JPH04168581A - Approximate curve generating method for hyperellipse - Google Patents

Abstract

PURPOSE:To generate a hyperellipse with high approximation and to respond to local shape change, etc., by dividing the hyperellipse into segments, and making them into approximate curve at every segment by the Bezier curve. CONSTITUTION:A circle setting the major axis of the hyperellipse as a radius is generated, and point groups are arranged at every segment at equal interval by dividing the circle into N segments. A corresponding secondary ellipse with a minor axis is generated by mapping the circle on an XY plane with the point group, and the intersection of a normal line at every point in the point group on the secondary ellipse with the hyperellipse is found as the point group on the hyperellipse. The Bezier curve approximating the hyperellipse from the point group and a tangent vector at the start and end points of each segment on the hyperellipse at every segment can be generated. Thereby, it is possible to obtain the hyperellipse with high approximation, and also, to respond to the local shape change.

Description

DETAILED DESCRIPTION OF THE INVENTION A. Field of Industrial Application The present invention is directed to converting an n-th hyperellipse into a Bezier curve (Bezier curve).
Regarding the method of generating an approximation curve for a hyperellipse,
For example, graphic display devices, CAD, CAM
B. Conventional Technology In technical fields such as graphic display devices, CAD, and CAM, it is necessary to generate not only straight lines but also various curves such as circles and ellipses. Generally, straight lines and curves used to draw figures on the xy plane are X.

This can occur by calculating the coordinates of points on the straight line or curve to draw using the function f (x, y) that indicates the above straight line or curve in the orthogonal coordinate (x, y) system with y as a variable. Research and development of algorithms for efficiently generating these various curves has been progressing. For example, Japanese Patent Laid-Open No. 60-252951 discloses a method of generating a quadratic ellipse with rotation of coordinate axes.

However, a higher-order ellipse, for example, n with the major axis a and the minor axis b
The next superellipse was generated by arithmetic processing on the equation a'b'', which indicates an n-th order superellipse in the orthogonal coordinate (x, y) system.

C Problems to be solved by the invention However, with the conventional method of expressing a superellipse that relies on a coordinate system, it is not possible to efficiently generate a superellipse, and it is not possible to deal with local changes in shape. There was a problem.

Therefore, an object of the present invention is to make it possible to generate a highly approximate superellipse using parametric expression, and to be able to cope with local shape changes.

D. Means for Solving the Problems The method for generating an approximate curve of a hyperellipse according to the present invention is based on rectangular coordinates (
x, y) system, with x and y as variables, for an n-th order superellipse with the major axis a and the minor axis b, denoted by a-b'', let the major axis a of the superellipse be the radius. Generate a circle, divide this circle equally into N segments, place a school of fish on the circle at equal intervals for each segment, map the circle together with the school of fish onto the xy plane, and calculate the quadratic of the minor axis. Generate an ellipse, find the intersection of the normal line at each point of the fish school on the secondary ellipse and the superellipse as the fish school on the superellipse, and calculate the intersection of the fish school on the superellipse and each segment on the superellipse. This method is characterized in that a Bezier curve that approximates the superellipse for each segment is generated from tangent vectors at the starting point and the ending point.

E Effect In the method of the present invention, a parametric approximation curve is generated by dividing a hyperellipse into N segments and approximating each segment with a Bezier curve.

F Example In the method for generating an approximate curve for a superellipse according to the present invention, a parametric approximate curve is generated by dividing a superellipse into N segments and approximating each segment with a Bezier curve, for example, according to the procedure shown in the flowchart of FIG. do.

That is, in the orthogonal coordinate (x, y) system, X.

Using y as a variable, divide the nth-order superellipse whose major axis is a and whose minor axis is b, as shown in Figure 2, indicated by a''b', into N segments, and approximate each segment with a Bezier curve. In order to do this, first, in the first step, as shown in Fig. 3, a circle whose radius is the major axis a of the superellipse to be generated is generated, and as shown in Fig. 4, this circle is created with equal centers. Divide into N segments having an angle θ.The end point Q"(X"+,)"I) of the j-th segment on the above circle is: The position is given by j=1.N).

In the next second step, one segment on the circle is divided into d parts at equal intervals to create (d-1) schools of fish, as shown in FIG. 1 segment on the above circle, i inh (D i
th) point qc(X'+ , yc+ ) is X'+
=r COSθ ycl =r S! Its position is given by nθ (j=l, d+1).

In the next third step, the above circle is xy with the above fish school.
This is mapped onto a plane to generate a quadratic ellipse with a short axis as shown in FIG.

The i-th point q*@(
X2°+ + V”+) is given its position by X”I=r CO3θ.

In the next fourth step, as shown in FIG. 7, the position of the intersection of the normal line and the superellipse at each point of the school of fish on the secondary ellipse is determined.

The position of the intersection of the normal line and the superellipse at each point of the school of fish on the quadratic ellipse can be found by solving the following simultaneous equations.

Note that the above equation 0 is an equation for an n-th order superellipse. Also, the above equation 0 is the point ql+(x!+,) on the quadratic ellipse.

This is the equation of a straight line in the normal direction passing through y2°1). moreover,
In the above equation 0, vl is the X component in the normal direction, and v
y is the X component. It can be found by taking

When the above equations 0 and 0 are expanded by Taylor, a'
a” b”■.

Because it is, It can be expressed as a matrix formula.

Therefore, the above simultaneous equations can be expressed as Newton-Rabson using this matrix formula.
It can be solved by law.

Let the solution obtained from the above simultaneous equations be '(x8°1.y3°1).

In the next fifth step, by differentiating the equation of the n-th superellipse, we obtain the
Find the tangent vector 'l+'d-1 between the starting point (i=1) and the ending point (i=d-1) of the segment.

Point q 8 m (x S m , ,
X component t of the tangent vector at y m + , )
lx becomes t'',=(x S m , )1, and y components l and F become "I"(y"+)"-1.

In the next sixth step, from the school of fish on the superellipse and the tangent vectors at the start and end points of the segment on the superellipse, a Bezier curve, which is a least squares approximation of one segment of the superellipse, is calculated as follows. and generate it.

That is, the four control points P, , P, , P, , P, at the Bezier curve R(t) (%) +t'p4 are expressed as P+=
4+ Pt ” ql + t + Xα P 2 = 9.41 + t 4+1XβP4=
q −+1 and first, the magnitude α β of the tangent vector is given as an initial value to temporarily generate a Bezier curve. Then, the statistics of the distances between the (d-1) points in one segment on the superellipse found in the fifth step and the closest point on the Bezier curve are found and minimized.

f−Σ((R”(tll q y,)!+ (R'l
Differentiate t, l Q'+)2 10(R'tt1+ q'I)'1 to 0.

aα 10 ・ ・ ・ ・■ 10 ・ ・ ・ ・■ When the above equations 0 and 0 are linearized by Taylor expansion, they become 10 ・ ・ ・ ・■′ 10 ・ ・ ・ ・■゛From, 8α 8α 8β 8α 8α 8β a R'fL+1 a R'fL++aα
8β, it can be expressed by the matrix formula:

Using the equation of the second matrix, solve the above simultaneous equations of equation 0 and equation 0 using the Newton-Labrun method, and the obtained Δα, Δ
Finally, α is obtained by adding β to α and β and performing a cutback calculation process.

The above four control points P, , P, , P, using β.

P4 is determined, and a Bezier curve R(t) is generated by least squares approximation of one segment on the hyperellipse.

In the next seventh step, for all the segments on the above hyperellipse, the least squares approximated Bezier curve Bezier curve R
(t), 1. Engineering, 12. If there is a segment for which a Bezier curve has not been generated, return to the second step above and perform the second step for the new segment. The processes from step to seventh step are repeated to generate Bezier curves R(t)+, R(t)t, and R(t)* that are least squares approximated for all segments, as shown in FIG.

Here, when generating an approximation of a hyperellipse on the xy plane with multiple segments, is there a way to divide it at equal angular intervals at the central angle or at equal intervals at the X coordinate? Therefore, it is difficult to obtain high approximation accuracy.

For example, as shown in Figure 1O, if the arc of the first quadrant of a superellipse on the , is longer than the first and second segments and includes a portion with a large curvature, so
It is difficult to express with a single Bezier curve. Therefore, although the approximation accuracy of the short first segment is high, the approximation accuracy of the third segment, which includes the characteristic portion of the superellipse, decreases. Similarly, as shown in Fig. 11, when the circle in the first quadrant of the arc hyperellipse on the xy plane is divided into three segments at equal intervals on the The approximation accuracy of the third segment is low.

In contrast, in the method of the present invention, as shown in FIG. 12, for an n-th order superellipse whose major axis is a and whose minor axis is b, a circle whose radius is the major axis a of the superellipse is generated. , divide this circle equally into N segments, place a school of fish on the circle at equal intervals for each segment, and map the circle together with the school of fish onto the xy plane to generate a quadratic ellipse with a short axis. By determining the intersection of the normal line at each point of the school of fish on the secondary ellipse and the superellipse as the school of fish on the superellipse, the length of each segment of the superellipse is approximately averaged, and, A Bezier curve with high approximation accuracy can be generated by using a portion with a large curvature as one segment.

For example, according to the method of the present invention, the arc of one quadrant of the superellipse on the xy plane is divided into three segments, and the superellipse is approximated by a Bezier curve for each segment, as shown in FIG. 13. '! a=70.00'. b=100. The approximate curve SE of the superellipse with n=4 and a=70. b=100. It was possible to generate an approximation SE of a superellipse with n=4°1. In this way, according to the method of the present invention, by changing the order after the decimal point, it is possible to accurately represent minute changes in shape at the four corners of a superellipse.

Furthermore, by dividing the arc of one quadrant of the superellipse on the xy plane into three segments, and approximating each segment of the superellipse with a Bezier curve, as shown in Fig. 14, at 60'60'' It was possible to generate the approximate curve SE of the superellipse with a=60.b=60.n=4 as shown.In this way, according to the method of the present invention, a=b, that is, the major axis and the minor axis are equal. A hyperelliptic approximation curve SE can also be generated.

Further, according to the method of the present invention, by combining a circle, a quadratic ellipse, and a superellipse, for example, as shown in FIG. (2) and a bottom (3) whose cross section is a quadratic ellipse (1
0) was able to be expressed and designed parametrically.

G. Effects of the Invention As described above, in the method for generating an approximate curve for a superellipse according to the present invention, it is possible to generate an approximate curve with a high degree of approximation by approximating a superellipse with a Bezier curve for each segment with high precision. can. The degree of the superellipse for generating an approximate curve by the method of the present invention can be not only an integer but also a real number, and in the method of the present invention, the curvature of the superellipse can be adjusted by changing the fractional part of the degree of the superellipse. It is possible to generate an approximate curve that accurately represents subtle changes in shape of a large portion.

Therefore, according to the method of the present invention, it is possible to generate a hyperelliptic approximate curve with a high degree of approximation by parametric expression, and it is also possible to deal with local shape changes.

[Brief explanation of the drawing]

FIG. 1 is a flowchart showing the procedure for generating an approximate curve of a superellipse according to the method of the present invention. FIGS. 2 to 9 are diagrams showing various figures used in the process of generating an approximate curve of a hyperellipse according to the flowchart shown in FIG. Fig. 4 shows a circle with radius a as the long axis of the superellipse, Fig. 4 shows a circle with radius a divided into segments, and Fig. The figure shows a quadrant of radius a with a school of fish in the l segment,
Figure 6 shows a quadratic ellipse whose major axis is a and whose minor axis is b, generated by tilting a circle with radius a and mapping it onto a plane, and Figure 7 shows the school of fish on the secondary ellipse shown in Figure 6. The relationship between the intersection point and the school of fish on the fish school superellipse is shown, Figure 8 shows the tangent vectors at the start and end points of the segments of the superellipse, and Figure 9 shows the approximate curves in which each segment of the superellipse is approximated by a Bezier curve. show. Figures 1θ and 11 are diagrams showing other examples of segment division when each segment of a superellipse is approximated by a Bezier curve. The figure shows a case where the image is divided into equal intervals along the X axis. FIG. 12 is a diagram showing the procedure for segmenting a superellipse in the method of the present invention. FIG. 13 is a diagram showing a superellipse with n=4 and a superellipse with n=4.1 generated by the method of the present invention. FIG. 14 is a diagram showing a superellipse with a=b and n=4 generated by the method of the present invention. FIG. 15 is a diagram showing an example of a bottle design produced by the method of the present invention.

Claims (1)

[Claims] In the orthogonal coordinate (x, y) system, where x and y are variables, the major axis is a and the minor axis is expressed as x^n/a^n+y^n/b^n=1. For the n-th superellipse, generate a circle whose radius is the major axis a of the superellipse, divide this circle equally into N segments, and place a group of points at equal intervals on the circle for each segment. Place the above circle along with the above point group x
Generate a quadratic ellipse whose minor axis is mapped onto the y plane, find the intersection of the normal at each point of the point group on the quadratic ellipse and the super ellipse as a point group on the super ellipse, and Approximate curve generation for a superellipse, characterized in that a Bezier curve that approximates the superellipse for each segment is generated from a group of points on the superellipse and tangent vectors at the start and end points of each segment on the superellipse. Method.