JPH06251161A - Curved line/curved surface display method and graphic display system - Google Patents

Abstract

PURPOSE:To provide a display method for a free curved line/curved surface without generating division by zero, the underfined number of partitions, system down, runaway, and erroneous display, etc., in the calculation of the number of partitions when a curved line/curved surface is approximated to a polygonal line or polyhedron even when any kind of set of weight with respect to a curved line/curved surface to be displayed is supplied. CONSTITUTION:The curved line/curved surface stored in a filing device 813 in advance or inputted from a mouse 804, etc., is stored in a memory 802. A program to decide the number of partitions used when the curved line/curved surface is approximated to the polygonal line or polyhedron is stored in the device 813. When the number of portions is decided, calculation is performed by separating the weight into positive, zero, and negative parts. The program is read out by the memory 802, and is executed by a CPU 801, then, the number of polygonal lines/polyhedrons is decided, and the graphic data on the curved line/curved surface is generated, and is stored in the memory 802, and it is transferred to a graphic subsystem 807, then, it is displayed on a CRT 811.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method and apparatus for displaying a curved line / curved surface when displaying the curved line / curved surface on a graphic display or the like.

[0002]

2. Description of the Related Art Recent graphic systems have a display function of more general primitives such as a curve, a curved surface (a free curve or a free curved surface) in addition to a text, a polygonal line, a polyhedron and the like. However, in a conventional graphic system, a method of displaying a free curve and a free curved surface is often performed as follows. That is, the free-form curve and the free-form surface are temporarily divided on the application side by an appropriate division number (usually a fixed value), an approximation is made with polygonal lines and polyhedra, and the approximate polyline and polyhedron are sent to the graphic system. The graphic system is a system that displays the polygonal lines and the polyhedron. However, this method has the following disadvantages.

(1) The data amount of a polygonal line and a polyhedron for approximating the curved line and the curved surface is larger than that of the free curve and the curved surface. Therefore, the amount of communication between the application and the graphic system increases.

(2) The application side cannot grasp the size on the display displayed by the graphic system. Therefore, it is necessary to make the number of divisions peculiar to the system as described above, and it is inevitable that the number of divisions is excessive or excessive.

In order to solve the above problems, recently, a graphic system has appeared, which receives definition equations or control points of free-form curves and free-form surfaces, calculates the number of divisions, divides them into polygonal lines and polyhedra, and displays them. It was In such a system, the above two problems are solved, and further,
Clipping judgment on free-form curves / curved surfaces having convex hullness such as NURB (non-uniform rational B-spline) curves / curved surfaces and Bezier curves / curved surfaces, which will be described later, can be performed only from the control points, and overall performance can be improved.

Here, the convex hull property means the following. First, in the linear space X, the convex hull co of the point set {x _i | _i ∈ I}
{x _i | _i ∈ I} is the smallest convex set containing all points x _i . Where I is the set of subscripts. Especially when I is a finite set {1, 2,…, n}

[0007]

[Equation 1]

Is given by Next, the convex hull property will be described.
A free-form curve whose domain is the interval [a, b] is a set of control points
When determined from {P _i | i = 1, 2,…, n}, the curve c
Has a convex hull property if any point on the curve is co {P _i | i =
It is included in 1, 2,…, n}. That is, any t
For ∈ [a, b], c (t) ∈ co {P _i | i = 1, 2,…, n}. Similarly, rectangular area [a, b] × [c,
The free-form surface whose domain is d] is the set of control points {P _ij | i =
When 1, 2,…, m; j = 1, 2,…, n}, the surface s has convex hullness and any point on the surface is
co {P _ij | i = 1, 2,…, m; j = 1, 2,…, n}. That is, for any (u, v) ∈ [a, b] × [c, d], s (u, v) ∈ co {P _ij | i = 1, 2,…, m; j = 1 , 2,
…, N} is established.

Generally, since it is necessary to perform a complicated process to directly generate a curved line as a dot row, the curved line is approximated by a polygonal line and displayed. To generate a line segment, for example, DDA (Digital Differential An
alizer). That is, in the polygonal line approximation of the curve, appropriate N (≧ 1) dividing points are taken on the curve, and the starting point and the first dividing point, the first dividing point and the second dividing point, ... , The (N-1) th division point and the Nth division point,
This is done by connecting the Nth division point and the end point with a line segment.

Similarly, since a complicated process is required to directly generate a curved surface as a dot array, it is approximated and displayed by an open polyhedron or a closed polyhedron composed of a plurality of continuous polygons. To generate a string of dots that fills a polygon,
Similar to the case of generating line segments, there is a method of using a three-dimensional DDA.

That is, in the open polyhedron approximation or the closed polyhedron approximation of a curved surface, appropriate M × N (M, N ≧ 1) dividing points are taken on the curved surface, and a grid formed in this way is formed into a quadrangle or a diagonal line. It is approximated as a set of divided triangles.

As a method of obtaining the dividing points of the curve and the curved surface,
In the case where each coordinate of a curve and a curved surface is expressed by a mathematical expression based on a parameter, there is a method in which the value of the parameter is equally divided from the start point to the end point and the point at each parameter value is used as a division point. The conventional method for obtaining this number of divisions is "Su
rface algorithms using bounds on derivatives ”(D.
Filip, R. Magedson, and R. Markot, 1986) and “Real-t
ime rendering of trimmed surfaces ”(A. Rockwood,
K. Heaton, and T. Davis, 1989).

[0013]

The above-described free-form curve / curved surface display device is roughly classified into a processing part for approximating the free-form curve and the free-form surface with polygonal lines and polyhedra, and a processing part for displaying the polyline and polyhedron. It The processing part that approximates with polygonal lines and polyhedrons is the calculation of the number of divisions that divide the domain of free-form curves and free-form surfaces equally, the calculation of the vertex coordinates of polygonal lines and polyhedra created from the division points, and the unit normal for curved surfaces. Including the calculation of.

All of the above-mentioned conventional techniques for calculating the number of divisions include division by the weight (detailed later) of the control point, and the case where the weight is 0 was not assumed. However, it is necessary to perform perspective transformation when calculating the number of divisions according to the display size on the display, for example, even if the weight is not 0 when the free curve and free curved surface are defined, the weight becomes 0 after the perspective transformation. It can happen. Therefore, if a weight of 0 occurs, the graphic system will be down, or the number of divisions will be indeterminate, causing problems such as runaway or incorrect display.

It is an object of the present invention to provide a free curve / curved surface display method and apparatus in which the above-mentioned problems do not appear even if the weights of the control points of the free curve and free curved surface to be displayed are zero. .

[0016]

According to the present invention for solving the above problems, a set of control points of a curve to be displayed,
Storage means for holding a set of weights applied to the control points and an allowable error value regarding accuracy when the curve is approximated by a polygonal line, and a polygonal line for approximating the curve defined by the set of control points and the set of weights with a polygonal line In the graphic display system including the approximating means and the displaying means for displaying the polygonal line, the polygonal line approximating means determines the number of polygonal lines when approximating the curve by the polygonal line based on the tolerance value, the set of control points and the set of weights. It is possible to provide a division number calculation means for determining.

This division number calculation means separates the set of weights into a subset with a weight of 0 and a subset with a non-zero weight, and performs an operation for determining the number of broken lines for each subset. It is possible to provide a separation calculation means for performing the above and a calculation processing means for integrating the calculation results calculated for the subset by the separation calculation means to determine the number of broken lines.

The separation calculating means may be a means that has a process of performing division by weight for a subset whose weight is not 0, and does not have a process of performing division by weight for a portion whose weight is 0.

Further, a set of curved surface control points to be displayed,
A storage means for holding a set of weights applied to the control points and a tolerance value regarding accuracy when approximating a curved surface with a polyhedron, a set of control points, and a curved surface defined by the set of weights are approximated with a polyhedron In a graphic display system comprising a polyhedron approximating means for displaying a polyhedron, the polyhedron approximating means is a polyhedron when approximating a curved surface by a polyhedron based on a tolerance value, a set of control points and a set of weights. It is also possible to provide a division number calculating means for performing an operation for determining the number of.

The division number calculation means separates the set of weights into a subset with a weight of 0 and a subset with a non-zero weight, and determines the number of polyhedra for each subset. It is also possible to provide a separation calculation means for performing a calculation and an integration calculation means for determining the number of polyhedra by integrating the calculation results calculated by the separation calculation means for the subset.

The separating operation means may have a process of performing division by weight for a subset whose weight is not 0, and may not have a process of performing division by weight for a part whose weight is 0.

[0022]

When the curve is displayed by the broken line, the number of broken lines is determined based on the tolerance value, the set of control points and the set of weights.

In determining the number of polygonal lines, a set of weights is divided into a subset having a weight of 0 and a subset having a weight of not 0, and an operation for determining the number of polygonal lines for each subset. Do. In this way, the calculation results calculated for the subsets are integrated to determine the number of broken lines.

In the above calculation, a process of performing division by weight is performed in a subset whose weight is not 0, and a process of performing division by weight is not performed in a subset whose weight is 0. In the calculation, in the portion where the weight is 0, the division by the weight is not performed, but the division is performed by the minimum absolute value of the weight in the portion where the weight is not 0.

When approximating with a polyhedron when displaying a curved surface, the number of polyhedrons is determined based on a tolerance value, a set of control points, and a set of weights.

In determining the number of polyhedra, a set of weights is divided into a subset having a weight of 0 and a weight of 0.
And a calculation for determining the number of polyhedra for each subset. In this way, the number of polyhedra is determined by integrating the calculation results calculated for the subsets.

In the above calculation, a process of performing division by weight is included in a subset whose weight is not 0, and a process of performing division by weight is not included in a subset whose weight is 0. In the calculation, in the portion where the weight is 0, the division by the weight is not performed, but the division is performed by the minimum absolute value of the weight in the portion where the weight is not 0.

Therefore, even if the weight is 0, there is no problem that the graphic system goes down, or the number of divisions becomes indefinite and a runaway or correct display cannot be obtained.

[0029]

First, an example of the equipment configuration of a graphic system will be briefly described with reference to FIG.

Generally, a graphics system is a CPU
801, a main memory 802, a graphic subsystem 807, a CRT 811, and other devices, and a keyboard 803 and a mouse 8 for inputting data.
04, a stylus pen 805, various input devices such as a dial 812, a filing device 813 for storing and saving graphic data, etc. are connected via a system bus 806,
Exchange data. Also, the graphics subsystem 80
7 is a graphic LSI 808 and a frame memory 8
10 and the like, which are connected via a frame bus 809 to exchange data.

Next, using the graphic system having such a configuration, a general primitive is added to the CRT 811.
The process of displaying in will be described.

First, a keyboard 803, a mouse 804,
Data is input by various input devices such as a stylus pen 805 and a dial 812, and graphic data is created in the main memory 802 via the system bus 806. Or
Preliminarily created graphic data is transferred from the filing device 813 to the main memory 802 via the system bus 806. The CPU 801 converts the graphic data in the main memory 802 into a keyboard 803 and a mouse 80 if necessary.
4. Changes are made in accordance with instructions sent via the system bus 806 from various input devices such as the stylus pen 805 and the dial 812. To change the figure data, for example,
It includes geometrical operations such as rotation, parallel movement of figures, and change of viewpoint, and polygonal line approximation calculation of free curve and polyhedral approximation calculation of free curved surface, which will be described later. The CPU 801 sequentially sends the graphic data in the main memory 802 created as described above via the system bus 806 to the graphic subsystem 8.
Transfer to 07. In the graphic subsystem 807, the graphic LSI 808 displays the graphic data sent thereto on the CRT 811, so that the graphic data geometrically (ie, as coordinate data) and the graphic type (line segment, polygon, etc.) are given. ), Create a dot image. This processing includes, for example, processing for generating a dot row by DDA from coordinate data of a line segment, processing for generating a dot row for filling the inside of a polygon by three-dimensional DDA from polygonal coordinate data, and the like. The created dot image is transferred to the frame memory 810 via the frame memory bus 809, and finally, the graphic subsystem 807 converts the dot image of the frame memory 810 into an RGB signal or the like, and the CRT 811 displays it. .

Next, the free curve and the free curved surface will be briefly described.

FIG. 9 shows a section [a,
It is a figure showing the free curve c as an image from [b] to XYZ space. That is, the free curve 901 is the one-dimensional space T
Is defined as a mapping 902 from a certain area (domain) 903 of the above to the three-dimensional space XYZ. Map 902
Is usually defined as a polynomial with respect to the parameter t or a polynomial in the four-dimensional XYZW space for ease of handling by a computer, and then this is viewed as homogeneous coordinates and W
A rational polynomial that divides by coordinates is used. That is, the map 90
2 is

[0035]

[Equation 2]

Or

[0037]

[Equation 3]

The following description will be given, assuming that it is in the form of. Where the mappings x, y, z, w are polynomials in t, respectively.

The domain 903 of the mapping 902 is usually a closed section because it is easy for a computer to handle. Also, the map 9
02 is continuous everywhere, and handles a sufficiently smooth one (that is, a high-order derivative becomes continuous) except for a finite number of points.

As an example of a free curve satisfying the above requirements,
NURB (non-uniform rational B-spline) curve and ration
There is an al Bezier curve. NURB curves are piecewise ration
It is known that it can be converted to al Bezier curve. Therefore, the rational Bezier curve will be described below as an example. The rational Bezier curve is

[0041]

[Equation 4]

A family of functions defined by {B _ik } (Bernst
ein function family)

[0043]

[Equation 5]

It is a free curve defined as follows. The domain of this curve is [0, 1], the degree is K, and {P
[i] ＝ (x [i], y [i], z [i]) | 0 ≦ i ≦ K} is a set of control points,
{w [i] | 0 ≦ i ≦ K} is a set of weights.

FIG. 10 is a diagram showing the free-form surface s as an image from the rectangular area [a, b] × [c, d] having the UV parameter space to the XYZ space. That is, the free-form surface 1001 is defined as a mapping 1002 from a certain area (domain) 1002 of the two-dimensional space UV to the three-dimensional space XYZ. The map 1002 is a rational polynomial that is usually defined as a polynomial with respect to the parameters u and v, or a polynomial in the four-dimensional XYZW space, which is then viewed as homogeneous coordinates and divided by W coordinates because of easy handling by a computer. Is used. That is, the map 1002 is

[0046]

[Equation 6]

Or

[0048]

[Equation 7]

The following description will be given, assuming that it is in the form of. Where the mappings x, y, z, w are u and v, respectively.
Is a polynomial with respect to.

The domain 1003 of the mapping 1002 is usually
A rectangular area is adopted because it is easy to handle on a computer.
The mapping 1002 is continuous everywhere, and is sufficiently smooth except for a finite number of points (that is, high-order differential becomes continuous).

As an example of a free-form surface satisfying the above requirements,
There are NURB surfaces and rational Bezier surfaces. Similar to the NURB curve, it is known that the NURB surface can be converted piecewise into a rational Bezier surface. Therefore, here
The rational Bezier curved surface will be described below as an example. rat
ional Bezier surface uses Bernstein function family

[0052]

[Equation 8]

It is a free-form surface defined as follows. The domain of this surface is [0, 1] × [0, 1], the order is uK in the U direction and vK in the V direction, and {P [i] [j] ＝ (x [i] [j], y
[i] [j], z [i] [j]) | 0≤i≤uK, 0≤j≤vK} is a set of control points, {w [i] [j] | 0≤i≤uK, 0 ≤j≤vK} is a set of weights.

The display procedure of the free curve and the free curved surface as described above will be briefly described with reference to FIG.

First, the graphic system receives an instruction to start a display procedure from an application and performs various initial processing for displaying (step 101). next,
Each primitive data and attribute data (or command) is received (step 102), and the processing is branched according to the type of primitive (further, depending on the attribute) (step 103). If it is an end command, various end processes of the display procedure are performed (step 10
8) The process ends. If the input primitive is a free-form curve, it is first approximated to a polygonal line (step 104) and displayed (step 105). Similarly, when the input primitive is a free-form surface, it is first approximated to a polyhedron (step 106) and displayed (step 1).
07). After that, the processing from step 102 is repeated until the end command is input.

Here, step 104 and step 10
The polygonal line approximation and the polyhedron approximation in 6 are performed by the polygonal line approximation means and the polyhedron approximation means, and are performed by the CPU 801 in the above graphic system. The display in step 105 and step 107 is performed by the polygonal line display means and the polyhedron display means, and is performed by the graphic subsystem 807 in the above graphic system. This approximation and display processing is the same as that described above for general primitives other than free-form curves and curved surfaces.

Next, the construction of the broken line approximation means will be described with reference to FIG.

The polygonal line approximation means receives the set of control points of the free curve and the set of weights of each control point and the polygonal line approximation parameter, calculates the number of divisions by the division number calculation means 201, and calculates the division points by the coordinate calculation means 202. The coordinates are calculated, the division points are connected by the polygonal line creating means 203 to create a polygonal line, and the polygonal line data is transferred to the polygonal line display means.

The division number calculation means 201 calculates the division number as follows.

As an equation for calculating, "The grarphic
al processing of B-splines in ahighly dynamic envi
ronment (SSAbi-Ezzi, 1989, Rensselaer Polytechnic
Research Center) ", page 154,

[0061]

[Equation 32]

Based on here,

[0063]

[Equation 9]

And

[0065]

[Equation 10]

It is In particular, the curve c is the rati of (Equation 5)
If it is an onal Beier curve,

[0067]

[Expression 33]

Therefore, if all w [i] are positive, by convex hullability,

[0069]

[Equation 34]

It is possible to Therefore, instead of (Equation 32),

[0071]

[Equation 35]

The above is sufficient. The above is the above P. 1
54 is mainly described.

However, the above fact is "all w
There is a restriction "when [i] is positive. So we will consider removing this restriction.

[0074]

[Equation 36]

Consider as. Using the triangular inequality, (33)

[0076]

[Equation 37]

It can be modified as follows.

Regarding the first term, it can be assumed that the second term of the denominator has a sufficiently small absolute value in the range (w [i] w [j]> 0) considered in the numerator. Therefore, (Equation 3
The first term of 7) is

[0079]

[Equation 38]

It can be modified as follows.

Similarly, the second term of (Equation 37) is

[0082]

[Formula 39]

It can be modified as follows.

Regarding the third term,

[0085]

[Formula 40]

Where the second of the denominator of (Equation 40) is
It can be assumed that the absolute values of the terms and the fourth term are sufficiently small in the range considered by the numerator, and instead of ignoring either the first or the third term, the absolute value of w [j] If you decide to take the minimum value (currently w [j]> 0),

[0087]

[Formula 41]

It becomes Similarly, the fourth, fifth, and sixth terms

[0089]

[Equation 42]

[0090]

[Equation 43]

[0091]

[Equation 44]

It becomes

For the seventh term, the numerator is the maximum of each term,
It can be assumed that the denominator can be approximated by taking the minimum value of each term. Therefore,

[0094]

[Equation 45]

It can be

From the above, the number of divisions N is

[0097]

[Equation 46]

It can be determined as follows. here,

[0099]

[Equation 47]

Then, the division number N is

[0101]

[Equation 11]

It can be determined as follows.

FIG. 3 is a functional block diagram of the division number calculation means 201.

The division number calculation means 201 using the configuration of the division number calculation means 201 of FIG. 3 follows the following flow.
The number of divisions N determined by (Equation 11) is calculated.

(Step 3-1) Two counters 301
Initialize (counter1 and counter2) to 0. The degree register 303 (K) is initialized with the degree of the free curve.
Operation register 307 (MAX1 to MAX7 and Winv1 and
Winv2) is initialized to 0.0. The polygonal line approximation parameter register 308 (TOL) is initialized with the polygonal line approximation parameter.

(Step 3-2) Set variable i to counter1,
When the variable j is counter2, x [i-1], y [i-1], z [i-1], w [i- read from K and counter1, the set 304 of control points and the set 305 of weights. 1]; x [i], y
[i], z [i], w [i]; x [i + 1], y [i + 1], z [i + 1], w [i + 1]; x
From [j], y [j], z [j], w [j], the following (step 3-2-1) to (step 3-2-8) are performed using the norm calculation device 306.

(Step 3-2-1) Three-dimensional vector A
Norm of [i] w [j] -B [i] P [j] (represent this value by NORM)
To calculate. Where A [i] and B [i] are

[0108]

[Equation 9]

And

[0110]

[Equation 10]

It is assumed that

(Step 3-2-2) When w [i] w [j]> 0, the value obtained by dividing NORM by w [i] w [j] is set again as NORM and is set to MAX1 in the operation register 307. If it is larger than the comparison value, NORM is stored in MAX1. When w [i]> 0, w [i]
If the reciprocal of is larger than Winv1 in the operation register 307, the reciprocal of w [i] is stored in Winv1. When w [i] <0, if the reciprocal of -w [i] is larger than Winv2 in the operation register 307, the reciprocal of -w [i] is stored in Winv2.

(Step 3-2-3) When w [i] w [j] <0, the value obtained by dividing NORM by -w [i] w [j] is set again as NORM and MAX2 in the operation register 307 is set. Greater than, NO
Store RM in MAX2. If w [i]> 0 and the reciprocal of w [i] is greater than Winv1 in the operation register 307, Win
Store the reciprocal of w [i] in v1. -w [i] when w [i] <0
If the reciprocal of is larger than Winv2 in the operation register 307, the reciprocal of -w [i] is stored in Winv2.

(Step 3-2-4) w [i]> 0, w [j] = 0
At this time, the value obtained by dividing NORM by w [i] is set again as NORM, and if it is larger than MAX3 in the operation register 307, if NORM is stored, NORM is stored in MAX3. If the reciprocal of w [i] is larger than Winv1 in the operation register 307, w [i] is set in Winv1.
Stores the reciprocal of.

(Step 3-2-5) w [i] <0, w [j] = 0
, The value obtained by dividing NORM by -w [i] is NORM again.
If it is larger than MAX4 in the operation register 307, NORM is stored in MAX4. -If the reciprocal of w [i] is greater than Winv2 in the operation register 307, then Winv2 is set to-
Stores the reciprocal of w [i].

(Step 3-2-6) w [i] = 0, w [j]> 0
At this time, the value obtained by dividing NORM by w [j] is set again as NORM, and if it is larger than MAX5 in the arithmetic register 307, if NORM is stored in NORM, it is stored in MAX5.

(Step 3-2-7) w [i] = 0, w [j] <0
, The value obtained by dividing NORM by -w [j] is NORM again.
If it is larger than MAX6 in the operation register 307, NORM is stored in MAX6.

(Step 3-2-8) w [i] = 0, w [j] = 0
At this time, if NORM is larger than MAX7 in the operation register 307, NORM is stored in MAX7.

(Step 3-3) counter1 and counte
The following processing is performed by using the increment device 302 for r2.

If counter2 <K, counter2 is incremented by 1.

If counter2 = K and counter1 <K, co
Initialize unter2 with 0 and increment counter1 by 1.

After the above processing, the processing is repeated from (Step 3-2).

If counter2 = K and counter1 = K, the process proceeds to step 3-4.

(Step 3-4) From MAX1 to MAX7, Winv1, Winv2, and TOL obtained as described above, the division number calculation unit 309 is used to calculate the division number N of the free curve by (Equation 11). be able to.

The division number calculating means 201 can also calculate the division number as follows.

As a formula for calculating, "Surface algo
rithms using bounds on derivatives ”(D. Filip, R.
Based on Theorem 2 by Magedson, and R. Markot, 1986). From this Theorem 2,

[0127]

[Equation 48]

It is easy to see that the number of divisions can be calculated with. here,

[0129]

[Equation 12]

And

[0131]

[Equation 13]

It is However, from the condition for satisfying Theorem 2, (Equation 48) is satisfied only when all w [i] are positive.

Here, with respect to (Equation 48), (Equation 32)
The same transformation as that of deriving (Equation 46) from

[0134]

[Equation 49]

[0135]

[Equation 50]

Putting the number of divisions N,

[0137]

[Equation 14]

It can be calculated by

FIG. 4 is a functional block diagram of the division number calculation means 201 for calculating the division number based on (Equation 14).

The division number calculation means 201 using the configuration of the division number calculation means of FIG. 4 calculates the division number according to the following flow.

(Step 4-1) Three counters 401
Initialize (counter1, counter2, counter3) to 0. The degree register 403 (K) is initialized with the degree of the free curve. Operation register 407 (MAX1 to MAX15
And Winv1 and Winv2) are initialized to 0.0. The polygonal line approximation parameter register 408 (TOL) is initialized with the polygonal line approximation parameter.

(Step 4-2) Set the variable i to counter1,
When the variable j is counter2 and the variable k is counter3, x [i-2], y [i-2], z [i read from K and counter1, the control point set 404 and the weight set 405.
-2], w [i-2]; x [i-1], y [i-1], z [i-1], w [i-1]; x [i],
y [i], z [i], w [i]; x [i + 1], y [i + 1], z [i + 1], w [i + 1];
x [i + 2], y [i + 2], z [i + 2], w [i + 2]; x [j], y [j], z [j],
w [j]; x [k-1], y [k-1], z [k-1], w [k-1]; x [k], y [k],
z [k], w [k]; x [k + 1], y [k + 1], z [k + 1], w [k + 1]
The following (step 4-
2-1) to (Step 4-2-16) are performed.

(Step 4-2-1) Three-dimensional vector
(C [i] w [j] -D [i] P [j]) w [k] -2 (A [i] w [j] -B [i] P [j]) B [k]
Calculate the norm of (expressed as NORM). Here, A [i] and B [i] are (Equation 9) and (Equation 10), and C [i] and D [i] are

[0144]

[Equation 12]

And

[0146]

[Equation 13]

It is assumed that

(Step 4-2-2) w [i] w [j] w [k]> 0
, NORM divided by w [i] w [j] w [k] again
NORM is set, and if it is larger than MAX1 in the operation register 407, NORM is stored in MAX1. When w [i]> 0, if the reciprocal of w [i] is larger than Winv1 in the operation register 407, the reciprocal of w [i] is stored in Winv1. w [i] <
When 0, the reciprocal of -w [i] is Win in the operation register 407.
If it is larger than v2, the reciprocal of -w [i] is stored in Winv2.

(Step 4-2-3) w [i] w [j] w [k] <0
Then NORM divided by -w [i] w [j] w [k] again
If NORM is set and is larger than MAX2 in the operation register 407, NORM is stored in MAX2. When w [i]> 0, w
If the reciprocal of [i] is larger than Winv1 in the operation register 407, the reciprocal of w [i] is stored in Winv1. When w [i] <0, the reciprocal of -w [i] is Winv2 in the operation register 407.
If larger, store the reciprocal of -w [i] in Winv2.

(Step 4-2-4) w [i] w [j]> 0, w
When [k] = 0, the value obtained by dividing NORM by w [i] w [j] is set again as NORM, and if it is larger than MAX3 in the operation register 407, NORM is stored in MAX3. When w [i]> 0, if the reciprocal of w [i] is larger than Winv1 in the operation register 407, the reciprocal of w [i] is stored in Winv1. w [i] <
When 0, the reciprocal of -w [i] is Win in the operation register 407.
If it is larger than v2, the reciprocal of -w [i] is stored in Winv2.

(Step 4-2-5) w [i] w [j] <0, w
When [k] = 0, the value obtained by dividing NORM by -w [i] w [j] is set again as NORM, and if it is larger than MAX4 in the operation register 407, NORM is stored in MAX4. When w [i]> 0, if the reciprocal of w [i] is larger than Winv1 in the operation register 407, the reciprocal of w [i] is stored in Winv1. w [i]
When <0, the reciprocal of -w [i] is W in the operation register 407.
If it is larger than inv2, the reciprocal of -w [i] is stored in Winv2.

(Step 4-2-6) w [k] w [i]> 0, w
When [j] = 0, the value obtained by dividing NORM by w [k] w [i] is set again as NORM, and if it is larger than MAX5 in the operation register 407, NORM is stored in MAX5. When w [i]> 0, if the reciprocal of w [i] is larger than Winv1 in the operation register 407, the reciprocal of w [i] is stored in Winv1. w [i] <
When 0, the reciprocal of -w [i] is Win in the operation register 407.
If it is larger than v2, the reciprocal of -w [i] is stored in Winv2.

(Step 4-2-7) w [k] w [i] <0, w
When [j] = 0, the value obtained by dividing NORM by -w [k] w [i] is set again as NORM, and if it is larger than MAX6 in the operation register 407, NORM is stored in MAX6. When w [i]> 0, if the reciprocal of w [i] is larger than Winv1 in the operation register 407, the reciprocal of w [i] is stored in Winv1. w [i]
When <0, the reciprocal of -w [i] is W in the operation register 407.
If it is larger than inv2, the inverse of -w [i] is stored in Winv2.

(Step 4-2-8) w [i] = 0, w [j] w
When [k]> 0, NORM is divided by w [j] w [k] to obtain NORM again, and if it is larger than MAX7 in the arithmetic register 407, NORM is stored in MAX7.

(Step 4-2-9) w [i] = 0, w [j] w
When [k] <0, the value obtained by dividing NORM by -w [j] w [k] is set as NORM again, and if it is larger than MAX8 in the operation register 407, NORM is stored in MAX8.

(Step 4-2-10) w [i] = 0, w [j]>
When 0, w [k] = 0, the value obtained by dividing NORM by w [i] is set again as NORM, and if it is larger than MAX9 in the operation register 407, NORM is stored in MAX9. If the reciprocal of w [i] is greater than Winv1 in the operation register 407, Win
Store the reciprocal of w [i] in v1.

(Step 4-2-11) w [i] <0, w [j] =
When 0, w [k] = 0, the value obtained by dividing NORM by -w [i] is set again as NORM, and if it is larger than MAX10 in the operation register 407, NORM is stored in MAX10. -w [i]
If the reciprocal of is larger than Winv2 in the operation register 407, the reciprocal of -w [i] is stored in Winv2.

(Step 4-2-12) w [i] = 0, w [j]>
When 0, w [k] = 0, the value obtained by dividing NORM by w [j] is set again as NORM, and if it is larger than MAX11 in the arithmetic register 407, NORM is stored in MAX11.

(Step 4-2-13) w [i] = 0, w [j] <
When 0, w [k] = 0, the value obtained by dividing NORM by -w [j] is set again as NORM, and if it is larger than MAX12 in the arithmetic register 407, NORM is stored in MAX12.

(Step 4-2-14) w [i] = 0, w [j] =
When 0 and w [k]> 0, the value obtained by dividing NORM by w [k] is set as NORM again, and if it is larger than MAX13 in the operation register 407, NORM is stored in MAX13.

(Step 4-2-15) w [i] = 0, w [j] =
When 0, w [k] <0, NORM is divided by -w [k] to obtain NORM again, and if it is larger than MAX14 in the arithmetic register 407, NORM is stored in MAX14.

(Step 4-2-16) w [i] = 0, w [j] =
When 0 and w [k] = 0, NORM is the M in the operation register 407.
If it is larger than AX15, store NORM in MAX15.

(Step 4-3) counter1 and counte
The following processing is performed on the r2 and counter3 using the increment device 402.

If counter3 <K, counter3 is incremented by 1.

If counter3 = K and counter2 <K, co
Initialize unter3 with 0 and increment counter2 by 1.

Counter3 = K and counter2 = K and coun
If ter1 <K, counter3 and counter2 are initialized to 0 and counter1 is incremented by 1.

After the above processing, the processing is repeated from (step 4-2).

Counter3 = K and counter2 = K and coun
If ter1 = K, proceed to (Step 4-4).

(Step 4-4) MAX1 to MAX15, Winv1, Winv2, and TO obtained as described above
From L, the division number N of the free curve can be calculated by (Equation 14) using the division number calculation device 409.

Next, the coordinate calculating means 202 determines the number of divisions N obtained by any one of the above-mentioned number of division calculating means 201.
, The parameters of the free curve t = 0, 1 / N, 2 / N,…, 1
Calculate the coordinate values of. Horne can be used to calculate the coordinate values.
de Caste for r method, forward difference method, Bezier curve
There is ljau algorithm.

For Horner's method, see “Curves and
surface for computer aided geometric design, secon
d edition ”(G. Farin, 1990, Academic Press), P.47, etc., and forward difference method, P.80-P.81, etc., de Ca
The steljau algorithm is described on page 29 of the same book.

Next, the polygonal line creating means 203 successively connects the division points, that is, the points of the free curve parameters t = 0, 1 / N, 2 / N, ... The polygonal line data is created together with the coordinates.

Finally, the polygonal line data created by the polygonal line creating means 203 is transferred to the polygonal line display means and displayed on the graphic display.

Next, the structure of the polyhedral approximation means will be described with reference to FIG.

The polyhedral approximation means receives the set of control points of the free-form surface, the set of weights of each control point and the polyhedral approximation parameter, calculates the number of divisions by the division number calculation means 501, and calculates the division points by the coordinate calculation means 502. , The unit normal calculation unit 503 calculates the unit normal of the division point, the polyhedral creation unit 504 connects the division points to create a polyhedron, and transfers the polyhedron data to the polyhedron display unit.

The division number calculation means 501 calculates the division number as follows.

As an equation for calculating, "The grarphic
al processing of B-splines in ahighly dynamic envi
ronment (SSAbi-Ezzi, 1989, Rensselaer Polytechnic
Research Center) ", P. 155. Based on the equations (32) to (11),
The above P. The following (Equation 19) and (Equation 20) can be derived from the equation of 155, and the number of divisions of the free-form surface in the U direction (this value is represented by uN) is (Equation 19).
The number of divisions in the V direction (this value is represented by vN) can be calculated by (Equation 20).

[0178]

[Formula 19]

[0179]

[Equation 20]

FIG. 6 is a functional block diagram of the division number calculation means 501.

The division number calculation means 501 using the configuration of the division number calculation means 501 of FIG. 6 calculates the division number according to the following flow.

(Step 6-1) Four Counters 601
(Ucounter1 and ucounter2 and vcounter1 and vco
unter2) is initialized to 0. Two degree register 60
Initialize 3 (uK and vK) by the degree of the free-form surface in the U and V directions. Operation register 607 (uMAX1 to uMA
X7 and vMAX1 to vMAX7 and Winv1 and Winv2)
Initialize with 0.0. The two polyhedron approximation parameter registers 608 (uTOL and vTOL) are initialized with the polyhedron approximation parameters stored in advance in the memory.

(Step 6-2) Set variable i to ucounter
1, variable j is vcounter1, variable k is ucounter2, variable
When l is vcounter2, uK and vK and ucounter
1 and vcounter1, x [i-1] [j], y [i-1] [j], which are read from the set 604 of control points and the set 605 of weights,
z [i-1] [j], w [i-1] [j]; x [i] [j-1], y [i] [j-1], z [i]
[j-1], w [i] [j-1]; x [i] [j], y [i] [j], z [i] [j], w [i]
[j]; x [i] [j + 1], y [i] [j + 1], z [i] [j + 1], w [i] [j + 1]; x
[i + 1] [j], y [i + 1] [j], z [i + 1] [j], w [i + 1] [j]; x [k]
From [l], y [k] [l], z [k] [l], w [k] [l], the following (step 6-2-1) to (step Perform 6-2-8).

(Step 6-2-1) Three-dimensional vector E
[i] [j] w [k] [l] -F [i] [j] P [k] [l] norm (this value is uN
ORM) and G [i] [j] w [k] [l] -H [i] [j] P [k] [l] norm (this value is expressed as vNORM). Where E
[i] [j] and F [i] [j] and G [i] [j] and H [i] [j] are

[0185]

[Equation 15]

And

[0187]

[Equation 16]

And

[0189]

[Equation 17]

And

[0191]

[Equation 18]

It is assumed that

(Step 6-2-2) w [i] [j] w [k] [l]>
When it is 0, the value obtained by dividing uNORM by w [i] [j] w [k] [l] is newly set as uNORM, and if it is larger than uMAX1 in the operation register 607, it is stored in uMAX1. At this time, the value obtained by dividing vNORM by w [i] [j] w [k] [l] is set again as vNORM, and if it is larger than vMAX1 in the operation register 607, vNORM is stored in vMAX1. w [i] [j]
When> 0, the reciprocal of w [i] [j] is stored in the arithmetic register 607.
If it is larger than Winv1, the reciprocal of w [i] [j] is stored in Winv1. When w [i] [j] <0, if the reciprocal of -w [i] [j] is larger than Winv2 in the operation register 607, Winw2 will show -w.
Stores the reciprocal of [i] [j].

(Step 6-2-3) w [i] [j] w [k] [l] <
When it is 0, the value obtained by dividing uNORM by -w [i] [j] w [k] [l] is set again as uNORM. If it is larger than uMAX2 in the operation register 607, uNORM is stored in uMAX2. or,
At this time, the value obtained by dividing vNORM by -w [i] [j] w [k] [l] is set again as vNORM, and if it is larger than vMAX2 in the operation register 607, vNORM is stored in vMAX2. w
When [i] [j]> 0, the reciprocal of w [i] [j] is the arithmetic register 60
If it is larger than Winv1 in 7, the reciprocal of w [i] [j] is stored in Winv1. If w [i] [j] <0, and the reciprocal of -w [i] [j] is greater than Winv2 in the operation register 607, Winv2
Store the reciprocal of -w [i] [j] in.

(Step 6-2-4) w [i] [j]> 0, w [k]
When [l] = 0, the value obtained by dividing uNORM by w [i] [j] is set again as uNORM, and if it is larger than uMAX3 in the operation register 607, uNORM is stored in uMAX3. Also, at this time, the value obtained by dividing vNORM by w [i] [j] is updated to vNOR.
M, and vNORM is stored in vMAX3 if larger than vMAX3 in the operation register 607. If the reciprocal of w [i] [j] is greater than Winv1 in the operation register 607,
Store the reciprocal of w [i] [j] in inv1.

(Step 6-2-5) w [i] [j] <0, w [k]
When [l] = 0, the value obtained by dividing uNORM by -w [i] [j] is set again as uNORM, and if it is larger than uMAX4 in the operation register 607, it is stored in uMAX4. In addition, at this time, the value obtained by dividing vNORM by -w [i] [j] is updated to vNO
RM, and vNORM is stored in vMAX4 if it is larger than vMAX4 in the operation register 607. -If the reciprocal of w [i] [j] is greater than Winv2 in the operation register 607, W
Store the reciprocal of -w [i] [j] in inv2.

(Step 6-2-6) w [i] [j] = 0, w [k]
When [l]> 0, the value obtained by dividing uNORM by w [k] [l] is set again as uNORM. If it is larger than uMAX5 in the operation register 607, uNORM is stored in uMAX5. Also, at this time, the value obtained by dividing vNORM by w [k] [l] is re-established as vNOR.
M, and vNORM is stored in vMAX5 if it is larger than vMAX5 in the operation register 607.

(Step 6-2-7) w [i] [j] = 0, w [k]
When [l] <0, the value obtained by dividing uNORM by -w [k] [l] is set as uNORM again, and if it is larger than uMAX6 in the operation register 607, uNORM is stored in uMAX6. Also, at this time, the value obtained by dividing vNORM by -w [k] [l] is updated to vNO
RM, and vNORM is stored in vMAX6 if larger than vMAX6 in the operation register 607.

(Step 6-2-8) w [i] [j] = 0, w [k]
When [l] = 0, uNORM is the uMAX value in the arithmetic register 607.
If it is greater than 7, store uNORM in uMAX7. Also, at this time, vNORM is v in the operation register 607.
If it is larger than MAX7, store vNORM in vMAX7.

(Step 6-3) ucounter1 and ucoun
The following processing is performed using the increment device 602 for ter2, vcounter1 and vcounter2.

If vcounter2 <vK, vcounter2 is incremented by 1.

If vcounter2 = vK and ucounter2 <uK, vcounter2 is initialized to 0 and ucounter2 is incremented by 1.

Vcounter2 = vK and ucounter2 = uK and
If vcounter1 <vK, set vcounter2 and ucounter2 to 0
Initialize with and increment vcounter1 by 1.

Vcounter2 = vK and ucounter2 = uK and
If vcounter1 = vK and ucounter1 <uK, vcounter2
, Ucounter2 and vcounter1 are initialized to 0, and uco
Increment unter1 by 1.

After the above processing, the processing is repeated from (Step 6-2).

Vcounter2 = vK and ucounter2 = uK and
If vcounter1 = vK and ucounter1 = uK (step 6)
Go to -4).

(Step 6-4) uMAX1 to uMAX7 and Winv1 and Winv2 and uTOL obtained as described above
Therefore, by using the division number calculation unit 609,
The number of direction divisions (uN) can be calculated by (Equation 19).

Similarly, from vMAX1 to vMAX7 and Winv1 and Winv2 and vTOL, the number of divisions (vN) of the free-form surface in the V direction is calculated by using the division number calculation unit 609 (Equation 20).
Can be calculated by

The division number calculation means 501 can also calculate the division number as follows.

As a formula for calculating, "Surface algo
rithms using bounds on derivatives ”(D. Filip, R.
Based on the formula in Section 3.1 of Magedson, and R. Markot, 1986).

[0211]

[Equation 27]

And

[0213]

[Equation 28]

And

[0215]

[Equation 29]

When

[0217]

[Equation 30]

And

[0219]

[Equation 31]

Then, in the same way as (Equation 14) is derived from (Equation 48), (Equation 30) is obtained from the equation in the above section 3.1.
And (Equation 31) can be derived, and the free-form surface U
The number of direction divisions (this value is represented by uN) is (Equation 30),
The number of V-direction divisions of a free-form surface (this value is represented by vN) can be calculated using (Equation 31).

FIG. 7 is a functional block diagram of the division number calculation means 501 for calculating the number of divisions in (Equation 30) and (Equation 31).

The division number calculation means 501 using the configuration of the division number calculation means of FIG. 7 calculates the division number according to the following flow.

(Step 7-1) Six counters 701
(Ucounter1 and ucounter2 and ucounter3 and vco
unter1 and vcounter2 and vcounter3) are initialized to 0. Two degree registers 703 (uK and vK) are initialized with the degree in the U direction and the degree in the V direction of the free-form surface. Operation register 707 (uuMAX1 to uuMAX15 and u
vMAX1 to uvMAX15vvMAX1 to vvMAX15 and Winv1
And Winv2) is initialized to 0.0. Two polyhedron approximation parameter registers 708 (uTOL and vTOL) are initialized with polyhedron approximation parameters.

(Step 7-2) Set variable i to ucounter
1, variable j is vcounter1, variable k is ucounter2, variable
l for vcounter2, variable p for ucounter3, variable q for vc
ounter3, uK and vK and ucounter1 and v
x [i-2] [j], y [i-2] [j], z [i-2] read from counter1, control point set 704, and weight set 705
[j], w [i-2] [j]; x [i-1] [j-1], y [i-1] [j-1], z [i-1] [j
-1], w [i-1] [j-1]; x [i-1] [j], y [i-1] [j], z [i-1] [j],
w [i-1] [j]; x [i-1] [j + 1], y [i-1] [j + 1], z [i-1] [j + 1],
w [i-1] [j + 1]; x [i] [j-2], y [i] [j-2], z [i] [j-2], w [i]
[j-2]; x [i] [j-1], y [i] [j-1], z [i] [j-1], w [i] [j-1];
x [i] [j], y [i] [j], z [i] [j], w [i] [j]; x [i] [j + 1], y
[i] [j + 1], z [i] [j + 1], w [i] [j + 1]; x [i] [j + 2], y [i] [j +
2], z [i] [j + 2], w [i] [j + 2]; x [i + 1] [j-1], y [i + 1] [j-
1], z [i + 1] [j-1], w [i + 1] [j-1]; x [i + 1] [j], x [i + 1]
[j], y [i + 1] [j], z [i + 1] [j], w [i + 1] [j]; x [i + 1] [j + 1],
y [i + 1] [j + 1], z [i + 1] [j + 1], w [i + 1] [j + 1]; x [i + 2] [j], y
[i + 2] [j], z [i + 2] [j], w [i + 2] [j]; x [k] [l], y [k] [l],
z [k] [l], w [k] [l]; x [p-1] [q], y [p-1] [q], z [p-1] [q],
w [p-1] [q]; x [p] [q-1], y [p] [q-1], z [p] [q-1], w [p]
[q-1]; x [p] [q], y [p] [q], z [p] [q], w [p] [q]; x [p] [q +
1], y [p] [q + 1], z [p] [q + 1], w [p] [q + 1]; x [p + 1] [q], y
From [p + 1] [q], z [p + 1] [q], w [p + 1] [q], the following (Step 7-2-1) to ( Perform step 7-2-16).

(Step 7-2-1) Three-dimensional vector
(I [i] [j] w [k] [l] -J [i] [j] P [k] [l]) w [p] [q] -2 (E [i] [j] w
[k] [l] -F [i] [j] P [k] [l]) The norm of F [p] [q] (this value is u
uNORM) and (K [i] [j] w [k] [l] -L [i] [j] P [k] [l])
w [p] [q]-(E [i] [j] w [k] [l] -F [i] [j] P [k] [l]) H [p] [q]-(G
[i] [j] w [k] [l] -H [i] [j] P [k] [l]) The norm of F [p] [q] (this value is represented by uvNORM) and (M [ i] [j] w [k] [l] -N [i] [j]
P [k] [l]) w [p] [q] -2 (G [i] [j] w [k] [l] -H [i] [j] P [k] [l]) H
Calculate the norm of [p] [q] (this value is represented by vvNORM). Where E [i] [j] and F [i] [j] and G [i] [j] and
H [i] [j] is Equation 15 and Equation 16 and Equation 17 and Equation 18, and I [i] [j] and J [i] [j] and K [i] [j] and L [i ]
[j] and M [i] [j] and N [i] [j] are

[0226]

[Equation 21]

And

[0228]

[Equation 22]

And

[0230]

[Equation 23]

And

[0232]

[Equation 24]

And

[0234]

[Equation 25]

And

[0236]

[Equation 26]

It is assumed that

(Step 7-2-2) w [i] [j] w [k] [l] w
When [p] [q]> 0, set uuNORM to w [i] [j] w [k] [l] w [p] [q]
The value divided by is set again as uuNORM, and the operation register 70
If it is larger than uuMAX1 in 7, uuNORM is uuMA
Store in X1. At this time, vNORM is set to w [i] [j] w [k]
The value divided by [l] w [p] [q] is set as uvNORM again, and if it is larger than uvMAX1 in the operation register 707, if it is larger, uv
Store NORM in uvMAX1. Also, at this time, vvNORM
The value divided by w [i] [j] w [k] [l] w [p] [q] is again vvNORM
Then, if it is larger than vvMAX1 in the operation register 707, vvNORM is stored in vvMAX1. w [i] [j]> 0
, The reciprocal of w [i] [j] is Win in the operation register 707.
If it is larger than v1, the reciprocal of w [i] [j] is stored in Winv1. When w [i] [j] <0, if the reciprocal of -w [i] [j] is greater than Winv2 in the operation register 707, Winv2 contains -w [i].
Stores the reciprocal of [j].

(Step 7-2-3) w [i] [j] w [k] [l] w
When [p] [q] <0, set uuNORM to -w [i] [j] w [k] [l] w [p] [q]
The value divided by is set again as uuNORM, and the operation register 7
If it is larger than uuMAX2 in 07, set uuNORM to uu
Store in MAX1. Also, at this time, set uvNORM to -w [i] [j] w
The value divided by [k] [l] w [p] [q] is set to uvNORM again, and if it is larger than uvMAX2 in the operation register 707,
Store uvNORM in uvMAX1. Also, at this time, vvNORM
The value obtained by dividing by -w [i] [j] w [k] [l] w [p] [q] is vvN
If it is ORM and it is larger than vvMAX2 in the operation register 707, vvNORM is stored in vvMAX2. w [i] [j]>
When 0, the reciprocal of w [i] [j] is W in the operation register 707.
If it is larger than inv1, the reciprocal of w [i] [j] is stored in Winv1. When w [i] [j] <0, if the reciprocal of -w [i] [j] is larger than Winv2 in the operation register 707, Winw2 displays -w.
Stores the reciprocal of [i] [j].

(Step 7-2-4) w [i] [j] w [k] [l]>
When 0, w [p] [q] = 0, the value obtained by dividing uuNORM by w [i] [j] w [k] [l] is set as uuNORM again, and the arithmetic register 707
UuNORM is larger than uuMAX3 in
To store. At this time, uvNORM is set to w [i] [j] w [k]
The value divided by [l] is set as uvNORM again. If it is larger than uvMAX3 in the operation register 707, if it is larger, uvNORM is
Store in uvMAX3. At this time, vvNORM is set to w [i] [j]
The value divided by w [k] [l] is set again as vvNORM, and if it is larger than vvMAX3 in the operation register 707, it is vvNORM.
Is stored in vvMAX3. When w [i] [j]> 0, w [i] [j]
If the reciprocal of is larger than Winv1 in the operation register 707, the reciprocal of w [i] [j] is stored in Winv1. w [i] [j] <0
, The reciprocal of -w [i] [j] is Wi in the operation register 707.
If it is larger than nv2, the reciprocal of -w [i] [j] is stored in Winv2.

(Step 7-2-5) w [i] [j] w [k] [l] <
When 0, w [p] [q] = 0, the value obtained by dividing uuNORM by -w [i] [j] w [k] [l] is set again as uuNORM, and the operation register 707
UuNORM is uuMAX4
To store. At this time, set uvNORM to -w [i] [j] w [k]
The value divided by [l] is set as uvNORM again, and if it is larger than uvMAX4 in the operation register 707, if it is larger, then uvNORM is changed to
Store in uvMAX4. Also, at this time, set vvNORM to -w [i] [j]
The value divided by w [k] [l] is set to vvNORM again, and if it is larger than vvMAX4 in the operation register 707, it is vvNORM.
Is stored in vvMAX4. When w [i] [j]> 0, w [i] [j]
If the reciprocal of is larger than Winv1 in the operation register 707, the reciprocal of w [i] [j] is stored in Winv1. w [i] [j] <0
, The reciprocal of -w [i] [j] is Wi in the operation register 707.
If it is larger than nv2, the reciprocal of -w [i] [j] is stored in Winv2.

(Step 7-2-6) w [p] [q] w [i] [j]>
When 0 and w [k] [l] = 0, the value obtained by dividing uuNORM by w [p] [q] w [i] [j] is set again as uuNORM, and the operation register 707
UuNORM is uuMAX5
To store. At this time, uvNORM is set to w [p] [q] w [i]
The value divided by [j] is set as uvNORM again, and if it is larger than uvMAX5 in the operation register 707, if uvNORM is larger,
Store in uvMAX5. At this time, vvNORM is set to w [p] [q]
The value divided by w [i] [j] is set to vvNORM again, and if it is larger than vvMAX5 in the operation register 707, it is vvNORM.
Is stored in vvMAX5. When w [i] [j]> 0, w [i] [j]
If the reciprocal of is larger than Winv1 in the operation register 707, the reciprocal of w [i] [j] is stored in Winv1. w [i] [j] <0
, The reciprocal of -w [i] [j] is Wi in the operation register 707.
If it is larger than nv2, the reciprocal of -w [i] [j] is stored in Winv2.

(Step 7-2-7) w [p] [q] w [i] [j] <
When 0, w [k] [l] = 0, the value obtained by dividing uuNORM by -w [p] [q] w [i] [j] is set again as uuNORM, and the operation register 707
UuNORM is uuMAX6
To store. At this time, set uvNORM to -w [p] [q] w [i]
The value divided by [j] is set as uvNORM again, and if it is larger than uvMAX6 in the operation register 707, if uvNORM is larger,
Store in uvMAX6. Also, at this time, vvNORM is -w [p] [q]
The value divided by w [i] [j] is set to vvNORM again, and if it is larger than vvMAX6 in the operation register 707 and is larger, vvNORM
Is stored in vvMAX6. When w [i] [j]> 0, w [i] [j]
If the reciprocal of is larger than Winv1 in the operation register 707, the reciprocal of w [i] [j] is stored in Winv1. w [i] [j] <0
, The reciprocal of -w [i] [j] is Wi in the operation register 707.
If it is larger than nv2, the reciprocal of -w [i] [j] is stored in Winv2.

(Step 7-2-8) w [i] [j] = 0, w [k]
When [l] w [p] [q]> 0, the value obtained by dividing uuNORM by w [k] [l] w [p] [q] is set again as uuNORM, and the operation register 707
UuNORM is larger than uuMAX7 in
To store. At this time, uvNORM is set to w [k] [l] w [p]
The value divided by [q] is set to uvNORM again, and if it is larger than uvMAX7 in the operation register 707, then uvNORM is
Store in uvMAX7. At this time, vvNORM is set to w [k] [l]
The value divided by w [p] [q] is set again as vvNORM, and if it is larger than vvMAX7 in the operation register 707, it is vvNORM.
Is stored in vvMAX7.

(Step 7-2-9) w [i] [j] = 0, w [k]
When [l] w [p] [q] <0, the value obtained by dividing uuNORM by -w [k] [l] w [p] [q] is set again as uuNORM, and the operation register 707
If it is larger than uuMAX8 in the above, set uuNORM to uuMAX8
To store. At this time, set uvNORM to -w [k] [l] w [p]
The value divided by [q] is set as uvNORM again, and if it is larger than uvMAX8 in the operation register 707, if it is larger, then uvNORM is set.
Store in uvMAX8. Also, at this time, set vvNORM to -w [k] [l]
The value divided by w [p] [q] is set again as vvNORM, and if it is larger than vvMAX8 in the operation register 707, it is vvNORM.
Is stored in vvMAX8.

(Step 7-2-10) w [i] [j]> 0, w [k]
When [l] = 0 and w [p] [q] = 0, the value obtained by dividing uuNORM by w [i] [j] is set again as uuNORM. If it is larger than uuMAX9 in the operation register 707, uuNORM UuMAX9
To store. At this time, the value obtained by dividing uvNORM by w [i] [j] is set as uvNORM again, and the value in the operation register 707
If it is larger than uvMAX9, store uvNORM in uvMAX9. Also, at this time, the value obtained by dividing vvNORM by w [i] [j] is set again as vvNORM, and v in the operation register 707
If it is larger than vMAX9, store vvNORM in vvMAX9. The reciprocal of w [i] [j] is Wi in the operation register 707.
If it is larger than nv1, the reciprocal of w [i] [j] is stored in Winv1.

(Step 7-2-11) w [i] [j] <0, w [k]
When [l] = 0 and w [p] [q] = 0, the value obtained by dividing uuNORM by -w [i] [j] is set again as uuNORM, and if it is larger than uuMAX10 in the operation register 707, uuNORM to uuMAX10
To store. At this time, the value obtained by dividing uvNORM by -w [i] [j] is set again as uvNORM, and if it is larger than uvMAX10 in the arithmetic register 707, if uvNORM is larger than uvMAX10,
To store. Also, at this time, the value obtained by dividing vvNORM by -w [i] [j] is set again as vvNORM. If it is larger than vvMAX10 in the operation register 707, if vvNORM is larger than vvMAX10,
To store. If the reciprocal of -w [i] [j] is larger than Winv2 in the operation register 707, the reciprocal of -w [i] [j] is stored in Winv2.

(Step 7-2-12) w [i] [j] = 0, w [k]
When [l]> 0 and w [p] [q] = 0, the value obtained by dividing uuNORM by w [k] [l] is set as uuNORM again, and if it is larger than uuMAX11 in the operation register 707, it is uuNORM. UuMAX11
To store. At this time, the value obtained by dividing uvNORM by w [k] [l] is set again as uvNORM, and if it is larger than uvMAX11 in the arithmetic register 707, if uvNORM is uvMAX11, then
To store. Also, at this time, the value obtained by dividing vvNORM by w [k] [l] is set again as vvNORM, and if it is larger than vvMAX11 in the arithmetic register 707, if vvNORM is larger than vvMAX11
To store.

(Step 7-2-13) w [i] [j] = 0, w [k]
When [l] <0 and w [p] [q] = 0, the value obtained by dividing uuNORM by -w [k] [l] is set as uuNORM again, and if it is larger than uuMAX12 in the operation register 707, uuNORM to uuMAX12
To store. At this time, the value obtained by dividing uvNORM by -w [k] [l] is set as uvNORM again. If it is larger than uvMAX12 in the operation register 707, if uvNORM is uvMAX12,
To store. Also, at this time, the value obtained by dividing vvNORM by -w [k] [l] is set again as vvNORM, and if it is larger than vvMAX12 in the operation register 707, if vvNORM is larger than vvMAX12,
To store.

(Step 7-2-14) w [i] [j] = 0, w [k]
When [l] = 0 and w [p] [q]> 0, the value obtained by dividing uuNORM by w [p] [q] is set again as uuNORM. If it is larger than uuMAX13 in the operation register 707, uuNORM UuMAX13
To store. At this time, the value obtained by dividing uvNORM by w [p] [q] is set again as uvNORM, and if it is larger than uvMAX13 in the operation register 707, uvNORM is set to uvMAX13.
To store. At this time, the value obtained by dividing vvNORM by w [p] [q] is set again as vvNORM. If it is larger than vvMAX13 in the operation register 707, if vvNORM is larger than vvMAX13
To store.

(Step 7-2-15) w [i] [j] = 0, w [k]
When [l] = 0 and w [p] [q] <0, the value obtained by dividing uuNORM by -w [p] [q] is set again as uuNORM, and if it is larger than uuMAX14 in the operation register 707, uuNORM to uuMAX14
To store. At this time, the value obtained by dividing uvNORM by -w [p] [q] is set again as uvNORM, and if it is larger than uvMAX14 in the operation register 707, then uvNORM is set to uvMAX14.
To store. At this time, the value obtained by dividing vvNORM by -w [p] [q] is set again as vvNORM, and if it is larger than vvMAX14 in the operation register 707, if vvNORM is vvMAX14,
To store.

(Step 7-2-16) w [i] [j] = 0, w [k]
When [l] = 0 and w [p] [q] = 0, uuNORM is the arithmetic register 7.
If it is larger than uuMAX15 in 07, uuNORM is set to u
Store in uMAX15. At this time, if uvNORM is larger than uvMAX15 in the arithmetic register 707, uvNOR
Store M in uvMAX15. At this time, if vvNORM is larger than vvMAX15 in the operation register 707, vvNORM is stored in vvMAX15.

(Step 7-3) ucounter1 and ucoun
ter2 and ucounter3 and vcounter1 and vcounter2
And vcounter3 using the increment device 702, the following processing is performed.

If vcounter3 <vK, vcounter3 is incremented by 1.

If vcounter3 = vK and ucounter3 <uK, vcounter3 is initialized to 0 and ucounter3 is incremented by 1.

Vcounter3 = vK and ucounter3 = uK and
If vcounter2 <vK, set vcounter3 and ucounter3 to 0
Initialize with and increment vcounter2 by 1.

Vcounter3 = vK and ucounter3 = uK and
If vcounter2 = vK and ucounter2 <uK, vcounter3
, Ucounter3 and vcounter2 are initialized to 0, and uco
Increment unter2 by 1.

Vcounter3 = vK and ucounter3 = uK and
vcounter2 = vK and ucounter2 = uK and vcounter1 <vK
Then, vcounter3 and ucounter3 and vcounter2 and ucounter2 are initialized to 0, and vcounter1 is incremented by 1.

Vcounter3 = vK and ucounter3 = uK and
vcounter2 = vK and ucounter2 = uK and vcounter1 = vK
If ucounter1 <uK, vcounter3 and ucounter
3 and vcounter2 and ucounter2 and vcounter1 to 0
Initialize with and increment ucounter1 by 1.

After the above processing, the processing is repeated from (Step 7-2).

Vcounter3 = vK and ucounter3 = uK and
vcounter2 = vK and ucounter2 = uK and vcounter1 = vK
If ucounter1 = uK, go to (Step 7-4). (Step 7-4) uuMAX1 to uuMAX15 and uvMAX1 to uvMAX15 and vvMAX1 to vvMAX15 and Winv1 and Winv2 and uTOL and vT obtained as described above
From the OL, using the division number arithmetic unit 709,
The number of divisions in the U direction (representing this value by uN) and the number of divisions in the V direction (representing this value by vN) are (Equation 30) and (Equation 3)
It can be calculated in 1).

Next, the coordinate calculation means 502 calculates the number of divisions uN obtained by any one of the above division number calculation means 501.
And vN, the free-form surface parameter u ＝ 0, 1 / uN, 2
Calculate the coordinates of / uN,…, 1; v = 0, 1 / vN, 2 / vN,…, 1. As the method of calculating the coordinates, as in the case of the free curve,
For Horner's method, forward difference method, Bezier surface, de
Casteljau algorithm etc.

For Horner's method, see “Currves and
surfaces for computer aided geometric design, sec
ond edition ”(G. Farin, 1990, Academic Press) P.47
Are detailed in. For the forward difference method, see P.80-P.81.
The de Casteljau algorithm is detailed in P.29.

The unit normal calculation means 503 uses the division numbers uN and vN obtained by the division number calculation means 501 to calculate the parameters u = 0, 1 / uN, 2 / uN, ..., Of the free-form surface.
1; Calculate the unit normal of v = 0, 1 / vN, 2 / vN,…, 1.
The unit normals are calculated by calculating the rate of change (partial differential coefficient) in each of the U and V directions using the Horner method, the forward difference method, the de Casteljau algorithm for Bezier surfaces, and then calculating the outer product. Then, unitize at the end.

Thereafter, the polyhedron creating means 504 divides the points, that is, the parameters u = 0, 1 / uN, 2 / uN, ..., Of the free-form surface.
1; v = 0, 1 / vN, 2 / vN, ..., 1 are sequentially connected, and the polyhedron data is obtained together with the coordinates calculated by the coordinate calculation means 502 and the unit normal calculated by the unit normal calculation means 503. create.

Finally, the polyhedron data created by the polyhedron creating means 504 is transferred to the polyhedron display means and displayed on the graphic display.

[0267]

According to the present invention, in the calculation of the number of divisions of a free curve and a free curved surface, a portion where the weight of the control point is positive,
Since the calculation is performed separately for the 0 part and the negative part, division by 0 does not occur in each calculation. Therefore, no matter what set of weights is given, division by 0 does not occur in the calculation of the number of divisions of the free-form curve / curved surface, and the number of divisions does not become indefinite. Therefore, the system of the graphic system goes down, runaway, or It is possible to prevent the occurrence of troubles such as not being able to obtain a correct display.

[Brief description of drawings]

FIG. 1 is a diagram showing an example of a display procedure of a graphic system.

FIG. 2 is a diagram illustrating a configuration example of a polygonal line approximation unit.

FIG. 3 is a diagram illustrating a configuration example of a free curve division number calculation unit.

FIG. 4 is a diagram illustrating another configuration example of a free curve division number calculation unit.

FIG. 5 is a diagram showing a configuration example of a polyhedral approximation unit.

FIG. 6 is a diagram illustrating a configuration example of a free-form surface division number calculation unit.

FIG. 7 is a diagram showing another configuration example of a free-form surface division number calculation unit.

FIG. 8 is a diagram showing a configuration example of a graphic system.

FIG. 9 is a diagram showing an example of a free curve.

FIG. 10 is a diagram showing an example of a free-form surface.

[Explanation of Codes] 201, 501 ... Division number calculation unit, 202, 502 ... Coordinate calculation unit, 503 ... Unit normal line calculation unit, 203 ... Fold line creation unit, 504 ... Polyhedron creation unit, 301, 401, 601,
701 ... Counter, 302, 402, 602, 702 ...
Increment device, 303, 403, 603, 703
... Degree registers, 304, 404, 604, 704 ... Control point set storage memory, 305, 405, 605, 7
05 ... Weight storage memory, 306, 406,
606, 706 ... Norm arithmetic unit, 307, 407, 6
07,707 ... Operation register, 308, 408 ... Broken line approximation parameter (allowable error value) storage register, 608, 7
08 ... Polyhedron approximation parameter (allowable error value) storage register, 309, 409, 609, 709 ... Division number calculation device, 801, ... CPU, 802 ... Main memory, 803 ...
Keyboard, 804 ... Mouse, 805 ... Stylus pen, 806 ... System bus, 807 ... Graphic subsystem, 808 ... Graphic LSI, 809 ... Frame memory bus, 810 ... Frame memory, 811 ... C
RT, 812 ... Dial, 813 ... Filing device.

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Toru Sakaihara Toru Sakaihara 292 Yoshida-cho, Totsuka-ku, Yokohama-shi, Kanagawa Inside the Hitachi, Ltd. Microelectronics Equipment Development Laboratory (72) Inventor Toshiyuki Kuwana Mika Oita, Ibaraki Prefecture 5-2-1, Machi, Ltd. Inside the Omika Plant, Hitachi, Ltd. (72) Inventor, Tatsuki Nakamura 5-2-1, Omika-cho, Hitachi-shi, Ibaraki, inside Hitachi Process Computer Engineering Co., Ltd.

Claims (10)

[Claims] 1. At least a storage device, a computing device, and a display device are provided, wherein a set of control points of a curve to be displayed, a set of weights applied to the control points, and a polygonal line for approximating the curve are used. When the storage unit holds the tolerance value for accuracy and the set of the control points, and the curve determined by the set of the weights is approximated to a polygonal line in the arithmetic device, and the polygonal line is displayed on the display device. In the approximation by the polygonal line, the number of the polygonal lines is determined based on the tolerance value, the set of control points, and the set of weights, and when the number of the polygonal lines is determined, the set of weights is weighted. Is divided into a subset in which the weight is 0 and a subset in which the weight is not 0, and an operation for determining the number of broken lines is performed for each subset. For the number of the polygonal lines by integrating the calculation results calculated for, and in the calculation, in the subset where the weight is not 0, there is a process of performing division by the weight, and in the subset where the weight is 0, Is a method of displaying a curve, which does not have a process of performing division by the weight. 2. At least a storage device, a computing device, and a display device, comprising a set of control points of a curved surface to be displayed, a set of weights applied to the control points, and a polyhedron for approximating the curved surface. When the storage unit holds the tolerance value for accuracy and the control point set, and a curved surface defined by the weight set, is approximated to a polyhedron in the arithmetic device, and the polyhedron is displayed on the display device. In the approximation by the polyhedron, based on the tolerance value, the set of control points and the set of weights, to determine the number of polyhedra, when determining the number of polyhedra, the set of weights, Separate into a subset having a weight of 0 and a subset having a non-weight of 0, and perform an operation for determining the number of the polyhedra for each subset. In the calculation, the number of the polyhedra is integrated by integrating the calculation results calculated for the subsets, and in the calculation, a division is performed by the weights in the subset where the weight is not 0. A curve display method characterized by not having a process of performing division by weight in a subset of 0. 3. The method according to claim 1 or 2, wherein, in the calculation, when the weight is 0, division is not performed by the weight, and the absolute value of the weight is the minimum value in the non-weight portion. A curve display method having a process of performing division. 4. A storage unit for holding a set of control points of a curve to be displayed, a set of weights applied to the control points, and an allowable error value regarding accuracy when the curve is approximated by a polygonal line. In a graphic display system comprising a polygonal line approximating means for approximating a curve defined by the set of the control points and the set of the weights with a polygonal line, and a display means for displaying the polygonal line, the polygonal line approximating means is the allowable error value, the Based on the set of control points and the set of weights, a division number calculation means for determining the number of polygonal lines when approximating the curve by a polygonal line, the division number calculation means, the set of weights, Separation performance in which a subset having a weight of 0 and a subset having a weight of not 0 are separated, and an operation for determining the number of broken lines is performed for each subset. Means and integrated operation means for integrating the operation results operated on the subsets by the separation operation means to determine the number of broken lines, the separation operation means for the subsets in which the weight is not 0 Has a process of performing division by weight, and does not have a process of performing division by weight for the portion where the weight is 0. 5. A storage unit for holding a set of control points of a curved surface to be displayed, a set of weights applied to the control points, and an allowable error value regarding accuracy when approximating the curved surface with a polyhedron. In the graphic display system comprising a set of the control points, and a polyhedral approximation means for approximating a curved surface defined by the set of weights with a polyhedron, and a display means for displaying the polyhedron, the polyhedral approximation means is the allowable error value. , A division number calculation means for performing an operation for determining the number of polyhedra when approximating the curved surface by a polyhedron based on the set of control points and the set of weights, and the division number calculation means, The set of weights is separated into a subset with a weight of 0 and a subset with a non-zero weight, and the number of polyhedra for each subset. Separation operation means for performing an operation for determining, and integration operation means for integrating the operation results operated on the subset by the separation operation means to determine the number of polyhedra, the separation operation means The graphic display system is characterized in that it has a process of performing division by weight for a subset whose weight is not 0, and does not have a process of performing division by weight for the part whose weight is 0. 6. The separation calculation means according to claim 4, wherein when the curve is represented by an expression with one parameter, (the number of broken lines) = (the equation representing the curve is first differentiated with the parameter). The absolute value of the maximum value of the expression within the domain of the curve) /
Based on the relational expression of the permissible error value, the number of the polygonal lines from the permissible error value, the set of control points, and the set of weights for each of the subset with weight 0 and the subset with weight 0. And a calculation process for performing a division by the weight for the subset whose weight is not 0, and a calculation process for performing the division for the part whose weight is 0. A graphic display system characterized by not having. 7. The separation calculating means according to claim 4, wherein, when the curve is expressed by an expression with one parameter, (the number of the broken lines) = √ ((the expression representing the curve is defined by the parameter). Based on the relational expression of the next-differentiated expression, the absolute value of the maximum value within the domain of the curve) / (8 × (the allowable error value))), the subset with weight 0 and the portion with non-zero weight For each of the sets, there is provided operation means for determining the number of the polygonal lines from the tolerance value set, the control point set and the weight set, wherein the operation means is for a subset in which the weight is not zero. Has a process for performing division by the weight, and does not have a process for performing division by the weight when the weight is 0. . 8. The separation calculation means according to claim 5,
When the curved surface is represented by an equation with two parameters, the number of the polyhedrons and the maximum value within the domain of the curve of the equation that is obtained by first differentiating the equation representing the curved surface with one of the parameters Absolute value, the absolute value of the maximum value within the domain of the curve, of the expression obtained by first differentiating the expression representing the curved surface with the other parameter of the parameters, and the allowable error value, the relationship established between Calculating means for determining the number of polyhedra from the allowable error value, the set of control points, and the set of weights for each of the subsets with a weight of 0 and the subsets with a non-zero weight. The calculation means has a process of performing division by the weight for the subset in which the weight is not 0, and a division process for the portion in which the weight is 0. A graphic display system characterized by having no process for division by weight. 9. The separation calculation means according to claim 5,
When the curved surface is represented by an equation with two parameters, the maximum value within the domain of the curve of the number of the polyhedrons and an equation obtained by quadratic differentiating the equation representing the curved surface with one of the parameters And an absolute value of the maximum value within the domain of the curve of an expression obtained by second-order differentiating the expression representing the curved surface with the other parameter of the parameters, and the expression representing the curved surface as the two parameters. Of the equation that is first-order differentiated by one of the parameters, the first-order derivative of the other parameter, the absolute value of the maximum value within the domain of the curve, and the allowable error value Based on the allowable error value, the set of control points, and the set of weights, the number of polyhedra is determined for each of the subset with weight 0 and the subset with weight 0. The calculation means has a process of performing division by the weight on the subset whose weight is not 0, and does not have a process of performing division by the weight on the part whose weight is 0. A graphic display system characterized in that 10. The separating / calculating means according to claim 4, 5, 6, 7, 8 or 9, wherein said weight is not 0 in a portion where said weight is 0 without performing division by said weight. A graphic display system having a process of performing division by a minimum absolute value of weights in a part.