JP2002117411A - Curve drawing method, curve drawing device, and storage medium with curve drawing program stored therein - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a curve drawing method and curve drawing device capable of precisely performing the segment approximation of a curve with an arithmetic quantity smaller than in the past to performing the drawing, and a storage medium with curve drawing program stored therein. SOLUTION: This method comprises acquiring the control point line of a Bezier curve, segment-approximating the Bezier curve by use of the acquired control point line, and performing the drawing of the approximated polygonal line data. In the segment approximation, whether subdivision is performed or not is evaluated, and the curve concerned is subdivided only when the necessity of subdivision is evaluated. The segment approximation is recursively executed. The evaluation uses the chord length estimated value of the Bezier curve as evaluation reference.

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

[0001] 1. Field of the Invention [0002] The present invention relates to a curve drawing method, a curve drawing device, and a recording medium storing a curve drawing program used in computer graphics, a printing device, and the like.

[0002]

2. Description of the Related Art Outline font design and C
In the field of AD (Computer Aided Design) and the like, it is required that a curve shape desired by a user or a font creator can be faithfully and easily expressed. In other words, it is preferable that the expressed curve and the input data for defining the curve are intuitively close curves. For example, TrueType font (Mi
Microsoft), PostScript (Ado)
Be) uses a Bezier curve to express the curved portion. Also in the CAD system,
Bezier curves are widely used.

FIG. 8 shows an example of a cubic Bezier curve, FIG. 9 shows a polygonal approximation method of the Bezier curve using the de Castello division method, and FIG. 10 shows another cubic Bezier curve. FIG. 11 is a diagram showing another example of a cubic Bezier curve, FIG. 12 is a diagram showing an example of an evaluation function used in a conventional subdivision evaluation step of a Bezier curve, and FIG. 13 is a conventional Bezier curve. FIG. 10 is a diagram showing an example of an evaluation function used in the subdivision evaluation step of FIG.

Here, a cubic Bezier curve will be described. 4 vertices P ₀ in FIG. 8, P _1, P and line consisting of ₂ and P _3, the end point P ₀ of the polygonal line _{P 0, P 1, P 2} , P 3
And the curve P (t) leading to the end point P ₃ is shown from. Among these, the curve P (t) is a cubic Bezier curve. The position vector of each point on the Bezier curve P (t) is expressed by the following equation (1) using the position vectors of the constituent points of the polygonal lines P ₀ , P ₁ , P ₂ , and P ₃ .

P (t) = (1−t) ³ P ₀ +3 (1−t) ² tP ₁ +3 (1−t) t ² P ₂ + t ³ P ₃ (1) (0 ≦ t ≦ 1) When the value of t is changed from 0 to 1, the position vector P (t) given by the above equation (1) becomes from the point P ₀ to the point P ₃
Move up to. That is, P ₀ is the start point of the Bezier curve P (t), and P ₃ is the end point of the Bezier curve P (t).
As is clear from the above equation (1), the Bezier curve P (t)
Is uniquely determined by the position vectors of the constituent points of the polygonal lines P ₀ , P ₁ , P ₂ , and P ₃ . Lines P ₀ , P ₁ , P ₂ , P ₃
Is called a control point sequence of the Bezier curve P (t), and each of the constituent points P _{0 to} P ₃ is called a control point. The straight lines P ₀ and P ₃ connecting the start point P ₀ and the end point P _{3 of} the Bezier curve P (t) are called a baseline.

In a CAD system or the like, a user specifies a position vector of a control point using a pointing device such as a mouse, calculates a Bezier curve corresponding to the control point sequence on the computer side, and displays the Bezier curve of the Bezier curve on a display. It is common practice to perform drawing.

Here, the shape of the Bezier curve substantially follows the polygonal shape of the control point sequence input by the user. The directions of the tangent vectors at the start point and end point of the Bezier curve are determined by the first side and the last side of the polygonal line of the control point sequence (side P in the example of FIG. 8).
_0, respectively coincide exactly with the direction of P ₁ and side P _2, P _3). Therefore, the user can input the control point sequence while predicting the shape of the completed Bezier curve to some extent,
A Bezier curve having a desired shape can be easily obtained. When the Bezier curve described above is displayed on a display of a computer or printed using a printing device, the Bezier curve is generally displayed and printed by approximating the Bezier curve with a polygonal line.

[0008] Here, several methods have been disclosed as a method of approximating a Bezier curve with a polygonal line. Particularly commonly known is de Castello (de Ca
This is a polygonal line approximation method that uses a Bezier curve division method developed by Stelijau). In this kind of broken line approximation method, a Bezier curve is divided into a plurality of Bezier curves, and, for example, each base line corresponding to each of the divided Bezier curves is connected in order, thereby performing a broken line approximation of the original Bezier curve.

In this case, if each of the divided Bezier curves substantially corresponds to each of the corresponding baselines, it can be said that each of the baselines sufficiently approximates the original Bezier curve. However, if each baseline contains a large gap between it and the original Bezier curve, then the polygonal line that such a baseline would make is probably far from the original Bezier curve. turn into. On the other hand, if the Bezier curve is divided too finely, there is a problem that the calculation time until the approximation is completed becomes enormous.

Therefore, in this kind of technique of dividing a Bezier curve to obtain an approximate polygonal line, a procedure of recursively repeating a polygonal line approximation process including an evaluation step and a division step is adopted. Hereinafter, the procedure will be described with reference to FIGS.
This will be described with reference to FIG.

First, a control point sequence P as shown in FIG.
Bezier curve P defined by ₀ , P ₁ , P ₂ , P ₃
Assuming that (t) is given, an error between this Bezier curve P (t) and the corresponding control point sequence is obtained using some evaluation function. Then, it is evaluated whether or not this error is within an allowable range. The above is the evaluation step.

Next, if the error exceeds the allowable range, the dividing step is executed. That is, the dividing method of the de Casteljau, as illustrated in FIG. 9 (b), determine the division points P ₄ on the Bezier curve P (t). Then, the original Bezier curve P (t) is set to a starting point P ₀ ,
A Bezier curve end point and P _4, the start point P _4, the end point P ₃
And a Bezier curve as follows.

Next, the above steps are executed again for each of the Bezier curves divided into two. That is, the error between the Bezier curve whose start point is P ₀ and the end point is P ₄ and the base lines P ₀ and P ₄ of the Bezier curve is obtained. If this error exceeds the allowable range, the error shown in FIG.
As illustrated (c), the calculated division points P _5, two dividing the Bezier curves. Similarly, an error between the Bezier curve whose start point is P ₄ and the end point is P ₃ and the base lines P ₄ and P ₃ of the Bezier curve is obtained. As illustrated in FIG. 9 (d), the dividing point P
₆ is obtained and the Bezier curve is divided into two.

As described above, a process consisting of the two steps of the evaluation step between the Bezier curve and the corresponding baseline and the step of dividing the Bezier curve when the error is outside the allowable range is recursively repeated, and the error is reduced. The division is terminated when it falls within the allowable range. Then, at the end of the division, the base line of each divided Bezier curve is set as an approximate broken line of the original Bezier curve.

In the above described Bezier function polygonal line approximation method, the evaluation function is used as a criterion for determining whether or not to terminate the Bezier curve polygonal line approximation.
How this is determined is extremely important. in this case,
A simple method of evaluating the distance between the point on the Bezier curve corresponding to t = 0.5 and the base line as an error is also conceivable. However, in the case of a Bezier curve of third order or higher, as shown in FIG. There is not only a simple shape in which a set of a control point sequence and a baseline forms a convex hull, but also a complicated and winding Bezier curve that does not form a convex hull as illustrated in FIGS. 10 and 11. .

If the selection of a point on the Bezier curve having such a complicated shape is inappropriate, it is evaluated that the error between the Bezier curve and the baseline is small even though they are far apart from each other. However, there is a possibility that the rough approximation ends without dividing the Bezier curve. Therefore,
When performing a polygonal line approximation of a Bezier curve, it is necessary to determine the end of the approximation by using an appropriate evaluation function that does not cause such inconvenience. Hereinafter, an example of a conventionally used evaluation function will be described.

First, JP-A-4-223577 discloses that
As illustrated in FIG. 12, the control point sequence P ₀ , P ₁ , P ₂ , P ₃
Given a Bezier curve P (t) defined by the following formulas, each of the _two control points P ₁ and P ₂ except for the starting point and the ending point of the Bezier curve P (t) extends from each of the control points P ₁ and P ₂ to the baselines P ₀ and P ₃ . A technique is disclosed in which both distances d ₁ and d ₂ are used as errors, and the approximation end is evaluated based on the errors.

As described above, the shape of the Bezier curve substantially follows the shape of the polygonal line formed by the control point sequence. Therefore, according to the technique described in this publication, it is necessary to appropriately evaluate the error between the Bezier curve and the baseline. It is thought that it is possible.

Japanese Patent Application Laid-Open No. Hei 5-223577 discloses that
A technique is disclosed in which the distance from the base line of a Bezier curve to the farthest point on the Bezier curve is used as an error, and the approximation end is evaluated based on the error. Here, it is generally not easy to find the farthest point from the baseline on the Bezier curve. Therefore, in this publication, the farthest point is obtained by a complicated process outlined below.

First, as illustrated in FIG. 13, it is assumed that a Bezier curve P (t) defined by a control point sequence P ₀ , P ₁ , P ₂ , P ₃ in an XY coordinate system is given. in this case,
The control point sequence and the Bezier curve P (t) are rotated so that the starting point of the Bezier curve P (t) is located at O and the baselines P ₀ and P ₃ coincide with the X axis.

Next, the value τ of t when dY / dX = 0 in the Bezier curve P (t) after the rotation is obtained.
Then, a point P (τ) on the Bezier curve corresponding to τ is set as the farthest point, and its Y coordinate is obtained as an error.

[0022]

SUMMARY OF THE INVENTION As outlined above,
In the conventional curve approximation data conversion method, a broken line approximation of a curve is performed by recursively repeating a process including an evaluation step and a division step. When a curved line approximation is performed by such a recursive process, it is necessary to avoid a problem that the approximation is terminated in an inappropriate state. For that purpose, it is required to accurately obtain an error between the Bezier curve and the control point sequence using an evaluation function as disclosed in the above-mentioned Japanese Patent Application Laid-Open Nos. Hei 4-223577 and Hei 5-223577. You.

However, on the other hand, the polygonal line approximation of the Bezier curve is performed by recursive repetition of the evaluation step and the division step. In order to shorten the time until the polygonal line approximation is completed, the evaluation step is not performed. It is required to minimize the execution time. However, for example, in the technique disclosed in Japanese Patent Laid-Open No. 4-223577, it is necessary to perform an operation to obtain a distance from two constituent points of a control point sequence to a baseline at the time of evaluation.
The calculation time per evaluation becomes long. Also,
According to the technique disclosed in Japanese Patent Application Laid-Open No. Hei 5-223577, one type of distance from the base line to the farthest point on the Bezier curve may be obtained, but a figure rotation process is performed as preparation for obtaining this farthest point. And the calculation time per evaluation becomes long.

As described above, in the prior art, if the approximation of the broken line of the curve is to be performed accurately, the calculation time in the evaluation step becomes longer, and the time required until the approximation of the broken line is completed and the drawing is performed is performed. There was a problem that would be long.

Accordingly, the present invention has been made in view of the circumstances described above, and a curve drawing method and a curve drawing method capable of accurately performing a broken line approximation of a curve with a smaller amount of calculation than in the past and performing the drawing. It is an object of the present invention to provide a recording device that records a drawing device and a curve drawing program.

[0026]

In order to solve this problem, a curve drawing method according to the present invention is a curve drawing method for drawing a Bezier curve by polygonal approximation, and obtains a control point sequence of the Bezier curve. Then, the Bezier curve is approximated by a broken line using the obtained control point sequence, and the approximated broken line data is drawn. More specifically, the polygonal line approximation evaluates whether or not subdivision is performed, and subdivides a target curve only when it is determined that subdivision is necessary. Then, the polygonal line approximation is performed recursively. The evaluation uses a chord length estimated value of the Bezier curve and an arc length estimated value of the Bezier curve as evaluation criteria.

Thus, when the Bezier curve to be drawn is approximated to an optimal polygonal line by recursive repetition of the evaluation step and the dividing step, the chord length of the curve is easily obtained by utilizing the control point sequence data of the Bezier curve. It is possible to establish a method of calculating the estimated value of the length and the estimated value of the arc length, and by using them, the evaluation step can be completed by a simple operation with a small calculation load.

[0028]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The invention according to claim 1 of the present invention is a curve drawing method for drawing a Bezier curve by polygonal line approximation, wherein the control point sequence of the Bezier curve is obtained, This is a curve drawing method characterized by approximating the Bezier curve with a polygonal line using a control point sequence and rendering the approximated polygonal line data, and performing high-speed, high-quality rendering with a simple operation with a small calculation load. It has the effect of being able to.

According to a second aspect of the present invention, in the first aspect of the present invention, the polygonal line approximation evaluates whether or not subdivision is performed, and only when it is evaluated that subdivision is necessary. A curve drawing method characterized by subdividing a target curve, and when a Bezier curve to be drawn is approximated to an optimal polygonal line by recursive repetition of an evaluation step and a division step, a control point of the Bezier curve is used. A method to easily calculate the estimated value of the chord length and the estimated arc length of the curve using the column data is established, and by using them, the evaluation step can be completed with a simple operation with a small calculation load. This makes it possible to perform high-speed drawing at high speed.

According to a third aspect of the present invention, there is provided the curve drawing method according to the second aspect, wherein the polygonal line approximation is performed recursively. When approximating the optimal polygonal line by recursive repetition of the evaluation step and the division step, the estimated chord length and the estimated arc length of the curve are easily calculated using the control point sequence data of the Bezier curve. By using these methods, the evaluation step can be completed by a simple calculation with a small calculation load by using these methods, and has the effect that high-quality drawing can be performed at high speed.

According to a fourth aspect of the present invention, in the second aspect of the present invention, the evaluation includes evaluating an estimated chord length of the Bezier curve and an estimated arc length of the Bezier curve. This is a curve drawing method characterized by using it as a reference, and when the Bezier curve to be drawn is approximated to an optimal polygonal line by recursive repetition of an evaluation step and a division step, control point sequence data of the Bezier curve is used. Then, a method for easily calculating the estimated chord length and the estimated arc length of the curve is established, and by using them, the evaluation step can be completed with a simple operation with a small calculation load. And has the effect that high-quality drawing can be performed at high speed.

According to a fifth aspect of the present invention, in the fourth aspect of the present invention, a length of a broken line connecting the control point sequence in order is used as the estimated chord length of the Bezier curve. And has an effect that it is possible to perform a simple calculation with a small calculation load.

According to a sixth aspect of the present invention, in the fourth aspect of the present invention, a length of a straight line connecting a start point and an end point of the control point sequence is used as the estimated arc length of the Bezier curve. This is a curve drawing method characterized by performing a simple calculation with a small calculation load.

According to a seventh aspect of the present invention, in the fourth aspect, the evaluation is performed by calculating an estimated chord length of the Bezier curve and an estimated arc length of the Bezier curve. This is a curve drawing method characterized by being performed by comparing an obtained result with a predetermined allowable error value, and has an effect that a simple operation with a small calculation load can be performed.

According to an eighth aspect of the present invention, there is provided a curve drawing method according to the seventh aspect, wherein the operation is a difference operation. It has the effect of being able to complete.

According to a ninth aspect of the present invention, there is provided a curve drawing device which draws a Bezier curve by approximating a polygonal line,
Input means for acquiring the control point sequence of the Bezier curve, polygonal line approximation means for polygonal approximation of the Bezier curve using the acquired control point sequence, and output means for drawing approximated polygonal line data This is a curve drawing device characterized by the fact that high-quality drawing can be performed at high speed with a simple calculation with a small calculation load.

According to a tenth aspect of the present invention, in the ninth aspect of the present invention, the broken line approximating means further evaluates whether or not to perform the subdivision. And a subdivision means for performing subdivision only when it is determined that the Bezier curve to be rendered is an optimal polygonal line by recursively repeating an evaluation step and a division step. When approximating, the method of easily calculating the estimated chord length and the estimated arc length of the curve using the control point sequence data of the Bezier curve is established, and the evaluation step is performed by using them. Can be completed by a simple operation with a small calculation load, and has an effect that high-quality drawing can be performed at high speed.

An eleventh aspect of the present invention is the curve drawing apparatus according to the ninth aspect, wherein the broken line approximating means is recursively executed.
When approximating the Bezier curve to be drawn to the optimal polygonal line by recursive repetition of the evaluation step and the division step,
A method for easily calculating the estimated chord length and the estimated arc length of the curve using the control point sequence data of the Bezier curve is established, and the evaluation step is simplified by using the calculated data. This makes it possible to perform high-quality drawing at high speed.

According to a twelfth aspect of the present invention, in the tenth aspect of the present invention, the evaluation means calculates the chord length estimated value of the Bezier curve and the arc length estimated value of the Bezier curve. A curve drawing device characterized by using as an evaluation criterion.When approximating a Bezier curve to be drawn to an optimal polygonal line by recursive repetition of an evaluation step and a division step, control point sequence data of the Bezier curve is used. Using this method, a method for easily calculating the estimated chord length and the estimated arc length of a curve can be established, and by using these methods, the evaluation step can be completed with a simple operation with a small calculation load. This has the effect that high-quality drawing can be performed at high speed.

According to a thirteenth aspect of the present invention, in the twelfth aspect, a length of a polygonal line connecting the control point sequence in order is used as the estimated chord length of the Bezier curve. And has an effect that it is possible to perform a simple calculation with a small calculation load.

According to a fourteenth aspect of the present invention, in the twelfth aspect of the present invention, a length of a straight line connecting a start point and an end point of the control point sequence is used as the estimated arc length of the Bezier curve. This is a curve drawing device characterized by performing a simple operation with a small calculation load.

According to a fifteenth aspect of the present invention, in the twelfth aspect of the invention, the evaluation means calculates a chord length estimated value of the Bezier curve and an arc length estimated value of the Bezier curve. This is a curve drawing device characterized by evaluating by comparing a result obtained by the above with a predetermined allowable error value, and has an effect that a simple calculation with a small calculation load can be performed.

According to a sixteenth aspect of the present invention, in the fifteenth aspect, the operation is a difference operation, wherein the operation is a difference operation. It has the effect of being able to complete.

According to a seventeenth aspect of the present invention, in a recording medium recording a curve drawing program for outputting a Bezier curve by approximating the Bezier curve by a polygonal line, the curve drawing program acquires a control point sequence of the Bezier curve. Input processing,
A recording medium characterized by comprising a polygonal line approximation process for performing a polygonal line approximation on the Bezier curve using the obtained control point sequence, and an output process for drawing approximated polygonal line data. The distributed user has an effect that the drawing program can be installed in a personal computer or the like to execute the curve drawing method.

Hereinafter, an embodiment of the present invention will be described with reference to FIG.
This will be described with reference to FIG. In these drawings, the same members are denoted by the same reference numerals, and duplicate description is omitted.

FIG. 1 is a block diagram showing the configuration of a curve drawing device according to an embodiment of the present invention. This curve drawing device has a function of performing a polygonal line approximation conversion of a Bezier curve to be rendered and rendering the approximated polygonal line data. The CPU 101, the ROM 102, the RAM 103,
It comprises an input unit 104, an output unit 105 and a bus 106 interconnecting them.

The input unit 104 is a means for receiving information for specifying a Bezier curve to be drawn (hereinafter referred to as “curve information”) from a host computer (not shown) via a communication path 107. Here, the curve information includes control point sequence data defining the Bezier curve. When performing the polygonal line approximation of the Bezier curve, the control point sequence data is provided to the input unit 104 as an input value.

The CPU 101 receives the control point sequence data via the input unit 104, performs a line approximation on the Bezier curve according to a curve drawing program stored in the ROM 102, and performs a process of drawing the approximate line.

The RAM 103 is used as storage means for information representing input control point sequence data and output approximate polygonal line data, and is used by the CPU 10 when performing polygonal line approximation conversion.
1 is also used as a work memory used for work.

The output unit 105 is a means for drawing approximate polygonal line data under the control of the CPU 101. The output unit 105 creates bitmap data for rendering from the approximate polygonal line data and outputs it as visible information. For example, an electrophotographic printer, an inkjet printer, or a display device such as a CRT that prints such visible information on recording paper is used as the output unit 105.

Next, the operation of the embodiment of the present invention will be described.

FIG. 2 shows the contents of the curve drawing process executed by the CPU 101. As shown in FIG. 2, the curve drawing process according to the present embodiment includes a curve data input process (step S1), a broken line approximation process (step S2), and an output process (step S3).

First, the curve data input processing (step S
In 1), curve information corresponding to a Bezier curve to be drawn is fetched from outside, control point sequence data is extracted therefrom, and temporarily stored in the RAM 103 as a control point sequence data file F1. Here, the control point sequence data includes position coordinates of control points necessary for expressing the Bezier curve.

Next, in the polygonal line approximation processing (step S2), the control point sequence data file F1 stored in the RAM 103 is read out, approximated to the polygonal line data, and the resulting polygonal line data is set as an approximate polygonal line data file F2. To save temporarily.

Next, in the output processing (step S3), R
The approximate broken line data file F2 stored in the AM 103 is read, and bitmap data for drawing is created based on the read data, and drawn as visible information. The above is the CPU
3 is an outline of a curve drawing process executed by the CPU 101;

FIG. 3 shows the broken line approximation processing (step S
This shows the detailed processing contents of 2). As shown in FIG. 3, in the present embodiment, the Bezier curve specified by the control point sequence data file F1 is divided into two in advance, and the first
And a second Bezier curve (step S21).

FIG. 4 is a flowchart for explaining the method of dividing the Bezier curve in step S21. Hereinafter, the Bezier according to the present embodiment will be described with an example in which a third-order Bezier curve P (t) defined by control point sequences P ₀ , P ₁ , P ₂ , and P ₃ illustrated in FIG. A method of dividing a curve will be described.

First, at step S101, the control point P ₀
~P following equation (2) using the position vector of the _3-
The calculation of (4) is performed.

L ₁ = (P ₀ + P ₁ ) / 2 (2) H = (P ₁ + P ₂ ) / 2 (3) R ₂ = (P ₂ + P ₃ ) / 2. (4) By this calculation, three sides of the control point sequence P ₀ , P ₁ , P ₂ , P ₃ except for the base lines P ₀ , P ₃ , that is, the sides P ₀ , P ₁ , P ₁ , P ₂ and each midpoint L ₁ of P ₂ and P ₃ ,
The position vectors of H and R ₂ are obtained.

Next, in step S102, step S
The following calculation is performed using the position vectors of the respective midpoints L ₁ , H and R ₂ obtained in step 101 to obtain the midpoints L ₁ and H
Obtain each position vector of the center point R ₁ of the side connecting the midpoints of the sides L ₂ and the middle point H and R ₂ connecting.

L ₂ = (L ₁ + H) / 2 (5) R ₁ = (H + R ₂ ) / 2 (6) Next, in step S103, each midpoint obtained in step S102 is obtained. the position vectors of L ₂ and the side of the midpoint L ₃ connecting the R ₁ obtained by calculation below.

L ₃ = (L ₂ + R ₁ ) / 2 (7) Here, the above-mentioned middle point L ₃ is represented by t = t on the Bezier curve P (t).
It exactly matches the point corresponding to 0.5. Therefore, the middle point L ₃ and division points, to divide the original Bezier curve P (t) on the left side of the Bezier curve LEFT division point L ₃ (t) and the right side of the Bezier curve RIGHT (t).

Next, in step S104, control point sequence data of the Bezier curve LEFT (t) and the Bezier curve RIGHT (t) obtained by the above division are created. Here, the control point sequence data of the Bezier curve LEFT (t) is
The Bezier curve is constituted by position vectors of control points L ₀ , L ₁ , L ₂ and L ₃ . Note that the control point L
₀ corresponds to the control point P ₀ of the original Bezier curve. Therefore, the position vector of the control point P ₀ is directly used as the control point L _0.
Used as the position vector of

The control point data of the Bezier curve RIGHT (t) is constituted by the position vectors of the control points R ₀ , R ₁ , R ₂ and R ₃ of the Bezier curve. Note that the control point R ₀ coincides with the control point L ₃ of the Bezier curve LEFT (t). Therefore, the position vector of a control point L ₃ is directly used as a position vector of a control point R _0. The control point R ₃ is the control point P ₃ of the original Bezier curve.
It corresponds to. Therefore, the position vector of a control point P ₃ is directly used as a position vector of a control point R _3.

As described above, according to the present division method, by performing an operation using only the position vector of each control point of the original Bezier curve, the Bezier curve is divided into two, and each of the divided Bezier curves is Control point sequence data composed of control point position vectors can be obtained. The above is the details of the Bezier curve splitting process performed in step S21 of FIG.

When the two-division processing of the Bezier curve to be drawn is completed in this way, in step S21, a first control point sequence consisting of the position vectors of the control points of the first Bezier curve obtained by the division is obtained. Data F11 and second control point sequence data F12 including the position vector of each control point of the second Bezier curve are stored in the RAM 103.

In step S21, the division into two is performed. However, this is merely an example, and does not prevent the division into three or more. Step S above
When dividing into three or more n Bezier curves at 21, each of the first to n-th Bezier curves is a processing target of step S22 described below.

Next, in step S22, step S2
For each of the first Bezier curve and the second Bezier curve obtained in step 1, a polygonal line approximation process is performed. Hereinafter, the content of the broken line approximation processing will be described with reference to the flowchart shown in FIG.

First, in step S201, an error between a curve to be approximated by a polygonal line and an approximate polygonal line approximating the curve is evaluated by a predetermined evaluation function. A specific example of the evaluation function will be described later.

Next, in step S202, step S202
It is evaluated whether the error obtained in 201 is within an allowable range. Here, when the error exceeds the allowable range, steps S203 to S205 described below are executed.

First, in step S203, the Bezier curve to be subjected to the polygonal line approximation is transformed into two Bezier curves LEFT.
(T) and RIGHT (t). The division of the Bezier curve into two can be performed by the division method already described with reference to FIGS. 4 and 5.

Next, in step S204, step S204
One Bezier curve LEFT divided at 203
The routine shown in FIG. 6 is called recursively with (t) as the target of the polygonal line approximation.

Next, in step S205, step S205
The other Bezier curve RIGHT divided at 203
The routine shown in FIG. 6 is called recursively with (t) as the target of the polygonal line approximation.

Such a recursive call is performed when the error between the Bezier curve to be broken-line approximation and the control point sequence of the Bezier curve is within an allowable range in the evaluation processing in step S201 after being called. Repeat until evaluated.

If the error falls within the allowable range after the recursive call is repeated, the process proceeds from step S202 to step S206. Then, as a result of the repetition of the division, the data of the polygonal lines obtained by sequentially connecting the base lines of the Bezier curves finally obtained is stored in the RAM 103 as the approximate polygonal data file F2, and the output processing (step S3 in FIG. 2) ). The above is Step S
22 is a detail of the polygonal line approximation processing in FIG.

As described above, in the present embodiment,
Since the target of the polygonal line approximation processing is the first and second Bezier curves obtained by dividing the Bezier curve as the original drawing target into two, it has a simple shape as illustrated in FIGS. 7 and 8. For this reason, even if a simple evaluation function as described in the above specific example is used, the error between the Bezier curve and the approximate broken line can be appropriately obtained. Therefore, according to the present embodiment, it is possible to accurately perform a polygonal line approximation of a curve with a small amount of calculation and to perform drawing.

Next, a specific example of the evaluation function will be described with reference to FIGS. 9 (a) to 9 (d). First, as exemplified in FIG. 7A, a cubic Bezier curve P (t) defined by control point sequences P ₀ , P ₁ , P ₂ , and P ₃ is given as an object of the polygonal line approximation. Shall be. Set the coordinates of each control point to P ₀
(X ₀ , Y ₀ ), P ₁ (X ₁ , Y ₁ ), P ₂ (X ₂ , Y ₂ ), P
₃ (X ₃ , Y ₃ ), the length of the polygonal line connecting the control point sequence in order is α, the length of the line segment (base line) connecting the start and end points of the control point sequence is β, and the square root operation of sqrt () as, α = Σ [sqrt {( X I + 1-X I) 2 + (Y I + 1-Y I) 2}] ··· (8) ( integer I is from 0 to 3) β = sqrt {( X ₃ −X ₀ ) ² + (Y ₃ −Y ₀ ) ² } (9)

FIGS. 7B to 7D show changes in the control point sequence and the base line corresponding to each of the divided Bezier curves when the re-division is repeated. As can be seen, each time the division is repeated, the shapes of the two approaches each other. This is based on the basic mathematical property of the Bezier curve.
α, β → “actual curve length of Bezier curve”
By using the difference between α and β as an evaluation function in the evaluation step, it is possible to easily determine whether or not to perform subdivision. The above is the description of the specific example of the evaluation function.

The above-described embodiment is merely an example of the present invention. The present invention can be implemented by adding various modifications to the above embodiment without departing from the spirit thereof. For example, the following other embodiments can be considered.

In the above embodiment, the division is performed by the de Castello division method using the Bezier curve as an example. However, the present invention is not limited to this, and other division methods may be used. May be used.

In the above embodiment, a cubic Bezier curve is described as an object to be drawn. However, the present invention may be applied to drawing a quaternary or higher-order Bezier curve. For example, if De Castello's division method is used as the division method, since this method is defined for general dimensions (third or higher), it can be easily extended.

Further, in the above embodiment, when the approximation of the polygonal line is completed, the base line of each finally divided Bezier curve is output as an approximate polygonal line. May be output as an approximate polygonal line.

Further, the drawing program in the above embodiment is recorded on a recording medium such as an FD (Floppy (registered trademark) disk) and distributed to the user, and the user installs the drawing program on a personal computer or the like, and May be implemented.

In the block diagram of the above embodiment, the CPU, the ROM, the RAM, the input unit, and the output unit are configured to be mounted on the same device. May be present above. For example, since a personal computer has a high capability at present, the CPU and RO used for the calculation of the polygonal line approximation processing are used.
The M and RAM are those on a personal computer, and the output unit for performing drawing is a separate device, which does not prevent a configuration in which the personal computer and the output unit are connected via a communication path.

[0085]

As described above, according to the present invention, when a Bezier curve to be drawn is approximated to an optimal polygonal line by recursive repetition of the evaluation step and the dividing step, the control point sequence data of the Bezier curve is obtained. Using this method, a method for easily calculating the estimated chord length and the estimated arc length of a curve can be established, and by using these methods, the evaluation step can be completed with a simple operation with a small calculation load. With such a configuration, an effective effect that high-quality drawing can be performed at high speed can be obtained.

[Brief description of the drawings]

FIG. 1 is a block diagram illustrating a configuration of a curve drawing device according to an embodiment of the present invention.

FIG. 2 is a flowchart showing processing contents of a curve drawing program according to one embodiment of the present invention;

FIG. 3 is a flowchart showing processing contents of a polygonal line approximation processing according to an embodiment of the present invention;

FIG. 4 is a flowchart illustrating a method of dividing a Bezier curve according to an embodiment of the present invention.

FIG. 5 is a diagram showing a method of dividing a Bezier curve according to an embodiment of the present invention.

FIG. 6 is a flowchart showing processing contents of a broken line approximation processing of a Bezier curve after division according to an embodiment of the present invention;

FIG. 7 is a diagram showing an evaluation function according to an embodiment of the present invention.

FIG. 8 is a diagram showing an example of a cubic Bezier curve.

FIG. 9 is a view for explaining a polygonal line approximation method for a Bezier curve using a de Castello division method;

FIG. 10 is a diagram showing another example of a cubic Bezier curve.

FIG. 11 is a diagram showing another example of a cubic Bezier curve.

FIG. 12 is a diagram showing an example of an evaluation function used in a conventional Bezier curve subdivision evaluation step.

FIG. 13 is a diagram illustrating an example of an evaluation function used in a conventional Bezier curve subdivision evaluation step.

[Explanation of symbols]

 101 CPU 102 ROM 103 RAM 104 Input unit 105 Output unit 106 Bus

Claims (17)

[Claims] 1. A curve drawing method for drawing a Bezier curve by a polygonal line approximation, obtaining a control point sequence of the Bezier curve, using the acquired control point sequence to perform a polygonal line approximation on the Bezier curve, A curve drawing method characterized by drawing approximated line data. 2. The method according to claim 1, wherein the polygonal line approximation evaluates whether or not subdivision is performed, and subdivides a target curve only when it is determined that subdivision is necessary. Curve drawing method. 3. The curve drawing method according to claim 2, wherein the polygonal line approximation is performed recursively. 4. The curve drawing method according to claim 2, wherein the evaluation uses an estimated value of a chord length of the Bezier curve and an estimated value of an arc length of the Bezier curve as evaluation criteria. . 5. The curve drawing method according to claim 4, wherein a length of a polygonal line connecting the control point sequence in order is used as the estimated chord length of the Bezier curve. 6. The curve drawing method according to claim 4, wherein a length of a straight line connecting a start point and an end point of the control point sequence is used as the estimated arc length of the Bezier curve. 7. The evaluation is performed by comparing a result obtained by calculating an estimated value of the chord length of the Bezier curve and an estimated value of the arc length of the Bezier curve with a predetermined allowable error value. The curve drawing method according to claim 4, wherein: 8. The method according to claim 7, wherein the calculation is a difference calculation. 9. A curve drawing device which draws a Bezier curve by approximating the Bezier curve by a polygonal line, comprising: input means for acquiring a control point sequence of the Bezier curve; and a polygonal line drawing the Bezier curve using the acquired control point sequence. A curve drawing device, comprising: a polygonal line approximation unit for approximation; and an output unit for rendering approximated polygonal line data. 10. A polygonal line approximation means for evaluating whether or not to perform further subdivision, and a subdivision means for executing subdivision only when the evaluation means determines that subdivision is necessary. The curve drawing device according to claim 9, comprising: 11. The curve drawing device according to claim 9, wherein said polygonal line approximating means is executed recursively. 12. The curve drawing according to claim 10, wherein said evaluation means uses a chord length estimated value of said Bezier curve and an arc length estimated value of said Bezier curve as evaluation criteria. apparatus. 13. The estimated chord length of the Bezier curve:
13. The curve drawing device according to claim 12, wherein a length of a polygonal line connecting the control point sequence in order is used. 14. An arc length estimation value of the Bezier curve,
13. The curve drawing device according to claim 12, wherein a length of a straight line connecting a start point and an end point of the control point sequence is used. 15. The evaluation means evaluates by comparing the result obtained by calculating the estimated chord length of the Bezier curve and the estimated arc length of the Bezier curve with a predetermined allowable error value. The curve drawing device according to claim 12, wherein the curve drawing is performed. 16. The curve drawing device according to claim 15, wherein the calculation is a difference calculation. 17. A recording medium on which a curve drawing program for outputting a Bezier curve by approximating the Bezier curve by a polygonal line is recorded, wherein the curve drawing program obtains a control point sequence of the Bezier curve, and the obtained control points A recording medium comprising: a polygonal line approximation process for approximating the Bezier curve using a column; and an output process for drawing approximated polygonal line data.