JPH0652533B2 - Method for displaying partial curved surface and graphic display device using the same - Google Patents

Description

Description: BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a three-dimensional partial curved surface generation method and display method, and more particularly to a partial curved surface generation method and display method suitable for a curved surface cutting operation.

[Conventional technology]

Conventionally, in order to represent a partial curved surface subjected to a cutting operation, a method of dividing a partial curved surface into triangles and approximating with a polyhedron is known, but since this method has a large error with the shape model,
There was a drawback that it was not suitable for high precision display and numerical control processing.

[Object of the Invention]

An object of the present invention is to provide a partial curved surface display method that can accurately display a curved surface which is divided by a cutting operation, from elementary geometric curved surfaces such as planes and cylinders to free curved surfaces.

[Outline of Invention]

In addition to the above polyhedral approximation method, the following method can be considered as a method of representing a partial curved surface.

(1) As shown in FIG. 1, a curved surface patch S1
The partial curved surface is approximated by dividing into S25.

(2) As shown in FIG. 2, a three-dimensional closed curve C is given on the curved surface S, and a vector V indicating the area of the partial curved surface at each point of this curve is defined.

The method (1) is similar to the above-mentioned approximation method for a polyhedron, but the accuracy is improved because it is approximated by a curved surface patch. However, this method has the drawback that the surface patch changes each time the surface segmentation operation is added, and the surface data changes. Therefore, when an erroneous operation is added,
Since the original shape cannot be recovered, there is a problem that it is not suitable for interactive processing which is a general form of a graphic processing system. The method (2) uses a three-dimensional curve as the boundary curve of the partial curved surface, and thus has the advantage that it is not necessary to prepare a special curve format that takes values in the curved surface parameter u, V space as in this method. There is.

However, when a complex curved surface such as a free curved surface is targeted, the reliability decreases due to the following circumstances.

For example, when u and V curves are generated on a curved surface to display the curved surface in a mesh shape, it is necessary to calculate the intersection of the boundary curve and the display mesh curve and to cut off the portion outside the partial curved surface. In this case, both curves that are the targets of intersection calculation are space curves and free curves, so they have an interpolation error and do not intersect in a strict sense. Finding the intersection becomes difficult.

The present invention aims to solve such a drawback. That is, regarding the drawback that the curved surface data changes with each segmentation operation as in the method (1),
The solution was not to change the curved surface data, but to attach the boundary curve and the area designation data to the curved surface data. Regarding the problem that the calculation of the intersection point between the three-dimensional curves is low like the method (2), the boundary curve of the partial curved surface and the mesh curve for display are both curved surface parameters u,
By generating it as a two-dimensional curve that takes a value in the V space, it resulted in a highly reliable two-dimensional intersection calculation process. The above is the motive and idea of the present invention.

Example of Invention

Next, an outline of the partial curved surface generation method and the display method will be described with reference to an embodiment. When the cut curved surface is cut by the cut curved surface, the cut curved surface is divided into two partial curved surfaces. Any of them may be used as the partial curved surface obtained as a result of cutting, but here, the partial curved surface located on the front side of the cutting curved surface is used. Here, the front side of the cut curved surface means a normal vector of the cut curved surface at any point on the intersection line between the cut curved surface and the cut curved surface, that is, (S 'is a method of representing a cut surface) is assumed to be the side on which it is located. The S'and the normal vector will be described later. The outline of this embodiment will be described below with reference to FIGS. 3 and 4. In general,
A partial curved surface obtained by cutting a certain curved surface is often cut by another curved surface. Therefore, the case where the curved surface to be cut is such a partial curved surface will be described below.

In FIG. 3, 1A represents a partial curved surface generated from the curved surface 1. Reference numeral 7 is the boundary curve. 2 is the curved surface to be cut 1
It is a cutting curved surface for A. Since the curved surface 1 is the curved surface to be cut with respect to the cut curved surface 2, both the curved surface 1 and its partial curved surface 1A may be referred to as the curved surface to be cut below.

The intersection line 3 between the cut surface 1 and the cut surface 2 is obtained by a curve having a value in the u and V parameter space of the cut surface (see FIG. 3).
(a)). ...... [I] Calculation of intersecting line At a point P on the intersecting line 3, a normal vector 4 facing the front side of the cut curved surface 2 is calculated, and a vector 6 projected onto the tangent plane 5 at P of the cut curved surface 1 is calculated. And the area designating data is determined by this (FIG. 3 (b)). …
[II] Generation of area designation data The intersection points P ₁ and P ₂ of the intersection line 3 and the partial curved surface boundary curve 7 of the cut surface are obtained, and the curve connecting the points P1 and P2 and the boundary curve 7 of the partial curved surface 1A are combined. Then, a new boundary curve 10 for a new partial curved surface 1B obtained by cutting the original partial curved surface 1A with the cutting curved surface 2 is generated. ... [III] Boundary curve generation There are wire frame display, curved mesh display, cross-section display, shadow display, etc. as the curved surface display method. Here, a method of displaying a curved surface as a mesh curve will be described. Here, as the partial curved surface to be displayed, the original curved surface 1A to be cut is used.
Will be described as an example. The case of the cut partial curved surface 1B generated in this embodiment will be described later in detail.

U-direction curves 21 ′ to 24 ′ of curved surface 1, V-direction curve 2
5'to 29 'are generated (Fig. 4 (a)). [IV] Generation of curved surface mesh curve The intersection points P _{1 to} P ₁₀ of the curves 21 ′ to 29 ′ and the partial curved surface boundary curve 10 are obtained (FIG. 4 (b)). Then curve 21 '
Respectively through 29 ', divided by the intersection _P 1 ~ P _10, only the portion of the interior of the portion curved surfaces. 1A, taken out display in accordance with area indication data (4th
(Figure (c)). ... [V] Cutting of curved surface mesh curve The above is the outline of the partial curved surface generation method and display method. The above items [I] to [V] will be described in detail below.

[I] Intersection Line Calculation Since various methods have been proposed for the intersection line calculation method, it will be explained here that the intersection line is obtained as a curve having a value in the curved surface parameter space. The cut surface is S (u, V) and the cut surface is S '(u', V '). S and S'can be written by a vector-valued function that takes u, V and u ', V'as parameters, and takes a value in a three-dimensional space.

That is, S (u, V) = ( _Sx (u, V), _Sy (u, V), _Sz (u, V)) S '(u', V ') = ( _S'x (u ′, V ′), S ′ _y (u ′, V ′), S ′ _z
(U ', V')).

The line of intersection is the solution of the simultaneous equations S (u, V) = S '(u', V ').

Since this equation has three equations and four unknowns, it has one degree of freedom. Therefore, by introducing the parameter t, u = f ₁ (t), V = f ₂ (t), u ′ = f ₃ It can be solved as (t), V '＝ f ₄ (t). The specific calculation method is as follows. Since there is no problem if it can be solved only by transforming the formula, the method to solve by the Newton method is explained here. First S,
S'is approximated by a polyhedron and the intersection line is approximated by a polygonal line. In this way, the initial values u ₀ , V ₀ required for the Newton method,
u ′ ₀ , V ′ ₀ and u, V, for solving the equation of the intersection line
A constraint condition between u ′ and V ′, for example, V = au + b can be obtained. Therefore The following simultaneous equations are solved: u _{i + 1} = u _i + Δu _i V _{i + 1} = a (u _i + Δu _i ) + b u ′ _{i + 1} = u ′ _i + Δu ′ _i V ′ _{i + 1} = V ′ _i + ΔV ′ _I (i = 0,1,2 ...), and repeat until the required accuracy is reached. In this way, one of the points in the U and V space corresponding to the point on the intersection line 3 which is used as the initial value and is close to one bending point on the polygonal line that approximates the intersection line 3 U and V coordinates are obtained. This calculated point is the intersection line 3
Is a point belonging to a curve in the U and V spaces corresponding to. U,
In order to obtain the entire curve image of this curve in the V space, the same processing as above is performed for each of the polygonal points on the approximate polygonal line used as the initial value, and the above convergence calculation is repeated to obtain the intersection line 3. Since the passing points of the corresponding curves in the U and V spaces are obtained as a point sequence, an appropriate curve interpolation formula is applied to this to obtain a smooth curve in the U and V spaces. The curve thus obtained is a curve in the U and V spaces corresponding to the intersection line 3. In the following, this curve is also often simply called the line of intersection.

[II] Generation of Area Designation Data A method of generating area designation data relating to the intersection line obtained in [I] will be described with reference to FIG. The area designation data is
It is data indicating which side of the boundary of the partial curved surface is the partial curved surface region. In the following explanation, the area designation data is
It is defined as a vector in the u, V parameter space that is orthogonal to the boundary curve in the u, V parameter space, and this is called a region pointing vector.

The method of generating the area designating vector will be described below.
As described in the outline, the part to be cut and removed is the part on the back side of the cut surface.

In FIG. 5, the line of intersection between the cut surface 1 and the cut surface 2 is 3,
Let N be a normal vector 4 pointing to the surface of the cutting plane 2 at a point P on the intersection line. A vector 6 obtained by projecting N on the tangent plane 5 of the cut surface at P is N ₁ . N
Image of ₁ cut plane u, V parameter space Is obtained by solving the following equation.

Here, u ₀ and V ₀ are u and V parameter values of the point P, and S is an equation of the cut surface 2. Therefore, the region designating vector R is Is an orthogonal component vector obtained when orthogonal decomposition is performed with u and V tangent vectors of the intersection line in P.

Further, if the area vector is defined by only one point P, it is uniquely determined at each point on the intersection line, so the above calculation only needs to be performed once.

[III] Generation of boundary curve When the surface to be cut has already been made into a partial surface,
A process of generating a new partial curved surface boundary curve by adding the intersection line obtained in [I] will be described. FIG. 6 is an explanatory diagram of the processing of [III], which is described in the u and V parameter space of the curved surface to be cut. The range of u and V is 0u and V
It is normalized to 1. Further, in FIG. 6, 30 [I]
A curve 70 in the UV space corresponding to the line of intersection 3 obtained in (in the following, this curve may be simply referred to as line of intersection) 70 is in the UV space corresponding to the boundary curve 7 (Fig. 3) of the curved surface 1A to be cut. Is a partial curved surface boundary curve, and 11 and 12 are area designating vectors of the curves 30 and 70. That is, each point in the curved surface 1 shown in FIG. 3 corresponds to any point in the definition area 8 of this UV space, and each point in the partial curved surface 1A shown in FIG. It corresponds to any point within the boundary curve 70.

First, the intersection point {P _i } i = 1, n of the curve 30 and the curve 70 is calculated. Then, as shown in FIG. 6 (b), the curve 30 is extended if necessary, and the curve 30 and the boundary line 8 of the u and V parameter domain are connected to generate a closed curve 9. Then, the curves 70 and 9 are divided at intersection points {P _i } i = 1, n,
Partial curves 70a, 70b of curves 70 and 9 and 9a, 9
b ... is generated (FIG. 6 (c)). Finally partial curve 7
0a, 70b ... Inside the closed curve 9 (the side toward which the region pointing vector is directed) and partial curves 9a, 9b.
The one inside the closed curve 70 is selected, and the sixth
A new partial curved surface boundary curve 100 is generated as shown in FIG. Here, the process of selecting the partial curves inside the curves 70 and 9 is performed as follows. As an example, a method of determining whether or not the partial curve 70b exists inside the curve 9 will be described. First, a point Q ₁ on the curve 70b that is not on the curve 9 is selected, and the shortest point Q ₂ on the curve 9 from Q ₁ is obtained. And the region pointing vector 11 and the vector in Q ₂ If the inner product of and is positive, the curve 70b is inside the curve 9, and if it is negative, the curve 70b is outside the curve 9.

The above is the detailed description of the partial curved surface generation method. The display method of the partial curved surface will be described in detail below.

[IV] Generation of curved surface mesh curve A mesh curve for displaying a curved surface is generated at a constant pitch in the u and V directions in the u and V parameter space of the curved surface. This curve may be a straight line in the u, V parameter space.
Lines 21 to 29 in FIG. 7 (a) are mesh curves in the u and V parameter spaces, and 21 'to 29' in FIG. 7 (b).
Is a mesh curve in the three-dimensional space corresponding to 21 to 29.

[V] Cutting of curved surface mesh curve In FIG. 7C, a closed curve 100 is a boundary curve of a partial curved surface, and a vector 11 is an area designating vector attached to the curve 100. First, each of the straight lines 21 to 29 and the closed curve 10
Find the intersection with 0. If there is no intersection, the straight line does not need to be displayed because it does not exist in the partial curved surface area.
If there is an intersection, divide the straight line at the intersection and
Only the part existing inside 00 is selected and displayed. Here, the method of selecting the straight line portion existing inside the closed curve 100 is based on the method described in [III]. in this way,
In the present embodiment, the boundary of the partial curved surface generated by cutting is accurately obtained as a curve in the UV space, the partial region in the UV space surrounded by the boundary curve is determined, and the curved surface to be cut is Since the partial curved surface including points having U and V belonging to the partial region as parameters is displayed, the partial curved surface can be displayed with high accuracy.

The above is the detailed description of the display method of the partial curved surface.

FIG. 8 is a block diagram of a three-dimensional graphic processing apparatus for realizing the present invention. Here, the part surrounded by a frame represents the device, the solid line is a line through which geometric data flows, and the broken line is a control signal line.

Reference numeral 101 is a curve data input device including a keyboard, tablet, stylus pen, function keys, and card reader, 102 is a curtain memory for storing curved surface data input from 101, 103 is a curve intersection line, a partial curved boundary curve, a curved surface A memory that stores a two-dimensional curve such as a mesh curve for display, and a memory 104 that stores area designation data associated with a curved line between curved surfaces and a partial curved surface boundary curve. 105
Is a partial curved surface generation control device for controlling the device for processing [I], [II], and [III], 106 is an intersection line calculation device for calculating an intersection line of two curved surfaces, and 106 is an intersection line between curved surface data A device for calculating area designating data (area designating vector) from the data, 108 is a device for cutting the partial curved surface boundary curve at the intersection line obtained in 106, and generating a new partial curved surface boundary curve, 109 is two two-dimensional curves A device for calculating an intersection point between 110 and 110 is a memory for temporarily storing the intersection data calculated at 109. Further, 111 is a partial curved surface display control device, 112 is a device that generates a mesh-shaped straight line in a [01] × [01] rectangular region at a constant pitch, 113 is a device that cuts the straight line from closed curve data and region designation data, 1
Reference numeral 14 is a display data generation device that generates a point sequence in a three-dimensional space along the line segment obtained in 113 based on curved surface data, and 116 is a display device that displays graphic data.
A display control device 15 controls the display device.

〔The invention's effect〕

 The effects of the present invention are as follows.

Since the partial curved surface can be accurately generated, the generated partial curved surface can be displayed with high accuracy.

Since the boundary curve of the partial curved surface is generated by a two-dimensional curve having a value in the u, V parameter space of the curved surface, the main part of the cutting processing and display processing of the partial curved surface is reduced to a two-dimensional problem. For this reason, the processing is facilitated and high reliability can be guaranteed.

Even if the curved surface is segmented one after another, only the boundary curve data of the partial curve changes, and the curved surface data does not change. Therefore, there is an advantage that the curved surface data is retained even if there is an erroneous operation.

[Brief description of drawings]

1 and 2 show an example of a partial curved surface representation method, FIG. 3 shows an outline of a partial curved surface generation method, and FIG. 4 shows an outline of a partial curved surface display method. FIG. 5 shows how to obtain the area designating vector, 6th.
FIG. 7 is a detailed explanatory diagram of the partial curved surface generation method, and FIG. 7 is a detailed explanatory diagram of the partial curved surface display method. FIG. 8 is a block diagram of a three-dimensional graphic processing apparatus for realizing the present invention. Curved surface patch to approximate the S1~S25 ...... portion curved surface, S ...... curved surface, C ...... three-dimensional closed _curve, V 1 ~V ₆ ......
Vector showing partial curved surface area, 1 ... Cut surface, 1A ...
... Partial curved surface of curved surface 1; 1B: Other partial curved surface obtained by cutting curved surface 1A with curved surface 2;
Line of intersection between 2 and 4 ... normal vector that faces the front side of the cutting plane,
5 ... a tangent plane at a point P of the cut surface, 6 ... a vector obtained by projecting 4 onto 5, 7 ... a partial curved boundary curve of the cut surface,
10 ... Newly generated partial curved boundary curve, 21'-
29 '... Mesh curve for curved surface display, 8 ... Boundary line of domain of curved surface parameters u, V, 9 ... Closed curve combining 30 and 8, 30 ... In UV space corresponding to intersection line 3 Curve, 70 ... Curve in UV space corresponding to boundary curve of partial curved surface 1A, 100 ... Newly generated partial curved boundary curve, 11, 12 ... Region designating vector, 21-29 ...
... display mesh generated in u, V parameter space,
21'-29 '... 21-29 three-dimensional image.

Claims (8)

[Claims] 1. A device for processing data representing a three-dimensional figure and displaying it on a two-dimensional screen of a display device, wherein the position of a point on a curved surface in the three-dimensional space is defined as a function of parameters U and V in the UV space. And the position of the point on the curved surface in the three-dimensional space is defined by the parameters U and V, and a three-dimensional curved surface for cutting the curved surface to be cut is defined. Data representing a plurality of points in the UV space corresponding to a plurality of points located on a line of intersection in the three-dimensional space between the curved surface to be cut and the cutting curved surface is generated based on the represented data. From data representing a plurality of points in the UV space, those points are interpolated to generate data representing a curve in the UV space corresponding to the intersection line, and the original surface in the U, V space of the curved surface to be cut is generated. Data representing the boundary curve and U corresponding to the above intersection line Data representing a closed curve in the U, V space formed by the original boundary curve and the curve corresponding to the intersection line is generated by the data representing the curve in the V space, and the curved surface to be cut is generated. Based on the data represented and the data representing the closed curve in the U, V space, the U, V
The partial curved surface having the values U and V belonging to the partial regions in the U and V spaces defined by the closed curved line in the space as the partial curved surface obtained by cutting the curved surface to be cut with the cutting curved surface is displayed. A method for displaying a partial curved surface displayed on the two-dimensional screen of the device. 2. When displaying the partial curved surface, based on the data representing the closed curve, a plurality of predetermined curves belonging to the U and V spaces, each of which is composed of a portion belonging to the partial area, is displayed. Data representing a line segment is generated, and based on the data representing the curved surface to be cut and the data representing the plurality of line segments, a plurality of three-dimensional line segments corresponding to the plurality of line segments represent the partial curved surface. The partial curved surface display method according to claim 1, wherein the partial curved surface is displayed as a figure on the two-dimensional screen. 3. The curve according to claim 2, wherein each of the plurality of curves in the U, V space comprises a straight line in the U, V space in which U has a constant value or V has a constant value. How to display a partial curved surface. 4. When generating data representing a closed curve in the U, V space for the partial curved surface, the original boundary line in the U, V space of the cut surface is defined by the U, V corresponding to the intersection line. V
The partial curved surface according to claim 1, wherein a portion having a predetermined positional relationship with respect to the curve in the U, V space is selectively used among a plurality of portions obtained by dividing the curve in the space. Display method. 5. The selectively used portion has a three-dimensional curve corresponding to the portion, which is seen from a point on the intersection line of the curved surface to be cut and the curved surface to be cut, in the front surface area or the back surface. The method for displaying a partial curved surface according to claim 4, wherein the partial curved surface is a portion belonging to one of predetermined areas of the areas in the direction. 6. The selectively used portion has a predetermined positional relationship with respect to a normal vector of the curve in the U, V space corresponding to the intersection line in the U, V space. The method for displaying a partial curved surface according to claim 4, wherein the partial curved surface is a portion belonging to a region. 7. When generating the data indicating the closed curve, when the curved surface to be cut is a partial curved surface of another curved surface, the data indicating the boundary line in the UV space of the other curved surface and the UV
U with respect to another partial curved surface obtained when the other curved surface is cut by the cutting curved surface from the data representing the intersection line in the space,
Data representing another closed curve corresponding to a boundary in the V space is generated, and as the data representing the closed curve for the partial curved surface, the original boundary line in the UV space of the cut surface, the cut surface, and The method for displaying a partial curved surface according to claim 1, wherein data representing a closed curve formed by the curve in the UV space corresponding to the intersection line with the cut curved surface and included in the other closed curves is generated. 8. A display device, an input device for curved surface data, and a position in a three-dimensional space of a point on the curved surface input from the input device is defined as a function of parameters U and V in UV space. Data representing a three-dimensional cut surface, and data representing a three-dimensional cut surface for cutting the cut surface, in which the positions of points on the curved surface in the three-dimensional space are defined by the parameters U and V. To generate data representing a partial curved surface obtained by cutting the curved surface by the cutting curved surface, and displaying the partial curved surface on the two-dimensional screen of the display device based on the data. And a processing device for generating data representing a plurality of points in the UV space corresponding to a plurality of points located on a line of intersection between the curved surface to be cut and the cutting curved surface in a three-dimensional space. And a curved surface to be cut and the curved surface to be cut Determining a plurality of points in the UV space corresponding to a plurality of points located on the intersection line in the three-dimensional space of, and interpolating the plurality of points in the UV space,
Corresponding to the boundary line of the partial curved surface by means for generating data representing the intersection line in the UV space, the original boundary curve in the UV space of the curved surface to be cut, and the determined intersection line in the UV space. Means for generating data representing a closed curve in the UV space, and parameters U and V that belong to the partial area in the UV space defined by the determined closed curve in the UV space, of the curved surfaces to be cut. A graphic display device having means for generating graphic data representing the partial curved surface and displaying the partial curved surface on the display device.