JPH06231261A - Method for displaying parametric trimming curve by polygonal line approximation and graphic display system - Google Patents

Abstract

PURPOSE:To provide a graphic system provided with a method for approximating a trimming curve by polygonal lines which can reduce an error on the curved face of the trimming curve to a specified value or less. CONSTITUTION:A trimming curve previously stored in a filing device 613 or inputted by a mouse 604 or the like is stored in a memory 604. An allowable error value specified from a keyboard 603 is stored in the memory 602. A program for determining the number of polygonal lines based upon an optional allowable error value is stored in the device 613 at the time of approximating a trimming curve by polygonal lines. The program is read out from the device 613 to the memory 602 and executed by a CPU 601, which determines the number of polygonal lines based upon the specified allowable error value, prepares graphic data for the trimming curve and stores the prepared data in the memory 602. The data are transferred to a graphic subsystem 607 and displayed on a CRT 611.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a graphic system for drawing a parametric curved surface on a graphic display or the like, and more particularly to a graphic system for trimming a parametric curved surface by a parametric trimming curve.

[0002]

2. Description of the Related Art Generally, when displaying a curve in a graphic system, complicated processing is required to display the curve as a row of dots (dots). Therefore, the curve is approximated by a polygonal line. To display. To generate a line segment, for example, DDA (Digital D
There is a method using an differential analyzer.

That is, in the polygonal line approximation of a curve, appropriate N (≧ 1) points (referred to as division points) are taken on the curve, and the start point of the curve, the first division point, and the first point Division point and second division point, ..., N-1th division point and Nth division point
This is done by approximating the th division point, the Nth division point, and the end point with a polygonal line connecting the line segments.

Similarly, in order to display a curved surface by a row of dots, since complicated processing is required, the curved surface is approximated by an open polyhedron or a closed polyhedron composed of a plurality of continuous polygons. In order to generate the above-mentioned polygon (correctly, a dot row for filling this polygon), there is a method of using DDA as in the case of displaying a line segment as described above (however, the line is A two-dimensional DDA that generates a minute
, And a DDA that generates a polygon is called a three-dimensional DDA).

That is, to approximate a curved surface with an open polyhedron or a closed polyhedron, appropriate M × N (M, N ≧ 1) dividing points are taken on the curved surface (M dividing points in one direction on the curved surface are taken. , N division points are taken in a direction different from this direction), and it is approximated by a rectangle of M × N lattices formed by the division points or a set of triangles obtained by further dividing this rectangle by diagonal lines.

As a method of obtaining the above-mentioned dividing points of a curve and a curved surface, when each coordinate of the curve and the curved surface is expressed by a mathematical expression with a parameter, the value of this parameter is equally divided from the start point to the end point. To find the dividing point of the parameter,
There is a method in which a curve and a point on the curved surface corresponding to the division points of each parameter are used as the division points. In this case, the conventional method for determining the number of division points (the number of divisions) is "Surfac".
e Algorithms using bounds
on derivings "(D. Filip,
R. Magedson, and R.M. Marko
t, 1986) and "Real-time render".
ring of trimmed surfaces "
(A. Rockwood, K. Heaton, an
d T. Davis, 1989) and the like.

[0007]

The above-mentioned prior art describes a method of determining the number of divisions of parametric (given as a function of parameters) curves and parametric curved surfaces.

On the other hand, there is a so-called trimming technique in which a parameter space of a parametric curved surface (this is referred to as a domain) is limited by a curve defined in the parameter space and only the limited portion is drawn. Trimming is specified by a closed curve (called a trimming loop) that is usually composed of one or more curves defined in the parameter space (called a trimming curve), and the left side (right side) of the closed curve is drawn. It is defined to leave as the side to do.

Generally, each trimming curve is often defined parametrically. In this case, this parametric curve (trimming curve) is also subjected to polygonal line approximation, but the method of dividing the trimming curve is not described in the above-mentioned conventional technique. However, normally, the parametric at each node (the point where two polygonal lines are connected) is obtained by performing polygonal line approximation in the parameter space of the parametric curved surface with the number of divisions fixed to the graphic system or passed to the system, and then polygonally approximating the trimming curve. Calculate the value on the curved surface or the value on the open polyhedron or the closed polyhedron approximated by the open polyhedron approximation or the closed polyhedron, and form the polygonal edge of the parametric curved surface by connecting the points on the curved surface thus calculated. (The end). However, in this way, an error occurs at the edge of the parametric curved surface caused by the trimming, but no consideration has been given to the means for evaluating the error. When it is desired to display a figure in which two trimmed curved surfaces are matched by an edge, if such an error is large, the edge portions do not match and it becomes unnatural.

According to the present invention, when an allowable error value is designated by a user, the edge error of the parametric curved surface caused by trimming can be suppressed to be equal to or less than the designated value. An object is to provide a graphic system having a polygonal line approximation method.

[0011]

In order to solve the above-mentioned problems, according to the present invention, at least an input device, a storage device, a calculation device, and a display device are provided, and a definition formula of a parametric curved surface or a control point A set, and a definition formula of a parametric trimming curve for trimming the curved surface or a set of control points are held in a storage device, and the parametric curved surface is approximated by an open polyhedron or a closed polyhedron in a computing device. When a parametric trimming curve is approximated by a polygonal line and the parametric curved surface and the parametric trimming curve are displayed on the display device, the tolerance of approximation of the trimming curve by the polygonal line is an allowable error of an arbitrary value specified from the outside. The value is received by the input device, stored in the storage device, and specified. The on the basis of the allowable error value, previously provided a trimming curve equation storage device for determining the number of the polygonal line when approximated by fold lines, it is possible to determine the number of fold lines by the arithmetical unit.

Further, a trimming curve is given by a map c with one variable t as a parameter and c (t) = (u (t), v (t)), and the parametric surface is given by two variables u and v. With s (u, v) = (x (u,
v), y (u, v), z (u, v)), and when the tolerance value is specified by L, determines the number of polygonal lines when the trimming curve is approximated by polygonal lines. The equation for doing the first derivative (s.
It is also possible to obtain a maximum value M1 of the size of c) ′ within the range of the parameter t of the trimming curve by an arithmetic device, and use it as an equation for determining the number of broken lines as M1 / L.

Further, the trimming curve has one variable t.
Is given as a mapping c that is c (t) = (u (t), v (t)), and the parametric surface has two variables u and v as parameters s (u, v) = (x ( u,
v), y (u, v), z (u, v)), and when the tolerance value is specified by TOL, determine the number of polygonal lines when approximating the trimming curve by polygonal lines. Is calculated by a computing device to find the maximum value M2 of the size of the second derivative (s · c) ″ of the composite map s · c by t within the range of the parameter t of the trimming curve. It can also be an equation for determining the number as √ (M2 / (8 × TOL)).

Further, when the parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section), the range of the mapping s of the parametric trimming curve c and each section of the parametric curved surface Whether or not the common part exists is determined by the arithmetic unit, and only when the common part exists, the range in which the common part exists is defined as the parameter range of the composite map s · c, (s · c) ′.
Alternatively, the maximum value of the size of (s · c) ″ can be obtained.

Further, a parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section),
In addition, when the parametric trimming curve is defined by a set of control points and has a convex hull property, the range of the mapping s of the set of control points defining the parametric trimming curve is common to each section of the parametric curved surface. It is also possible to determine whether or not there is a portion by an arithmetic device, and to determine whether or not there is a common portion between the range of the parametric trimming curve and each section of the parametric curve.

Further, a definitional expression of a parametric curved surface or a set of control points and a definitional expression of a parametric trimming curve for trimming the curved surface or a set of control points is held in a storage device, and the parametric curved surface is stored in a computing device. Is approximated by an open polyhedron or a closed polyhedron, the parametric trimming curve is approximated by a polygonal line in a computing device, and the parametric curved surface and the parametric trimming curve are displayed on a display device. Regarding the accuracy of the approximation, input means for inputting the tolerance value of any value specified from the outside,
First storage means for holding an input allowable error value, and first holding an equation for determining the number of broken lines when approximating the trimming curve with the broken line based on the input allowable error value. It is also possible to provide a graphic display system comprising two storage means and a determination means for determining the number of broken lines based on this formula.

Further, in this graphic display system, a trimming curve is given by a map c with one variable t as a parameter and c (t) = (u (t), v (t)), and a parametric surface is Two variables u and v
As a parameter, s (u, v) = (x (u, v), y
(U, v), z (u, v)) given by the map s and the tolerance value is specified by L, an expression for determining the number of polygonal lines when the trimming curve is approximated by polygonal lines. The maximum value M1 of the magnitude of the first-order derivative (s · c) ′ of the composite map s · c with respect to t within the range of the parameter t of the trimming curve is calculated by the arithmetic device, and the number of broken lines is set to M1 / L. It can also be a formula for determining.

Furthermore, in the graphic display system, the trimming curve has one variable t as a parameter and c
Given a mapping c that is (t) = (u (t), v (t)), the parametric surface takes two variables u and v as parameters s (u, v) = (x (u, v) , Y (u,
v), z (u, v)), and an allowable error value is specified by TOL, an equation for determining the number of polygonal lines when approximating the trimming curve by polygonal lines is given by Second derivative of composite map s · c by t (s · c) ″
The maximum value M2 of the trimming curve within the range of the parameter t of the trimming curve is calculated by a computing device, and the number of broken lines is √
It may be an equation for determining (M2 / (8 × TOL)).

Furthermore, when the parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section), the mapping s of the parametric trimming curve c
Determining means for determining whether or not there is a common part between each range of the parametric curved surface and each section of the parametric curved surface, and when the determining means determines that the common part exists, the range in which the common part exists is synthesized map s It is also possible to provide a graphic display system provided with a maximum value calculating means for obtaining the maximum value of the magnitude of (s · c) ′ or (s · c) ″ as the parameter range of c.

Further, the parametric surface is given as a union of a plurality of surfaces (each surface is referred to as a section),
In addition, when the parametric trimming curve is defined by a set of control points and has convex hullability, as a determination means,
It is also possible to provide a graphic display system including a range based on a map s of a set of control points that defines a parametric trimming curve, and control point determination means that determines whether or not there is a common part with each section of the parametric curved surface. it can.

Here, the convex hull property will be described. When the parametric curve having the interval [a, b] as the domain is determined from the set of control points {P _i | i = 1, 2, ..., N}, the curve c has a convex hull property. It means that any point on the curve is included in co {P _i | i = 1, 2, 3, ..., N} (minimum convex set including all points P _i ). That is, for any t ∈ [a, b], c (t) ∈ co
That is, {P _i | i = 1, 2, ..., N} holds.
Similarly, a parametric surface having a rectangular region [a, b] × [c, d] as a domain is a set of control points {P _ij | i =
1, 2, ..., M; j = 1, 2, ..., N}, it means that the curved surface s has a convex hull property if any point on the curve is co {P _ij | i = 1. , 2, ..., m; j = 1,2, ...,
n} (the smallest convex set that includes all points P _ij ). That is, any (u, v) ε [a, b] ×
For [c, d], s (u, v) εco {P _ij | i =
1, 2, ..., M; j = 1, 2, ..., N}.

[0022]

[Function] An example of determining the division method for the polygonal line approximation of the parametric trimming curve so that the error on the parametric curved surface becomes a certain value or less is an example of the definition equation of the given parametric curved surface or the given control point. Let s be the definition formula defined by the set and c be the definition formula of the given parametric trimming curve or the set of the given control points, and make the length of each line segment of the polygonal line approximation of the parametric trimming curve constant. In order to suppress the value L or less, the maximum value M1 within the parameter range of the magnitude of the first-order derivative (s · c) ′ by the parameter of the parametric trimming curve of the composite map s · c is obtained, and the broken line approximation of the parametric trimming curve is performed. Is to set the number of divisions to M1 / L.

In another example, s is defined as a definition equation of a given parametric surface or a set of control points, and a definition equation of a given parametric trimming curve or a set of control points is given. When the defining equation to be defined is c, in order to suppress the error from the polygonal line approximation of the parametric trimming curve and the true value of the parametric trimming curve to a fixed value TOL or less, the second derivative with the parameter of the parametric trimming curve of the composite map s · c. Maximum value M within the parameter range of size (s · c) ″
2 is obtained, and the division number for the polygonal line approximation of the parametric trimming curve is set to √ (M2 / (8 × TOL)).

Further, in the case where the parametric curved surface is given piecewise, the division method for the polygonal line approximation of the parametric trimming curve is performed by the polygonal line approximation method of the parametric trimming curve for each segment, and the composite map s · c The parameter range of is determined whether there is a common part between the range of the parametric trimming curve c and each section of the parametric curved surface, and the range in which the common part exists is set as the parameter range (s
・ C) ′ or (s · c) ″ is calculated as the maximum value.

Further, in the case where the parametric curved surface is piecewise given, and the parametric trimming curve is defined by a set of control points and has a convex hull property, the range of the parametric trimming curve and each section of the parametric curve are common. Whether or not a part exists is determined by whether or not there is a common part between the range of the set of control points that defines the parametric trimming curve and each section of the parametric curved surface.

[0026]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS First, the block diagram shown in FIG. 6 and FIG.
An example of the graphic system will be briefly described with reference to the functional block diagram shown in FIG.

Generally, a graphic system has a control unit 701, a storage unit 702, and an input unit 70 as shown in FIG.
3, image processing unit 707, display unit 711, external storage device 7
It is composed of 13 and others. Each of these units shown in FIG. 7 is specifically configured as shown in the block diagram of FIG. 6, for example. That is, the control unit 701 controls the central processing unit (CP
U) 601 and the storage unit 702 are main memory (MM) 60
2. The image processing unit 707 is the graphic subsystem 60.
7, the display unit 711 is composed of a CRT 611. The input unit 703 is a keyboard (KB) for inputting data.
603, a mouse (MOUSE) 604, a stylus pen (PEN) 605, a dial (DIAL) 612, and the like, and an external storage device 713 includes a filing device (FILE) 613, which stores and saves graphic data and the like. The control unit 701 (CPU 601), the input unit 7
03 (keyboard 603, mouse 604, stylus pen 605, dial 612), image processing unit 707 (graphic subsystem 607), external storage device 713
(Filing device 613) has a system bus 706.
Data is exchanged by being connected via (606). or,
The graphic subsystem 607 is a graphic LS.
It is composed of an I 608, a frame memory (FM) 610 and the like, and is connected via a frame bus 609 to exchange data.

The process of displaying a graphic on the CRT 611 using the graphic system shown in FIG. 6 will be briefly described.

First, the keyboard 603, the mouse 604,
Data is input by various input devices such as a stylus pen 605 and a dial 612, and graphic data is created in the main memory 602 via the system bus 606. Alternatively, the filing device 61 uses previously created graphic data.
3 through main bus 602 via system bus 606
Transfer to. The CPU 601 modifies the graphic data in the main memory 602 according to instructions sent from various input devices such as a keyboard 603, a mouse 604, a stylus pen 605, and a dial 612 via the system bus 606, if necessary. For changing the figure data, for example, geometrical operations such as rotation of the figure, parallel movement, change of viewpoint, etc., calculations for approximating the parametric curved surface by polygons described later, calculations for approximating the parametric trimming curves by polygonal lines, Etc. are included. The CPU 601 sequentially transfers the graphic data in the main memory 602 created as described above to the graphic subsystem 607 via the system bus 606. In the graphic subsystem 607, the graphic LSI 608 geometrically (ie, as coordinate data) and graphic type (line segment, polygon, etc.) given geometrically (that is, as coordinate data) in order to display the sent graphic data on the CRT 611. ), An image is formed by dots (pixels). This process is performed, for example, from the coordinate data of the line segment to the DDA (Digit
al Differential Analyzer)
To generate a dot row, and to generate a dot row that fills the inside of the polygon with three-dimensional DDA from the polygon coordinate data. The created dot image is transferred to the frame memory 610 via the frame memory bus 609, and finally, the graphic subsystem 607 causes the dot image of the frame memory 610 to be an RGB signal, which is a signal indicating red, green, and blue. Converted to C
RT611 displays it.

The trimming of the parametric curved surface by the parametric trimming curve will be briefly described below.

FIG. 3 shows a region 301 of a two-dimensional coordinate space (UV parameter space) having two variables U and V as parameters, and a three-dimensional coordinate space (XYZ space) defined by three variables X, Y and Z. ) Represents a parametric curved surface 302 as an image. That is, the parametric curved surface 30
2 is a certain area of the two-dimensional space (this is called the domain)
Is defined as a mapping s from s to a three-dimensional space. The mapping s is usually defined as a polynomial for u and v for ease of handling by a computer, or a mapping once defined as a mapping into a four-dimensional XYZW space, and then divided as W in terms of homogeneous coordinates. Is used (this allows
Transformations that combine basic transformations such as parallel movement, scaling, and rotation movement have the advantage that they can be expressed as determinant multiplications that express these basic transformations). That is, the map s is

[0032]

[Equation 1]

Or

[0034]

[Equation 2]

The following description will be given, assuming that it has a shape of. Here, the maps x, y, z, w are polynomials with respect to u and v.

The domain of the mapping s is usually a rectangle for easy handling by a computer. However, this means that the order of the mappings x, y, z, and w must be limited because of the performance of the graphic system for display.
By extension, the shape of the parametric surface is restricted. In order to solve this restriction, a method is used in which the domain of the map s is cut using a closed loop drawn in the domain, and one side of the closed loop (right side or left side of the closed loop) is cut off, and only the remaining portion is defined as the domain. Has been devised. This method is called trimming. After that, since it is the same for all of them, when the counterclockwise direction on the closed loop is the positive direction, the right side of the closed loop is the side to be cut off and the left side is to be the side to be left. Moreover, this closed loop is called a trimming loop.

The above trimming will be described with reference to FIG. The trimming loop 401 limits the domain 301 and sets the left side (inner side) of the loop 401 as the limited domain. Then, the parametric curved surface 302, which is an image of the mapping s from the domain 301, is also limited, and a part of the parametric curved surface 302 (this is called a trimmed curved surface) 40
Only 2. By using this method, it is possible to create a relatively free shape by using a relatively low-order polynomial or rational polynomial.

Normally, the trimming loop is also parametric (as a function of variables) because it is easy to handle on a computer.
Given. The trimming loop does not necessarily have to be a single closed loop, and may be defined as a set of curves that are connected to each other only at the start point and the end point. Each one is called a trimming curve. FIG. 5 is a diagram showing a parametric trimming curve 502 as an image from a certain region 501 of the T parameter space, which is the domain of the variable T, to the UV parameter space. That is, the parametric trimming curve 502 is defined as an image by the mapping c from a certain area (domain) of the one-dimensional space to the two-dimensional space. The map c is a polynomial with respect to t, or a three-dimensional UV
A rational polynomial that is defined as a mapping to H space and is then regarded as homogeneous coordinates and divided by H coordinates is used (this allows conversion that combines basic conversions such as parallel movement, scaling, and rotation movement). , Has the advantage that it can be expressed as a determinant multiplication that expresses these basic transformations). That is, the map c is

[0039]

[Equation 3]

Or

[0041]

[Equation 4]

It is assumed that it has the form of. Here, the maps u, v, h are polynomials with respect to t.

Next, an example of processing for drawing a trimmed curved surface will be described. For this explanation, NURB
(Non-uniform relational B-s
Although the trimming of the curved surface with the NURB curve is taken as an example, the present invention is not an invention peculiar to NURB. A NURB surface and a NURB curve are parametric surfaces and curves defined by a set of control points, a non-decreasing sequence called a knot sequence, and an order. For these control points and knot sequences, refer to "The Grap"
hical Processing of B-Spl
ines ina Highly Dynamic E
nvironment "(Salim S. Abi-Ez
zi, 1989) and the like.

For the NURB curved surface s, the orders in the U direction are p, V
The order of direction is q, and the set of control points is P _ij (1 ≦ i ≦ m, 1
≤j≤n), and the knot sequence in the U direction is u ₁ ≤u ₂ ≤ ... ≤u
_{When m + p + 1}

[0045]

[Equation 5]

When the mapping family {η _ik } defined by and the knot sequence in the V direction are v ₁ ≤v ₂ ≤ ... ≤v _{n + q + 1}

[0047]

[Equation 6]

Using the mapping family {θ _ik } defined by

[0049]

[Equation 7]

It is a piecewise parametric surface defined as follows. The domain is [u _{p + 1} , u _{m + 1} ] × [v _{q + 1} ,
v _{n + 1} ]. If the control point P _ij is a non-homogeneous coordinate (x _ij ,
y _ij , z _ij ) or the homogeneous coordinates (x _ij , y _ij ,
z _ij , w _ij ) depending on whether

[0051]

[Equation 8]

Or

[0053]

[Equation 9]

It can be rewritten as follows. u _i <u
_{When i + 1} and v _j <v _{j + 1} , [u _i , u _{i +1} ] × [v _j , v
One segment corresponds to [ _{j + 1} ]. This is called a curved surface span.

Similarly, the NURB curve c has a degree r, a set of control points Q _k (1≤k≤l), and a knot sequence t ₁ ≤t _2.
When ≦ ... ≦ t _{l + r + 1}

[0056]

[Equation 10]

Using the mapping family {ζ _lk } defined by

[0058]

[Equation 11]

It is a piecewise parametric curve defined as follows. The domain is [t _{r + 1} , t _{l + 1} ]. Control point Q
Depending on whether _k is a non-homogeneous coordinate (u _k , v _k ) or a homogeneous coordinate (u _k , v _k , h _k ), the above equation becomes

[0060]

[Equation 12]

Or

[0062]

[Equation 13]

It can be rewritten as follows. t _k <t
_{When k + 1} , one section corresponds to [t _k , t _{k + 1} ]. This is called a curve span.

FIG. 1 shows NUR, which is an embodiment of the present invention.
The flowchart which shows the procedure of the display process of the trimmed surface including the trimming by the NURB curve of B curved surface is shown. Further, FIG. 8 shows a functional block diagram of a processing unit for performing this display processing (these functional blocks may be provided in the CPU 601 of FIG. 6).

First, the overall flow of the procedure will be outlined. Typically, NURB surfaces and other primitives (points, lines, polygons, or strings) are modeled in the modeling coordinate system (Mode).
Ling Coordinate System. M
Each primitive is defined in a separate space called C) and thereafter all primitives are in the same world coordinate system (World).
Coordinate System. (Abbreviated as WC)
Is converted to. Next, by designating the viewpoint and the field of view (usually, a field of view pyramid is designated to indicate the field of view), the line of sight extends from the positive direction to the negative direction of the Z coordinate, and the field of view pyramid is defined in the XYZ space. Normalized projected coordinate system (Nor
malized Projection Coordi
nateSystem. (Abbreviated as NPC). Next, in consideration of the ratio with the display screen, a device coordinate system (Device Coordinate System)
m. DC) and is displayed according to the attributes and the attributes such as brightness calculated during the conversion process. Although all coordinate systems are three-dimensional in XYZ, it may be more convenient for processing to consider four-dimensional in XYZW, such as when control points are given in homogeneous coordinates. In general, it is convenient to consider homogeneous coordinates when considering only control points, and inhomogeneous coordinates when considering coordinates on an actual curved surface. The following is also considered in this way unless otherwise specified.

Next, the above outline will be described in detail with reference to FIGS. 1 and 8, but this embodiment is merely an example, and it is not necessary to execute the processing exactly as described.

First, the control unit 801 converts the control points into the world coordinate system (step 101). Next, the conversion unit 802 copies the control points and converts the copied control points into a normalized projection coordinate system (step 10).
2). Next, the checking unit 803 determines whether the control point copy is within the visual field area (step 103). Next, the process proceeds to step 120, and it is determined whether the processes of steps 101 to 103 have been completed for all control points. If not completed, the process returns to step 101, and steps 101 to 103 are repeated. If it is finished, go to step 121,
It is determined whether all the control points are within the visual field area. When all the control points are not included, the display processing is ended. If there is a control point within the visual field, the process proceeds to step 104.

In step 104, the expanding unit 804 then expands the NURB curved surface into a Bezier curved surface in the curved surface span. To develop a NURB surface (or curve) into a Bezier surface (or curve), use Cox-deBoo
A method called the r algorithm is known. The expansion to the Bezier curved surface is for simplifying the subsequent processing, and is not always necessary processing.

For a Bezier surface, the degree in the U direction is p, the degree in the V direction is q, and the set of control points is P _ij ( _1≤i≤p + 1,1
≦ j ≦ q + 1)

[0070]

[Equation 14]

Using the mapping family {B _ik } defined by

[0072]

[Equation 15]

It is a parametric curved surface defined as follows. The domain is usually [0,1] × [0,1],
An arbitrary rectangle may be formed by appropriately converting the parameters. The control point P _ij is a non-homogeneous coordinate (x _ij , y _ij , z _ij ).
Or the homogeneous coordinates (x _ij , y _ij , z _ij , w _ij ).
Depending on

[0074]

[Equation 16]

Or

[0076]

[Equation 17]

It can be rewritten as follows.

Similarly, for the Bezier curve c, when the degree is r and the set of control points is Q _k (1 ≦ k ≦ l), the above mapping family {B _ik } is used.

[0079]

[Equation 18]

It is a parametric curve defined as follows. The domain is usually [0, 1], but it may be an arbitrary closed section by appropriately converting the parameters. Depending on whether the control point Q _k has inhomogeneous coordinates (u _k , v _k ) or homogeneous coordinates (u _k , v _k , h _k ), the above equation is

[0081]

[Formula 19]

Or

[0083]

[Equation 20]

It can be rewritten as follows.

Next, the conversion unit 805 copies the Bezier control points and converts the copied control points into a normalized projection coordinate system (step 105). After this, check section 8
According to 06, it is determined whether or not the copy of the control point is within the visual field area (step 106). Next, the process proceeds to step 130, and it is determined whether the processes of steps 104 to 106 have been completed for all control points. If not completed, return to step 104, step 1
04 to step 106 are repeated. If it has ended, the routine proceeds to step 131, where it is judged whether all the control points are within the visual field area. If all control points have not been entered, the process proceeds to step 132.

If even one control point is within the field of view, the calculation unit 807 calculates the number of divisions of this curved surface span (step 107). To calculate the number of divisions, for example, "Surface algorithm" described above is used.
hms using bounds on deriv
atives ”and“ Real-time render ”
The calculation may be performed by the method described in "ing of trimmed surfaces", etc. Step 13
In 2, step 10 for all surface spans
It is checked whether or not the processes of steps 4 to 107 have been performed. If there is a curved surface span that has not been processed, the process returns to step 104. If all the curved surface spans have been processed, the process proceeds to step 108.

In step 108, the number of divisions of the trimming curve is calculated, and the approximating unit 808 approximates it by a broken line. Details of the process in step 108 will be described later. Next, in step 133, step 108
It is determined whether or not the processing of (3) has been performed on all trimming curves. If there is a trimming curve that has not been processed, the process of step 108 is repeated.

Next, the domain [u _{p + 1} , u _{m + 1} ] × of the mapping s
[V _{q + 1} , v _{n + 1} ] is divided according to the number of divisions of each curved surface span and the bending point of the trimming curve (step 10).
9). Each of the divided small pieces is called a cell. A cell is a rectangle or a triangle obtained by further dividing the rectangle by a diagonal line. The inflection point of the trimming curve means when the slope of the U coordinate changes from positive to negative and the slope of the V coordinate is negative, or
When the inclination of the U coordinate changes from negative to positive and the inclination of the V coordinate is positive, and when the inclination of the V coordinate changes from positive to negative and the inclination of the U coordinate is positive, or V It is the U coordinate when the inclination of the coordinate changes from negative to positive and the inclination of the U coordinate is negative. The division at the bending point is performed so that each side of the cell after trimming is monotonous.

Next, the trimming unit 809 trims each cell while calculating the intersection of each trimming curve and each cell (step 110). The trimmed cell becomes a polygon as described above. Subsequent processing is performed on this polygon. Alternatively, it may be divided into triangles by performing appropriate subdivision.

Next, the clipping unit 810 clips the image of the trimmed cell in the XYZ space (normalized projection coordinate system) (step 111). Finally,
The conversion unit 811 converts the cell into the device coordinate system (step 112), and displays the cell according to the attribute and the attribute such as brightness calculated during the conversion process (not described above) (step 113). In step 134, it is determined whether or not the processing of step 111 to step 113 has been performed for all cells, and if there is a cell for which processing has not been performed, the process returns to step 111. If the processing has been performed on all the cells, the display processing ends.

FIG. 2 shows a flowchart for explaining the process of step 108 in detail. Further, FIG. 9 shows a functional block diagram of a processing unit for executing this processing (this functional block is provided in the approximation unit 808 of FIG. 8). Hereinafter, referring to FIG. 2 and FIG. 9, step 10
The process of No. 8 will be described.

First, the expansion unit 901 expands a NURB curve into a Bezier curve in the curve span (step 201). The expansion to the Yahari and Bezier curves is for simplifying the subsequent processing, and is not always necessary. Next, the determination unit 902 determines whether the Bezier curve intersects the curved surface span (step 202). If they do not intersect, the subsequent processing is not performed. When crossing, the processing of the following steps is performed.

Here, the determination of crossover will be described.

Generally (when the curve and the curved surface are not Bezier), the composite map s · c is used to determine the intersection.
In the parameter range of, it is determined whether or not there is a common part between the range of the parametric trimming curve c and each section of the parametric curved surface. Then, the range in which the common part exists is defined as the parameter range (s
・ C) ′ or (s · c) ″ is found.

When the curve and the curved surface are Bezier as in this embodiment, it is determined that the control points of the Bezier curve do not intersect unless all the control points are within the curved surface span. If there is even one, the following processing is performed as a cross, but it may not cross. However, even if they do not cross over, they will be processed well in the subsequent processing, so there is no problem here.

Next, the synthesizing unit 903 substitutes the Bezier curve for the Bezier curved surface in the curved surface span to create a synthetic mapping (step 203). The domain of this composite mapping is such that the curve span is [t _k , t _{k + 1} ], the Bezier curve is c, and the surface span is [u _i , u _{i + 1} ] × [v _j , v _{j + 1} ]. When [t
_k , t _{k + 1} ] ∩ {tε [t _k , t _{k + 1} ] | c (t) ε
[U _i , u _{i + 1} ] × [v _j , v _{j + 1} ]}. Let this domain be T. The domain T can be obtained by calculating an intersection at the boundary between the Bezier curve c and the surface span [u _i , u _{i + 1} ] × [v _j , v _{j + 1} ].

Next, the calculation unit 904 calculates the number of divisions of the Bezier curve c in the domain T (step 20).
4).

Here, a method of calculating the number of divisions will be described. Therefore, the curves span _{[t k, t k + 1} ], the order of the Bezier curve r, the set of control points _{Q k (1 ≦ k ≦ l} ),
The Bezier curve is c and the surface span is [u _i , u _{i + 1} ] ×
[V _j , v _{j + 1} ], p is the degree in the U direction of the Bezier surface, q is the degree in the V direction, and P _{ij is} the set of control points (1 ≦ i ≦ p + 1,
1 ≦ j ≦ q + 1) Let s be a Bezier curved surface and T be the domain of the composite map s · c.

In the first calculation method of the number of divisions, in order to keep the length of each line segment of the polygonal line approximation of the Bezier curve c to be equal to or less than the constant value L, the first derivative (s · c) ′ of the composite map s · c. The maximum value M1 at T of the size of is calculated, and the number of divisions for the polygonal line approximation of the Bezier curve c is set to M1 / L. The theoretical basis for the fact that the length of each line segment in the polygonal line approximation can be suppressed to a fixed value L or less by determining the number of divisions in this way is "The g
radial processing of B-
spline in a highly dynami
c environment, S.C. S. Abi-E
by Zzi, Rensellar polytechni
c institute, published in 1989, p. 15
4 is mainly described. If the Bezier curved surface s and the Bezier curve c are both polynomials, the maximum value M1 is

[0100]

[Equation 21]

With being

[0102]

[Equation 22]

It may be determined as follows. At this time, the domain T of the composite map s · c is no longer relevant. Therefore,
When the maximum value M1 is evaluated (determined) in this way, the Bezier curve c and the surface span [u _i , u _{i + 1} ] × [v
_It is not necessary to calculate the intersection at the boundary of _j , v _{j + 1} ]. Even when the Bezier curved surface s or the Bezier curve c is a rational polynomial, it can be similarly calculated by dividing the numerator and the denominator and calculating the minimum value instead of the maximum value for the denominator by the above method.

In the second calculation method of the number of divisions, in order to suppress the error from the polygonal line approximation of the Bezier curve and the true value of the Bezier curve c to a fixed value (this is referred to as TOL) or less, the composite map s · c The maximum value M2 of the magnitude of the second derivative (s · c) ″ of T at T is obtained, and the number of divisions for the polygonal line approximation of the Bezier curve c is defined as √ (M2 / (8 × TOL)). The theoretical basis that the length of each line segment of the polygonal line approximation can be suppressed to a fixed value TOL or less by determining the number of divisions is
e graphical processing of
B-spline in a highly dyn
The description is centered on page 157 of "amic environment". The maximum value M2 is the case where both Bezier curved surface s and Bezier curve c are polynomials.

[0105]

[Equation 23]

By using

[0107]

[Equation 24]

It may be determined as follows. At this time, the domain T of the composite map s · c is no longer relevant. Therefore,
When the maximum value M2 is evaluated (determined) in this way, the Bezier curve c and the surface span [u _i , u _{i + 1} ] × [v
_It is not necessary to calculate the intersection at the boundary of _j , v _{j + 1} ]. Even when the Bezier surface s or the Bezier curve c is a rational polynomial, it can be similarly calculated by dividing the numerator and the denominator, calculating the numerator by the above method, and calculating the minimum value instead of the maximum value. .

Finally, the processing unit 905 approximates the Bezier curve in the curved surface span with a broken line by the number of divisions obtained in step 204 (step 205). Step 21
At 0, it is determined whether or not the processes of steps 202 to 205 have been performed for all the curved surface spans. If not, the process returns to step 202.
If all the curved spans have been processed, it is determined in step 211 whether or not all the curved spans have been processed. If not, the process returns to step 201. When the processing is performed for all the curve spans, the display processing ends.

In the above, the image of each Bezier curve in the XYZ space is considered, but the XYZ space may be any one of the world coordinate system, the normalized projection coordinate system, and the device coordinate system. However, it is preferable to consider the device coordinate system that is finally displayed.

Further, when considered in the normalized projection coordinate system or the device coordinate system, the Z coordinate has little meaning. Therefore,
When considering the size of (s · c) ′ or (s · c) ″, the Z coordinate may be ignored.

By the method described above, the number of divisions of the curved surface and the trimming curve is determined, and as a result of trimming, the trimmed curved surface is divided into cells. The divided cells undergo variable conversion and are sent to the graphic subsystem 607 shown in FIG. Graphic subsystem 6
In 07, the graphic LSI 608 creates dot image data. At this time, the true value of the trimming curve and the polygonal line input to the graphic LSI 608 are suppressed by a certain value of error TOL by the above method. The coordinate data sent to the graphic LSI 608 is generally a floating point number, but the created dot data is rounded to a whole number (or rounded down). The maximum error due to rounding is 1 / √2. Therefore, the maximum error between the true value of the trimming curve and the displayed dot image is TOL + 1 / √2.

On the other hand, in the graphic LSI 608, there is an anti-aliasing technique in which the brightness is dispersed and displayed in the surrounding dots in consideration of the existence ratio of the dots.
In general, anti-aliasing disperses around one adjacent dot, so in the present invention, TOL + 1 / √2 <
By selecting TOL so as to be 1 and using the antialiasing method, a result with no display error can be obtained.

Further, when the present invention is used to display a figure in which two trimmed curved surfaces are combined by edges, if both are evaluated by the error of TOL, the two edges are made into dot images respectively. Then the maximum error between them is 2 (TOL + 1 / √2). At this time, the antialiasing is performed by, for example, adjoining two (> 2)
(TOL + 1 / √2)) By performing the dispersion around the dots, it is possible to obtain a result with no display gap.

[0115]

As described above, according to the present invention, when an allowable error value is specified by the user, the edge error of the parametric curved surface caused by trimming can be suppressed to the specified value or less. Therefore, it is possible to provide a graphic system provided with a method for approximating a parametric trimming curve using a polygonal line.

[Brief description of drawings]

FIG. 1 is a flowchart illustrating an example of a trim curved surface display procedure.

FIG. 2 is a flowchart showing a procedure of polygonal line approximation of a trimming curve.

FIG. 3 is a diagram for explaining a parametric curved surface.

FIG. 4 is a diagram for explaining trimming.

FIG. 5 is a diagram for explaining a parametric trimming curve.

FIG. 6 is a block diagram showing an example of a graphic system.

FIG. 7 is a functional block diagram showing an example of a graphic system.

FIG. 8 is a functional block diagram of a processing unit that performs trim curved surface display processing.

FIG. 9 is a functional block diagram of a processing unit that performs a polygonal line approximation process of a trimming curve.

[Explanation of symbols]

601 ... Central processing unit (CPU), 602 ... Main memory (MM), 603 ... Keyboard (KB), 604 ... Mouse (MOUSE), 605 ... Stylus pen (PE)
N), 606 ... System bus, 607 ... Graphic subsystem, 608 ... Graphic LSI, 609 ... Frame bus, 610 ... Frame memory (FM), 611
... CRT, 612 ... Dial, 613 ... Filing device (FILE).

 ─────────────────────────────────────────────────── ─── Continuation of front page (72) Inventor Toshiyuki Kuwana 5-2-1 Omika-cho, Hitachi-shi, Ibaraki Hitachi Ltd. Omika factory

Claims (10)

[Claims] 1. A parametric curved surface defining equation or a set of control points, and a parametric trimming curve defining equation for trimming the curved surface, which comprises at least an input device, a storage device, a computing device, and a display device. Alternatively, a set of control points is stored in a storage device, the parametric curved surface is approximated by an open polyhedron or a closed polyhedron in the arithmetic device, and the parametric trimming curve is approximated by a polygonal line in the arithmetic device to generate the parametric curved surface and the parametric surface. When displaying a trimming curve on the display device, the accuracy of approximation by the broken line of the trimming curve,
The input device receives an allowable error value of an arbitrary value specified from the outside, holds it in a storage device, and based on the specified allowable error value, the number of polygonal lines when approximating the trimming curve by a polygonal line. A method for displaying a parametric trimming curve by polygonal line approximation, characterized in that the storage device is provided in advance with an equation for determining, and the arithmetic unit determines the number of polygonal lines. 2. A method for displaying a parametric trimming curve by polygonal line approximation according to claim 1, wherein the trimming curve uses one variable t as a parameter c
Given a mapping c that is (t) = (u (t), v (t)), the parametric surface takes two variables u and v as parameters s (u, v) = (x (u, v ), Y (u,
v), z (u, v)), and when the tolerance value is specified by L, the formula for determining the number of polygonal lines when approximating the trimming curve by polygonal lines is , The maximum value M1 of the magnitude of the first-order derivative (s · c) ′ of the composite map s · c with respect to t within the range of the parameter t of the trimming curve is obtained by the arithmetic unit, and the number of the broken lines is M
A method for displaying a parametric trimming curve by polygonal line approximation, which is an equation for determining 1 / L. 3. A method of displaying a parametric trimming curve by polygonal line approximation according to claim 1, wherein the trimming curve has one variable t as a parameter c
Given a mapping c that is (t) = (u (t), v (t)), the parametric surface takes two variables u and v as parameters s (u, v) = (x (u, v ), Y (u,
v), z (u, v)), and when the tolerance value is specified by TOL, the formula for determining the number of polygonal lines when approximating the trimming curve by polygonal lines is , The maximum value M2 of the size of the second derivative (s · c) ″ of the composite map s · c with respect to t within the range of the parameter t of the trimming curve is calculated by the arithmetic unit, and the number of the polygonal lines is √.
A method for displaying a parametric trimming curve by polygonal line approximation, which is an equation for determining (M2 / (8 × TOL)). 4. The range according to the mapping s of the parametric trimming curve c when the parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section) according to claim 2 or 3. And whether there is a common part with each section of the parametric curved surface is determined by the arithmetic unit, and only when the common part exists, the range in which the common part exists is the parameter range of the composite map s / c. As a method of displaying a parametric trimming curve by polygonal line approximation, the maximum value of the size of (s · c) ′ or (s · c) ″ is obtained. 5. The parametric curved surface according to claim 4, wherein the parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section), and the parametric trimming curve is defined by a set of control points and has a convex hull. If there is a property, the arithmetic unit determines whether or not there is a common part between the range by the mapping s of the set of control points defining the parametric trimming curve and each section of the parametric curved surface, and A method for displaying a parametric trimming curve by polygonal line approximation, which comprises determining whether or not there is a common part between the range of the parametric trimming curve and each section of the parametric trimming curve. 6. A parametric curved surface defining equation or a set of control points, and a parametric trimming curve defining equation for trimming the curved surface or a set of control points are held in a storage device, and the parametric surface is stored in the computing device. In the graphic display system for approximating a curved surface by an open polyhedron or a closed polyhedron, approximating the parametric trimming curve by a polygonal line in the arithmetic device, and displaying the parametric curved surface and the parametric trimming curve on the display device, the trimming curve About the accuracy of approximation by the broken line of
Input means for inputting an allowable error value of an arbitrary value specified from the outside, first storage means for holding the input allowable error value, and the trimming curve based on the input allowable error value A second storage means for preliminarily holding an equation for determining the number of polygonal lines when approximating by the polygonal line, and a determining means for determining the number of polygonal lines based on the equation. Display system. 7. The graphic display system according to claim 6, wherein the trimming curve has one variable t as a parameter.
Given a mapping c that is (t) = (u (t), v (t)), the parametric surface takes two variables u and v as parameters s (u, v) = (x (u, v ), Y (u,
v), z (u, v)), and when the tolerance value is specified by L, the formula for determining the number of polygonal lines when approximating the trimming curve by polygonal lines is , The maximum value M1 of the size of the first-order derivative (s · c) ′ of the composite map s · c with respect to t within the range of the parameter t of the trimming curve is obtained by the arithmetic unit, and the number of the broken lines is M
A graphic display system characterized by being an expression for determining 1 / L. 8. The graphic display system according to claim 1, wherein the trimming curve has one variable t as a parameter c.
Given a mapping c that is (t) = (u (t), v (t)), the parametric surface takes two variables u and v as parameters s (u, v) = (x (u, v ), Y (u,
v), z (u, v)), and when the tolerance value is specified by TOL, the formula for determining the number of polygonal lines when approximating the trimming curve by polygonal lines is , The maximum value M2 of the size of the second derivative (s · c) ″ of the composite map s · c by t within the range of the parameter t of the trimming curve is calculated by the arithmetic unit, and the number of the polygonal lines is √.
A graphic display system characterized by being an expression for determining (M2 / (8 × TOL)). 9. The range according to the mapping s of the parametric trimming curve c when the parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section) according to claim 7 or 8. , A determining unit that determines whether or not there is a common part with each section of the parametric curved surface, and if the determining unit determines that a common part exists, a range in which the common part exists is combined with the mapping map s. A graphic display system comprising: a maximum value calculating means for obtaining a maximum value of a size of (s · c) ′ or (s · c) ″ as a parameter range of c. 10. The parametric curved surface according to claim 9, wherein the parametric curved surface is given as a union of a plurality of curved surfaces (each curved surface is referred to as a section), and the parametric trimming curve is defined by a set of control points and has a convex hull. Control point that determines whether or not there is a common part between the range of the mapping s of the set of control points defining the parametric trimming curve and each section of the parametric curved surface, as the determining means. A graphic display system comprising: a determination unit.