
Used as The present invention is also applied to a graphic printing device. In this case, the display controller 5 and the CRT 6 shown in FIG. 4 are the print controller and the printer.

FIG. 5 is a block circuit diagram showing an embodiment of a curved line approximation apparatus according to the present invention. In Figure 5, the data input circuit 501 inputs handle four curves defined coordinate Q ₁₁ to Q ₁₄ representing the cubic Bezier curve. The curve division circuit 502 divides the curve of the input coordinates from the data input circuit 501 or the curve of the curve data returned from the error determination circuit 504 by ts: 1-ts. The error calculation circuit 503 calculates the coordinate difference between the input coordinate from the data input circuit 501 or the divided curve and the line segment approximating it. The error determining circuit 504 compares the coordinate difference calculated by the error calculating circuit 503 with a preset coordinate difference threshold value, and when the coordinate difference is within the threshold value, passes the data to the data output circuit 505. If it is not within the threshold value, the data is returned to the curve dividing circuit 502 again. The data output circuit 505 performs a process of outputting the curve-approximated curve data.

 Hereinafter, each of the circuits 502 and 503 will be described in detail.

The curve division circuit 502 obtains a point on the curve that divides the curve at the ratio ts: 1-ts from the property of a cubic Bezier curve (reference: Fujio Yamaguchi, "Shape processing engineering by computer display (2 ) ”Nikkan Kogyo p10〜1
(5, Showa 57) is used to divide the curve. As shown in FIG. 6, the curve division circuit 502 includes registers 601-
604, coordinate division circuits 605 to 610, and registers 611 to 618. That is, as shown in FIG. 2, if Q _{11 to} Q ₁₄ are given as coordinates representing a curve, these coordinates are stored in the registers 601 to 604. Coordinates Q ₁₁ ,, Q
₁₄ is stored in the registers 611 and 618 as it is. The coordinate division circuit 605 divides the coordinates Q ₁₁ and Q ₁₂ with the ratio ts: 1-ts.
Generates Q ₂₁ and stores it in register 612, and coordinate division circuit 606
The ratio ts between coordinates Q _12, Q ₁₃ are: the coordinates Q ₂₂ divided occurred in 1-ts, coordinates dividing circuit 607 ratio between coordinates _{Q 13, Q 14 ts: 1} -ts
The coordinate Q ₂₃ divided by is generated and stored in the register 617.
Further, the coordinate division circuit 608 calculates the ratio ts: 1−ts between the coordinates Q ₂₁ and Q _22.
The coordinate Q ₃₁ divided by is generated and stored in the register 613, and the coordinate dividing circuit 609 generates the coordinate Q ₃₂ which is obtained by dividing the sitting Q ₂₂ and Q ₂₃ by the ratio ts: 1−ts and stored in the register 616. To do. further,
The coordinate dividing circuit 610 generates the coordinate Q ₄₁ by dividing the coordinates Q ₃₁ and Q ₃₂ by the ratio ts: 1−ts and stores them in the registers 614 and 615. At this time, as shown in FIG. 2, the coordinate Q ₄₁ is Q ₁₁ , Q ₁₂ , Q ₁₃ , Q ₁₄
It becomes a coordinate on the curve divided by the curve ratio ts: 1-ts given by, and at the same time, the curve has two curves (Q ₁₁ , Q ₂₁ , Q ₃₁ , Q ₄₁ and Q ₄₁ , Q ₃₂ , It is divided into Q ₂₃ and Q ₁₄ ).

The input coordinates Q of the registers 601 to 604 in FIG.
_{11 to} Q ₁₄ data input circuit 501 or error determination circuit 504
The switch to is performed by a switch (not shown).

The coordinate division circuits 605 to 610 shown in FIG.
As shown in the figure, it is composed of multipliers 701 and 702 and an adder 703.

Midpoint R and line segment of line segment ▲ ▼ in Fig. 3
▼ When calculating the distance from the midpoint P, assuming that the coordinates of the two midpoints are (X ₁ , Y ₁ ), (X ₂ , Y ₂ ), the distance between the two points can be calculated by the equation (1). It is expressed as shown in.

Distance = In ^{_{{(X 1 -X 2) 2}} + (Y 1 -Y 2) 2} 1/2 ... (1) Equation (1), since it requires multiplication and division, if required high-speed processing Is at a disadvantage. In order to perform this processing only by addition and subtraction, the coordinate difference between two points is used as shown in equation (2).

Coordinate difference = | X ₁ −X ₂ | + | Y ₁ −Y ₂ |… (2) By doing this, the operation can be executed simply and at high speed, and it is calculated by the equation (2). The obtained coordinate difference is always longer than the distance obtained by the equation (1). For this reason, when the obtained coordinate difference is compared with the coordinate difference threshold value for determination, the accuracy of the approximated curve does not decrease, and the approximated curve is smooth.

Details of the error calculation circuit 503 of FIG. 5 for calculating the coordinate difference shown in Expression (2) are shown in FIG. In FIG. 8, 801, 802, 803, 804, 808 are registers, 805, 806 are midpoint arithmetic circuits for obtaining the midpoints of the two input coordinates, and 807 is the calculation formula shown in the equation (2) from the two midpoint coordinates R, P input. It is a coordinate difference calculation circuit for calculating the coordinate difference obtained by The midpoint arithmetic circuits 805 and 806 are composed of an adder 901 and a divider 902, for example, as shown in FIG.

First, Q ₁₁ , Q ₁₂ , Q ₁₃ , and Q ₁₄ necessary for coordinate difference calculation output from the data input circuit 50 are respectively registered in the registers 801-
Stored in 804. Where Q ₁₁ is the start point, Q ₁₄ is the end point,
Q ₁₂ and Q ₁₃ are control points. As a result, Q ₁₂ and Q ₁₃ are input to the midpoint arithmetic circuit 805, and the midpoint R of the line segment connecting the control points Q ₁₂ and Q ₁₃ is obtained using the midpoint arithmetic circuit 805 as an output. Further, Q ₁₁ and Q ₁₄ are input to the midpoint arithmetic circuit 806, and the midpoint P of the line segment connecting the start point Q ₁₁ and the end point Q ₁₄ is obtained as the output of the midpoint arithmetic circuit 806. Furthermore, the sitting difference calculation circuit 80
The midpoint R obtained by the midpoint arithmetic circuit 805 and the midpoint P obtained by the midpoint arithmetic circuit 806 are input to 7 and the difference between the two midpoints is output as the output of the coordinate difference arithmetic circuit 807. Is required. The calculation result is stored in the register 808 as the error e.

The input coordinates Q of the registers 801 to 804 in FIG.
_{11 to} Q ₁₄ data input circuit 501 or curve division circuit 502
The switch to is performed by a switch (not shown).

The above-mentioned error e is compared with the threshold value by the error determination circuit 504 in FIG. 5, and if the coordinate difference is smaller than the threshold value, the line segment ▲
▼ is output from the data output unit 806. If the coordinate difference is larger than the threshold value, the curves Q ₁₁ , Q ₂₁ , Q
₃₁ and Q ₄₁ are again input to the curve division circuit 502 and divided. At this time, another curve Q ₄₁ , Q ₃₂ , Q ₂₃ , Q ₁₄
Are temporarily stored in the error determination circuit 504, and when the curves Q ₁₁ , Q ₂₁ , Q ₃₁ , and Q ₄₁ are divided, they are input to the curve division circuit 502 and processed.

Through this series of processing, the cubic Bezier curve is approximated to a broken line and output.

The above-described embodiment can be realized as a program using a general-purpose microcomputer. For example, as shown in FIG. 10, in step 1001, cubic Bezier curve data Q _{11 to} Q ₁₄ are input, in step 1002 the coordinate difference e is calculated based on the above equation (2), and in step 1003
At, it is determined whether the coordinate difference e is less than or equal to the threshold value. As a result,
If the coordinate difference e ≦ threshold value, a line segment ▲ ▼ is output as broken line data in step 1006. If the coordinate difference e> threshold value, the process proceeds to steps 1004 and 1005.

In step 1004, the curve is divided by generating cubic Bezier curve data.
Store Q ₁₁ , Q ₂₁ , ..., Q ₁₄ in the memory. And step
Return to 1002.

That is, curve division is repeated. Step 100
At steps 1002 to 1008, steps 1002 to 1006 are repeated until all polygonal line approximations of the curve are completed.

〔The invention's effect〕

As described above, according to the present invention, the number of divisions, that is, the number of approximated polygonal lines changes according to the curvature of the curved line, so that the curved line can be approximated efficiently and smoothly without increasing the processing time and the amount of data. be able to.

[Brief description of the drawings]

FIG. 1 is a diagram showing a basic configuration of the present invention, FIG. 2 is a diagram illustrating division of a cubic Bezier curve, FIG. 3 is a diagram illustrating an operation of the present invention, and FIG. 4 is a curve relating to the present invention. FIG. 5 is a block circuit diagram showing a figure display device to which the broken line approximation device of FIG. 5 is applied. FIG. 5 is a block circuit diagram showing an embodiment of a curved line approximation device according to the present invention. FIG. 7 is a block diagram showing a circuit, FIG. 7 is a circuit diagram showing an example of the coordinate division circuit shown in FIG. 6, FIG. 8 is a circuit diagram showing an error calculation circuit shown in FIG. 5, and FIG. 9 is a central calculation shown in FIG. FIG. 10 is a circuit diagram showing a circuit, and FIG. 10 is a flow chart showing another embodiment. 501 ... Data input circuit, 502 ... Curve division circuit, 503 ... Error calculation circuit, 504 ... Error determination circuit, 505 ... Data output circuit.

Claims (1)

(57) [Claims] 1. Cubic Bezier curve coordinates for inputting a starting point (Q ₁₁ ), a first control point (Q ₁₂ ), a second control point (Q ₁₃ ), and an end point (Q ₁₄ ) of a cubic Bezier curve. A line segment connecting the input means and the start point (Q ₁₁ ) and the end point (Q ₁₄ ) ▲
▼ Midpoint (P) and the first control point (Q ₁₂ )
And the line segment connecting the second control point (Q ₁₃ ) (▲
▼) middle point (R), an error calculating means for calculating a coordinate difference, an error determining means for determining whether the coordinate difference is a predetermined value or more, and an error determining means for determining whether the coordinate difference is less than the predetermined value. A polygonal line data generating means for generating a line segment (▼) connecting the start point and the end point of the curve as polygonal line data of the curve, and dividing the cubic Bezier curve when the coordinate difference is a predetermined value or more, and dividing A polygonal line approximation device for a cubic Bezier curve, comprising: a curve dividing unit that repeats the operations of the error calculating unit and the error determining unit for each cubic Bezier curve.