JPH0632043B2 - Shading display method - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION The present invention will be described in the following order.

A Industrial field of use B Outline of the invention C Conventional technology (Figs. 11 and 12) D Problems to be solved by the invention (Figs. 6 to 10) E Means for solving problems (Figs. 1 and 2) F action (Figs. 1 and 2) G embodiment (G1) First embodiment (Figs. 1 and 2) (G2) Second embodiment (Fig. 3) ) (G3) Shading display processing procedure (Figs. 1, 2, and 4) (G4) Examination of division processing result of cutout area (Figs. 1, 2, and 5) (G5) Other Embodiment H Effect of the Invention A Field of Industrial Application The present invention relates to a shadow display method, for example, CAD (computer).
aided design) or CAM (computer aided manufa
cturing) and other free-form surfaces generated,
This is suitable when applied to the shading processing.

B Outline of the Invention The present invention provides a shadow display method in which a triangle unit area is cut out from patches that are respectively placed in a frame space and a shadow process is performed, and a plurality of second patches are provided via a boundary line of one side. When the triangular unit area is cut out from the first patch connected so as to be adjacent to the first patch, the parameters used in the corresponding patches of the plurality of second patches are used so that the first patch And it is possible to leave no area between the second patches that cannot be shaded.

C Conventional Technology For example, when designing the shape of an object having a free-form surface using a CAD method (giometric modeling), a designer generally uses a plurality of points in a three-dimensional space through which the surface should pass (this is called a node). Is specified, and a boundary curve network connecting the specified plurality of nodes is calculated by a computer using a predetermined vector function, thereby creating a curved surface represented by a so-called wire frame. Thus, it is possible to form a large number of framework spaces surrounded by boundary curves (such processing is called framework processing).

The boundary curve network formed by such framework processing itself has a rough shape that the designer intends to design, and can be represented by a predetermined vector function using the boundary curves surrounding each framework space. If the curved surface can be interpolated, it is possible to generate a free-form surface (which means one that cannot be defined by a quadratic function) designed by the designer as a whole. Here, the curved surface stretched in each frame space forms a basic element that constitutes the entire curved surface, and this is called a patch.

Conventionally, in this type of CAD system, a cubic tensor product, which is a Bezier equation or a B-spline equation, which is easy to calculate, is used as a vector function expressing a boundary curve network. Therefore, it is considered to be optimal for mathematical expression of a free-form surface having no special feature in terms of shape.

That is, a free-form surface that has no special features in terms of shape
When the points given to the space are projected on the xy plane, the projected points are often regularly arranged in a matrix, and when the number of the projected points is represented by m × n, The framework space can be easily extended by using a quadrilateral patch represented by a cubic Bezier equation.

However, when trying to stretch a smooth free-form surface into a framework space (for example, having a large distorted shape) that is characteristic in terms of shape, it is difficult to connect the patches to each other, and sophisticated mathematical operation processing is executed. Since it is necessary to do so, the calculation processing by the computer has become complicated and huge in the past, and
There was a problem that the calculation time became long.

As a method of solving this problem, regarding the shared boundary of adjacent framework spaces, an internal control point that satisfies the condition of tangential plane continuity is obtained, and a vector function expressing a free-form surface determined by the internal control point is obtained. Therefore, a method of forming a patch with a free-form surface has been proposed (JP-A-60-277448).
No. 60-290849, Japanese Patent Application No. 60-298638, Japanese Patent Application No. 61
-33412, JP-A-61-59790, Japanese Patent Application No. 61-64560, Japanese Patent Application No. 61-69368, Japanese Patent Application No. 61-69385).

By the way, if the free-form surface represented by the free-form surface data generated by such a method can be shaded by performing a shading process, the curved surface is three-dimensionally displayed on the display. It is considered that a high-quality image can be provided as a graphic image because it can be displayed, and a method for realizing such shading processing by linear interpolation has been conventionally proposed (Japanese Patent Application No. 60-37077).
issue).

This shading processing method is, for example, as shown in FIGS. 11 and 12, for the quadrilateral patch and the triangle patch S _{(u, v)} , the u and v directions that represent the coordinates of the patch S _{(u, v).} about,
Each is divided into a predetermined number of divisions (for example, 4 × 4 divisions), the triangular unit area UA is cut out for each divided area, and data representing the brightness of the free-form surface for the three vertices of the triangular unit area UA is obtained. The linear luminance calculation is performed on the luminances of all the pixels included in the triangular unit area UA based on the luminance plane extending over the triangular unit area by the three data.

By doing so, it is possible to dramatically execute the shadowing process in a short time, as compared with the case where the brightness is calculated for every pixel included in the triangular unit area UA.

D. Problem to be solved by the invention However, when shading processing is performed by this method, a portion (this is called a gear tap) that cannot be shading occurs at the boundary position of adjacent patches due to improper framework processing. There is a risk.

As a result of various studies on the cause, it is considered that when the framework space is framed, the segments of the shared boundary of two adjacent framework spaces do not match each other.

For example, four quadrilateral patches S _{(u, v) 1} , S _{(u, v) 2} , S represented by cubic Bezier equations in four adjacent framework spaces
_{When (u, v) 3} and S _{(u, v) 4} are stretched, first, as shown in FIG. 6, first and second patches S _{(u, v) 1} and S _{(u, v) 2 are provided.} By setting the internal control points so that the condition of tangential plane continuity is established via the shared boundary COM1, two patches S
_{Assume that (u, v) 1} and S _{(u, v) 2} are connected.

However, the second patch S _{(u, v) 2} has the third and fourth patches S _{(u, v) 3} and S _{(u , v)} via the shared boundaries COM2 and COM3 which are boundary curves that form one side thereof. _{, v) 4} adjacent to each other, and with respect to these two patches S _{(u, v) 3} and S _{(u, v) 4} , the patch S _In order to connect _{(u, v) 2,} as shown in FIG. 7, the second patch S _{(u, v) 2 is connected} to two quadrilateral patches S _{(u, v)} by the shared boundary COM4. _{twenty one}
And S _{(u, v) 22} and then one of the divided patches.
S _{(u, v) 21} is shared by the third boundary S2 via the boundary COM2.
_The other partition S _{(u, v) 22} which is connected to _{(u, v) 3} and is also connected to the fourth patch S _{(u, v) 4} via the shared boundary COM3, and the shared boundary COM4 is connected. It suffices to connect the two patches S _{(u, v) 21} and S _{(u, v) 22} that are divided via the above-mentioned method. In this way, a practically smooth free-form surface can be obtained by frame processing. It is thought that it can be set up in one framework space.

However, if the triangular unit area is cut out and the shading processing is performed using the linear interpolation method for the free-form surface formed as shown in FIG. 7, the first and second connected areas are connected via the shared boundary COM1. Patches S _{(u, v) 1} and S _{(u, v) 2}
There is a possibility that a portion (which is called a gear gap) where the shading processing cannot be performed occurs during the period.

That is, in FIG. 8, the first patch S _{(u, v) 1} and 2
The cutout of the triangular unit area with respect to the shared boundary COM1 between the patches S _{(u, v) 21} and S _{(u, v) 22} divided into two is performed by dividing each patch S _{(u, v) 1} , S _{(u, v) 21} and S _{(u, v) 22} are executed separately.

In the first patch S _{(u, v) 1} , for example, three division points (P ₁₁ , P ₁₂ , P ₁₃ ) are set on the shared boundary COM1 for the parameter v, and thus each of the four division ranges is divided. The four triangular unit areas UA1 are cut out and subjected to shading processing.

On the other hand, in the two patches S _{(u, v) 21} and S _{(u, v) 22} obtained by dividing the second patch S _{(u, v) 2} , on the shared boundaries COM11 and COM12, respectively. Are set to three division points (P ₂₁₁ , P ₂₁₂ , P ₂₁₃ ) and (P ₂₂₁ , P ₂₂₂ , P ₂₂₃ ) and four triangle unit areas UA21 and UA22 are cut out and subjected to shading processing.

Here, the triangle unit areas UA1, UA21, UA22 are
Using the parameter v in the v direction in each patch, 0.25 is divided at even intervals between v = 0 to 1 and cut out.

However, in this way, the cutout of the triangular unit area UA1 of the first patch S _{(u, v) 1} is performed by the nodes P _{00 to} P _{03 at} both ends of the shared boundary COM1 formed by the frame processing.
It is divided into four as one segment. On the other hand, because of the need to connect patches
S _{(u, v) 21} and S _{(u, v) 22} have a shared boundary COM11 between points P _{M and} P ₀₃ as a first segment with a point P _M where the shared boundary COM4 abuts the shared boundary COM1 as a boundary. At the same time, the division processing is executed with the shared boundary COM12 between the points P _{M and} P ₀₀ as the second segment.

For sharing boundary COM1 between node P ₀₀ to P ₀₃ In this way, the first Patsuchi S _{(u, v)} with respect to cutting out four triangular unit areas UA1 in ₁ side, new Patsuchi
On the S _{(u, v) 21} and S _{(u, v) 22} sides, the shared boundary COM1
The triangles UA21 and UA22 are formed by dividing the shared boundaries COM11 and COM12, which are line segments, into four, respectively, so that the shared boundary COM1 is divided into eight triangle unit areas UA21 and UA.
The result is to cut out 22.

Therefore, the triangular unit area U that divides the nodes P _{00 to} P ₀₃ into four
Sharing boundary COM1 position one side passes along the A1 likewise node P ₀₀ to P ₀₃ triangular unit areas between which is divided into eight UA21
The position where one side along the shared boundaries COM11 and COM12 of UA22 and UA22 does not become the same, and a gap GUPX occurs between them.

Therefore, the gear GUPX portion is not shaded, so that there is a portion that is not shaded so that a so-called hole is formed.

In FIGS. 6 to 8, the case where the first patch S _{(u, v) 1 is connected} to the second patch S _{(u, v) 2} under the condition of continuous tangential plane is described. As shown in FIG. 9 corresponding to FIG. 6, the first patch S _{(u, v) 1} is replaced with the second patch S _{(u, v) 2.}
As shown in FIG. 10 corresponding to FIG. 7, the second patch S _{(u, v) 2 is connected} to the two patches S _{( u, v) 21} and S
_{Also in the case} of dividing into _{(u, v) 22} and connecting to adjacent patches under the condition of tangential plane continuity, the result is that the gear GUPX is generated in the same manner as described above with reference to FIG.

If there is a portion that cannot be shaded in this way, the shaded free-form surface will appear unnatural.

The present invention has been made in consideration of the above points, and provides a shadow display method capable of cutting out a triangle unit area for shadow processing so that the gear GUPX as described above with reference to FIG. 8 does not occur. It is a proposal.

E Means for Solving the Problems In order to solve the problems, according to the present invention, a large number of frame spaces surrounded by boundary curves are formed by the frame processing, and parameters representing positions are respectively set in the frame spaces. With respect to the free-form surface generated by stretching the patch represented by the vector function, the triangular unit area is cut out from the respective patches stretched in the frame space, and based on the luminance information of the three vertex positions of the triangular unit area,
In the shading display method in which the luminance information about the pixels included in the triangular unit area is interpolated to shade the free-form surface, the boundary curve C of one side is
A plurality of second patches S _{(u, v) 21} , S _{(u, v) 22} via OM1.
Prior to cutting out the triangular unit area from the first patch S _{(u, v) 1} connected so as to be adjacent to the first patch S
_{(u, v) 1} is divided into a plurality of cutout regions corresponding to the plurality of second patches S _{(u, v) 21} and S _{(u, v) 22,} and a plurality of cutout regions are divided in each of the divided cutout regions. 2 patches S _{(u, v) 21} , S _{(u, v) 22}
The triangle unit area is cut out using the parameter v used in the corresponding patch.

F action In the cut-out region divided and formed in the first patch S _{(u, v) 1} , the corresponding second patches S _{(u, v) 21} , S _{(u, v) 22}
By cutting out the triangular unit area using the parameter v used in
Patches S _{(u, v) 1} and second patches S _{(u, v) 21} , S _{(u, v) 22}
Triangular unit regions can be cut out so as not to cause a gear gap at the shared boundary COM1 between them.

In this way, it is possible to give a natural shadow to the entire free-form surface without leaving a portion of the free-form surface that cannot be shaded as a hole.

G Embodiment One embodiment of the present invention will be described in detail below with reference to the drawings.

(G1) First Embodiment FIG. 1 shows an embodiment in which the present invention is applied to a free-form surface formed as described above with reference to FIGS. 6 and 7, and a patch S _{(u, v ) 21} and S _{(u, v) 22} , the triangular unit areas UA21 and UA22 are cut out in the same manner as described above with reference to FIG.

When executing the first Patsuchi S _{(u, v) 1} of shading cope with this, the first Patsuchi S _{(u, v) 1} and two adjacent Patsuchi S _{(u, v) 21} and Two cutout areas S are cut out so that a triangle unit area is cut out using the same parameters as S _{(u, v) 22.
It} is divided into _{(u, v) 11} and S _{(u, v) 12} . That is, as shown in FIG. 2 corresponding to FIG. 8, the common boundary COM4 between the two patches S _{(u, v) 21} and S _{(u, v) 22} passes through the point P _M where the common boundary COM1 abuts. The first boundary S _{(u, v) 1} is divided into two cut-out regions S _{(u, v) 11} by the shared boundary COM5 extending in the vertical direction.
And S _{(u, v) 12} .

The first cut-out region S _{(u, v) 11} is the range from the node P ₀₃ to the point P _M , using the same parameter v as the parameter of the patch S _{(u, v) 21} adjacent to the shared boundary COM11. Is divided into four, and the triangular unit area UA11 is cut out for each divided range.

Further, the second cutout area S _{(u, v) 12} is the shared boundary COM12.
The range from the node P ₀₀ to the point P _M is divided into four parts by using the same parameter v as that of the patch S _{(u, v) 22} adjacent to each other via the triangle unit area UA12.

By doing so, the triangular unit area UA cut out on the patch S _{(u, v) 21} side with the shared boundary COM 11 as a boundary.
21 in the direction along the shared boundary COM11
The position where the side passes is exactly the same as the position where one side passes in the direction along the shared boundary COM11 of the triangular unit area UA11 that is cut out on the first cutout area S _{(u, v) 11} side.

Similarly, for the shared boundary COM12, the patch S
Triangle unit area UA2 cut out on the _{(u, v) 22} side
The position where one side in the direction along the shared boundary COM12 of 2 passes is the position where one side in the direction along the shared boundary COM12 of the triangular unit area UA12 cut out on the second cutout area S _{(u, v) 12} side passes. Is the same as As a result,
Between the triangular unit areas UA21 and UA11 cut out on both sides of the shared boundary COM11 and COM12, UA
There is no gear gap between 22 and UA12.

Thus, the first and second cutout regions S _{(u, v) 11} and S
_Between the first patch S _{(u, v) 1} that is divided into _{(u, v) 12} and two adjacent patches S _{(u, v) 21} and S _{(u, v) 22} . There is no so-called hole that is not shaded.

According to the above-mentioned embodiment, when one of the adjacent patches is divided based on the condition for putting the patch on the frame space,
When shading is applied to undivided patches and adjacent divided patches, the shadow on the free-form surface looks unnatural by preventing the possibility of forming a so-called hole at the boundary position between them. It is possible to effectively prevent the possibility of a result.

(G2) Second Embodiment FIG. 3 shows an embodiment in which shading processing is performed on the free-form surface generated as described above with reference to FIG. 10, and in this case as well as in the case of FIG. Similarly, the first patch S _{(u, v) 1} is set to the first patch S _{(u, v) 1} using the same parameters as those of the adjacent patches S _{(u, v) 21} and S _{(u, v) 22} via the shared boundaries COM11 and COM12. 1st and 2nd cut-out area S
_It is divided into _{(u, v) 11} and S _{(u, v) 12} .

By doing so, the same effect as described above with reference to FIG. 1 can be obtained.

(G3) Shading display processing procedure The division processing of the triangle cutout region associated with such shading processing can be realized by executing the shading display processing procedure shown in FIG. 4 by a computer.

In FIG. 4, when the shadow display processing program is started in step SP1, the computer is in step S.
At P2, the patch data is read and wire frame display is performed. This patch data can be obtained by, for example, separately designing a free-form surface by a designer by framing a boundary curve network in a three-dimensional space.

In this display state, the operator has an inconvenient patch when performing shadow processing on the free-form surface displayed in wireframe (as shown in FIG. 6 or 9).
If it is determined that there is a defect by visually checking in Step SP3 whether there is any, Step SP3
After moving to step 4, the defective patch is divided into a plurality of cutout regions, and then the process proceeds to step SP5.

As described above with reference to FIGS. 7 and 10, the processing procedure of steps SP3 and SP4 remains unchanged when two adjacent patches are configured to cut out a triangular unit area by different parameters. If shading processing is performed on each patch, a region where shading processing cannot be performed may be left as a hole at the position of the boundary curve of adjacent patches. Therefore, it is prevented that a hole between the patches is generated.

If no defective patch can be found at step SP3, the process immediately proceeds to step SP5.

This step SP5 is a step for designating the value of the error δ between the shared boundary and one side along the shared boundary of the triangle unit area to be cut out. When the operator designates the error δ, the computer moves to step SP6. Then, the number of divisions of the triangular unit area with respect to the error δ is determined and the division processing is executed.

The processing of steps SP5 and SP6 is a subdivision processing so that the patch does not lose the curvilinear feeling originally possessed by the boundary curve when the triangular unit area is cut out from one patch. The operator visually confirms the processing result at step SP7,
If the curve is still lost in practical use, the process returns to step SP5 to re-specify the value of the error δ, and the triangular unit area division processing for the error δ is repeated.

When such processing ends, the computer moves to step SP8 and ends the shade display processing program.

(G4) Examination of division processing result of cutout area (G4-1) Boundary curve COM12 As described above with reference to FIGS. 1 and 2, the parameter v of the first patch S _{(u, v) 1} is changed from 0 to 0. For the shared boundary COM1 designated by changing it to 1, the shared boundary COM1 is divided into two shared boundaries COM11 and COM12, and the parameters v of the adjacent patches S _{(u, v) 21} and S _{(u, v) 22} are respectively divided. When four triangular unit areas UA11 and UA12 are respectively cut out while changing the value from 0 to 1, the cutout result is directly the original shared boundary C.
The same result is obtained as when eight triangular unit areas are cut out while changing from 0 to 1 using the parameter v of the patch S _{(u, v) 1} for OM1. This can be confirmed as follows.

The patch S _{(u, v)} that extends in each frame space is expressed by the following formula S _{(u, v)} = (1-u + uE) ³ (1-v + vF) ³ P ₍₀₀₎ ... (1) If it is expressed by an equation, the common boundary COM1 of the patch S _{(u, v) 1} in FIGS. 1 and 2 is represented by the following equation R (t) = (1-t + tE) ³ P ₀ (2) As shown in FIG. 5, while the parameter t changes from 0 to 1, the shared boundary COM1 from the nodes P ₀ to P _{3 is set} to the space curve R in the xyz coordinate space.
It can be represented as (t).

By the way, the common boundary COM4 (Fig. 2) is the common boundary COM.
When it is considered that the first patch S _{(u, v) 1} is divided by the shared boundary COM5 at the point P _M that abuts 1, the node P that can be designated when the parameter t changes from 0 to t _{0 The} common boundary COM12 consisting of the line segment from ₀ to the point P _M is
A space curve R ₁ having a parameter t ₁ (= 0 to ₁ ) as a variable, as in the following equation R ₁ (t ₁ ) = (1-t ₁ + t ₁ E) ³ P _{(0) 1} (3) It can be represented by (t ₁ ).

Since the shared boundary COM12 is a line segment of the shared boundary COM1, the space curve R ₁ (t ₁ ) after the division should be equal to the space curve R (t) before the division, and the equation (2) and ( 3)
From the formula, (1-t ₁ + t ₁ E) ³ P _{(0) 1} = (1-t + tE) ³ P ₀ ......
The relationship of (4) holds.

Here, the parameter t ₁ of the space curve R ₁ (t ₁ ) after the division is as follows when the shared boundary COM12 is specified from the node P ₀ to the point P _M.
Similarly, while changing from 0 to 1, nodes P ₀ to P _M
In order to specify the position of the space curve R (t), the parameter t is changed from 0 to t ₀ . If this relation is expressed by a proportional expression, t: 4 = t: t ₀ (5), and the relation between the parameters t and t ₁ can be expressed by t = t ₀ t ₁ (6).

By substituting the equation (6) into the right side of the equation (4) and adjusting the form of the equation and expanding it, (1-t-tE) ³ P ₀ = (1-t ₀ t ₁ + t ₀ t ₁ E) ³ P ₀ = {(1-t ₁ + t ₁ (1-t ₀ + t ₀ E)} ³ P ₀ = (1-t ₁ ) ³ P ₀ +3 (1-t ₁ ) ² t ₁ (1-t ₀ + t ₀ E) P ₀ +3 (1-t ₁ ) t ₁ ² (1-t ₀ + t ₀ E) ² P ₀ + t ₁ ³ (1-t ₀ + t ₀ E) ³ P ₀ (7) , For the parameter t ₁ the term (t−t ₁ ) ³ and 3
It may represent a sum of the _(1-t 1) of ² t ₁ claim, and _{3 (1-t 1) t} 1 2 term and t ₁ ³ sections.

Expanding the left side of the hand (4), the following equation _{(1-t 1 + t 1} E) 3 P (0) 1 = (1-t 0) 3 P (0) +3 (1-t 1) ² t ₁ EP _{(0) 1} +3 (1-t ₁ ) t ₁ ² E ² P _{(0) 1} + t ₁ ³ E ³ P _{(0) 1} = (1-t ₁ ) ³ P _{(0) 1} +3 ( 1-t ₁ ) ² t ₁ P _{(1) 1} ＋3 (1-t ₁ ) t ₁ ² P _{(2) 1} ＋ t ₁ ³ P _{(3) 1}・ ・ ・ Equation (7) as shown in (8) An expression having a term corresponding to the term is obtained.

In (8) _{P (0) 1, P (} 1) 1, P (2) 1, P (3) 1 , as shown in FIG. 5, the control points that define the shared boundary COM12, P _{(1 ) 1} = EP _{(0) 1} …… (9) P _{(2) 1} ＝ E ² P _{(0) 1} …… (10) P _{(3) 1} ＝ E ³ P _{(0) 1} …… (11) I have a relationship. Therefore, in order to satisfy the equation (4),
It suffices if the corresponding terms of the expressions (7) and (8) are equal to each other. Then, the relation of P _{(0) 1} = P ₀ ...... (12) holds for the term of (1-t ₁ ) ³ and this means that in FIG. 5, the control of the node P ₀ of the common boundary COM1 and the common boundary COM12 is performed. Point P
Indicates that _{(0) 1} is equal.

Next, for the term of 3 (1-t ₁ ) ² t ₁ , P _{(1) 1} = (1-t ₀ ) P ₀ + t ₀ P ₁ (13) holds. In the equation (13), P ₀ and P ₁ together with P ₂ and P ₃ form a control point defining the common boundary COM1 in FIG. 5, and P ₁ = EP ₀ (14) P ₂ = E ² P ₀ …… (15) P ₃ ＝ E ³ P ₀ …… (16) Therefore, in the equation (13), the control point P _{(1) 1} of the common boundary COM ₁₂ is a point where the control P ₀ and P ₁ of the common boundary COM ₁ are internally divided by the parameters (1-t ₀ ) and t ₀ . It means that there is (Fig. 5).

The _{3 (1-t 1) t} 1 for ² term _{P (2) 1 (1-} t 0 + t 0 E) 2 P 0 ...... (17) holds. If it deformed by expanding the right hand side of (17) _{(1-t 0 + t 0} E) 2 P 0 = (1-t 0) 2 P 0 +2 (1-t 0) t 0 P 1 + t 0 2 P ₀ = (1-t ₀₎ can be modified as _{{(1-t 0) P} 0 + t 0 P 1} + t 0 {(1-t 0) P 1 + t 0 P 2} ... (18). In formula (18), (1-t ₀ ) P ₀ + t ₀
P ₁ is the control point P ₀ and P ₁ as described above for the equation (13).
It is the point P _{(1) 1} that divides the space internally. Also, (1-t ₀ ) P ₁ + t ₀ P ₂
Is the control point P ₁ of the common boundary COM1 as shown in FIG.
And P ₂ are internally divided by the parameter (1-t ₀ ) and t ₀
Therefore, it can be expressed by the following equation (1-t ₀ ) P ₁ + t ₀ P ₂ ≡Q (19). Therefore, Eq. (17) can be expressed as P _{(2) 1} = (1-t ₀ ) P _{(1) 1} + t ₀ Q ...... (20), and eventually the second control point P of the shared boundary COM12
It can be seen that _{(2) 1} is at a position where the control point P _{(1) 1} and the internally divided point Q are further internally divided by the parameters (1-t ₀ ) and t ₀ .

Further, in the expressions (7) and (8), the following expression P _{(3) 1} = (1-t ₀ + t ₀ E) ³ P ₀ (21) holds for the term of t ₁ ³ and expands the right side thereof. In summary, (1-t ₀ + t ₀ E) ³ P ₀ = (1-t ₀ ) ³ P ₀ +3 (1-t ₀ ) ² t ₀ P ₁ +3 (1-t ₀ ) t ₀ ² P ₂ + t ₀ ³ P ₃ = (1-t ₀ ) ² {(1-t ₀ ) P ₀ + t ₀ P ₁ } +2 (1-t ₀ ) t ₀ {(1-t ₀ ) P ₁ + t ₀ P ₂ } + T ₀ ² {1-t ₀ ) P ₂ + t ₀ P ₃ } ... (22) It can be transformed. In equation (22), the first term (1-
t ₀ ) P ₀ + t ₀ P ₁ is the control point P _{(1) 1} from the equation (13), and the second term (1-t ₀ ) P ₁ + t ₀ P ₂ is the equation (19) described above. It can be seen that the point Q is internally divided.

On the other hand, the third term (1-t ₀ ) P ₂ + t ₀ P ₃ has a parameter (1-t ₀ ) between the control points P ₂ and P ₃ of the shared boundary COM 1 as shown in FIG. and it represents a point obtained by internally dividing at t _0, which following equation as represented by _{(1-t 0) P 2} + t 0 P 3 ≡P (2) 2 ...... (23), the second shared boundary COM11 Control point
Let P _{(2) 2} .

By doing so, the control point P _{(3) 1 in the} equation (21) is expressed by the following equation P _{(3) 1} = (1-t ₀ ) {(1-t ₀ ) P _{(1) 1} + t ₀ Q} + t ₀ {(1-t ₀ ) Q + t ₀ P _{(2) 2} } …… (24) can be arranged. However, in this equation (24), the first term (1-t ₀ ) P _{(1) 1} + t ₀ Q is the control point P _{(2) 1} as described above in equation (20), and the second term Of (1-t
₀ ) Q + t ₀ P _{(2) 2} is the internally divided point Q and control in FIG.
P _{(2) 2} between the parameters (1-t ₀₎ represents a point obtained by internally dividing at and t _0, which _{(1-t 0) Q +} t 0 P (2) 2 ≡P (1) 2 ...... (25 ), The first control point of the shared boundary COM11
Let P _{(1) 2} .

In this way, the control point P _{(3) 1 in} Eq. (24) is eventually expressed as P _{(3) 1} = (1-t ₀ ) P _{(2) 1} + t ₀ P _{(1) 2} …… (26) , The second control point of the shared boundary COM12
P _{(2) 1} and the first control point P _{(1) 2} of the shared boundary COM11
It can be seen that the point between and is internally divided by the parameter (1-t ₀ ) and t ₀ .

In this way, the shared boundary COM12 has four control points P _{(0) 1} , while the parameter t ₁ changes from 0 to ₁ .
It can be defined as a curve defined by P _{(1) 1} , P _{(2) 1} , and P _{(3) 1} , and if specified in this way, the condition of equation (4) can be satisfied, Eventually, the shared boundary COM12 represented while the parameter t ₁ changes from 0 to 1 is
The shared boundary COM1 before division coincides with the boundary curve represented when the parameter t is changed from 0 to t ₀ .

(G4-2) Boundary Regarding COM11 above, regarding the common boundary COM1 before division, the parameter t
Then verify the curve designated when changing from t ₀ to 1, that the curve designated matches when the parameter t ₂ for sharing boundary COM11 after division is changed from 0 to 1 To do.

When the space curve R ₂ (t ₂ ) representing the shared boundary COM11 after division is represented by the following equation R ₂ (t ₂ ) ＝ (1-t ₂ −t ₂ E) ³ P _{(0) 2} …… (27) In order for this curve R ₂ (t ₂ ) to be equal to the above R (t) representing the shared boundary COM1 described above for the following equation, the following equation (1-t ₂ −t ₂ E) ³ P _{(0) 2} = It is necessary that (1-t-tE) ³ P ₀ (28) holds.

Here, the parameter t ₂ is 0 between the parameters t and t _2.
Point P of common boundary COM11 when changing from 1 to 1
_While the curve from _{(0) 2} to the point P _{(3) 2} is specified,
The same curve has a relationship that can be specified by changing the parameter t from t ₀ to 1. Therefore, the proportional equation of the following equation t ₂ = 1 = (1-t ₀ ) ... (29) is obtained, and from this proportional equation, the parameter t is the following equation t = (1-t ₀ ) t ₂ ...... (30) Can be represented by the parameter t ₂ .

(30) Equation (28) and rearranging are substituted into the right side of equation (28) the right side of the equation ^{(1-t + tE) 3} P 0 = _{[1- {t 0 + (1-} t 0) t 2} + _{{t 0 (1-t 0} ) t 2} E ] _{^{3 P 0 = 1-t 0}} -t 2 + t 0 t 2 + t 0 E + t 2 E-t 0 t 2 E) 3 P 0 = {(1- _{t 2) - (1-t} 2) t 0 + (1-t 2) t 0 E + t 2 E} 3 P 0 = {(1-t 2) (1-t 0 + t 0 E) + t 2 E} 3 _{P 0 = {(1-t} 2) 3 (1-t 0 + t 0 E) 3 +3 (1-t 2) 2 (1-t 0 + t 0 E) 2 t 2 E +3 (1-t 2) ( 1-t ₀ + t ₀ E) t ₂ ² E ² + t ₂ ³ E ³ } P ₀ ... (31) can be expanded.

On the other hand, if the left side of Eq. (28) is expanded, (1-t ₂ + t ₂ E) ³ P ₍₀₎ 2 = (1-t ₂ ) ³ P ₍₀₎ 2 +3 (1-t ₂ ) ² t ₂ P _{(1) 2} ＋3 (1-t ₂ ) t ₂ ² P _{(2) 2} ＋ t ₂ ³ P _{(3) 2} …… (32) and (1-t ₂ ) ³ term and 3 (1-t ₂ ) ² t ₂ term,
3 (1-t ₂ ) t ₂ ² term and t ₂ ³ term can be arranged in the form of a sum, and by setting each term equal to the corresponding term in Eq. (31), The control points P _{(0) 2} , P _{(1) 2} , P _{(2) 2} , and P _{(3) 2} that define the curve of the common boundary COM11 can be obtained, respectively. Here, in equation (32), P _{(1) 2} = EP _{(0) 2} …… (33) P _{(2) 2} ＝ E ² P _{(0) 2} …… (34) P _{(3) 2} ＝ E ³ P There is a relationship of _{(0) 2} …… (35).

First, the following equation P _{(0) 2} = (1-t ₀ + t ₀ E) ³ P ₀ (36) is established from the terms (1-t ₂ ) ^{3 in} the equations (31) and (32). By deploying the right side _{(1-t 0 + t 0} E) P 0 = (1-t 0) 3 P 0 +3 (1-t 0) 2 t 0 P 0 +3 (1-t 0) t 0 2 P ₂ + t ₀ ³ P ₃ = (1-t ₀ ) ² {(1-t ₀ ) P ₀ + t ₀ P ₁ } +2 (1-t ₀ ) t ₀ {(1-t ₀ ) P ₁ + t ₀ P ₂ } + T ₀ ² {(1-t ₀ ) P ₂ + t ₀ P ₃ } ... (37) can be expressed and (1-t ₀ ) P ₀ + t ₀ P _{1 of the first term}
Is the control point P _{(1) 1} as described above with respect to equation (13), and (1-t ₀ ) P ₁ + t ₀ P ₂ of the _{second term} is internally divided as described above with respect to equation (19). Point Q, and further the third term (1-
t ₀ ) P ₂ + t ₀ P ₃ is the control point P as described above for Eq. (23).
₍₂₎ It is ₂ .

Therefore, the control point P _{(0) 2 in} Eq. (36) is P _{(0) 2} = (1-t ₀ ) {(1-t ₀ ) P _{(1) 1} + t ₀ Q} + t ₀ {(1-t ₀ ) Q + t ₀ P _{(2) 2} } ...... (38) can be rewritten. However, in the equation (38), the first term (1-t ₀ ) P _{(1) 1} + t ₀ Q is the control point P _{(2) 1} as described above for the equation (20), and the second term (1-t ₀ ) Q + t ₀ P _{(2) 2} is the control point P _{(1) 2} as described above for the equation (25). Therefore, the control point P _{(0) 2 in} equation (38) is expressed by the following equation P _{(0) 2} = (1-t ₀ ) P _{(2) 1} + t ₀ P _{(1) 2} ...... (39) , The control point P _{(2) 1} of the shared boundary COM12
And the control point P _{(1) 2 of the} common boundary COM11 can be obtained as a point internally divided by the parameters (1-t ₀ ) and t ₀ .

Next, from the term of 3 (1-t ₂ ) ² P ₂ in Eqs. (31) and (32), P _{(1) 2} = (1-t ₀ + t ₀ E) ² P ₁ ...... (40) . By deploying the right side _{(1-t 0 + t 0} E) 2 P 1 = (1-t 0) 2 P 1 +2 (1-t 0) t 0 P 2 + t 0 2 P 3 = (1-t 0 ) {(1-t ₀ ) P ₁ + t ₀ P ₂ } + t ₀ {(1-t ₀ ) P ₂ + t ₀ P ₃ } ... (41) (1-t ₀ )
P ₁ + t ₀ P ₂ is the point Q internally divided from Eq. (19), and the second term (1-t ₀ ) P ₂ + t ₀ P ₃ is the control point P _{(2) 2} from Eq. (23). Is. Therefore, the control point P _{(1) 2 in} Eq. (40) is internally divided as shown in the following equation P _{(1) 2} = (1-t ₀ ) Q + t ₀ P _{(2) 2} (42) It can be seen that the point Q and the control point P _{(2) 2} are internally divided by (1-t ₀ ) and t ₀ .

Further, from the term of 3 (1-t ₂ ) t ₂ ^{3 in} the equations (31) and (32), P _{(2) 2} = (1-t ₀ + t ₀ E) P ₂ = (1-t ₀ ) P ₂ ＋ t ₀ P ₃ …… (43) is established, and the control point P _{(2) 2} is finally the control points P ₂ and P.
₃ while it is a parameter (1-t ₀₎ and a point have been conducted under the internal division in t ₀ can be seen.

Furthermore, P _{(3) 2} = P ₃ (44) holds from the term t ₂ ³ in Eqs. (31) and (32). As a result, the control point P _{(3) 2} becomes a common boundary COM.
It can be seen that it is node P ₃ of 1.

Thus, the common boundary COM11 is the control point P _{(0) 2} P _{(1) 2} , P
It can be obtained by changing the parameter t ₂ from 0 to 1 according to _{(2) 2} and P _{(3) 2} . Then, this space curve can be expressed as an internally divided point based on the control points P ₀ , P ₁ , P ₂ , P ₃ of the shared boundary COM1, and thus the parameter t for the shared boundary COM1 from t _0. 1
It is possible to match the space curve drawn when changing up to.

(G4-3) Comprehensive evaluation Shared boundaries COM12 and CO drawn when the parameters t ₁ and t ₂ are changed from 0 to 1 as described above.
The spatial curve of M11 can be made to coincide with the spatial curve of the shared boundary COM1 drawn when the parameter t is changed from 0 to 1, thus, as described above with reference to FIG. Using the same parameters as those used when cutting out the area, the cut-out area S of the first patch S _{(u, v) 1
When a} triangle unit area is cut out from _{(u, v) 12} and S _{(u, v) 11} , the effect can produce the same effect as cutting out the triangle unit area about the line segment of the shared boundary COM1. As a result, it is possible to effectively prevent that there is a hole that is not shaded at the position of the shared boundary COM1.

(G5) Other Embodiments (1) In the above-mentioned embodiment, the frame space is set to 3
Although the case where the patch represented by the following Bezier formula is put up has been described, the order of the formula is not limited to this and may be a fourth order or higher.

(2) Furthermore, in the above-mentioned embodiment, the case where the patch represented by the Bezier formula is stretched is described, but the present invention is not limited to this, and the spline type, Coons
Other vector functions such as a formula and a Furgason formula may be used.

(3) Further, in the above-mentioned embodiment, the case of connecting between the quadrilateral patches was described, but the present invention is not limited to this, and the present invention can be similarly applied to the case of connecting patches of other shapes. .

H Effect of the Invention As described above, according to the present invention, when a triangle patch for shading processing is cut out from a patch connected to a plurality of patches via one boundary curve, it is used in the adjacent patch. Since the triangular unit area is cut out using the given parameters, it is possible to easily realize the shading display method that can perform shading processing on the entire free-form surface without creating a hole where shading processing cannot be performed. .

[Brief description of drawings]

FIG. 1 is a schematic diagram showing an embodiment of a shading display method according to the present invention, FIG. 2 is a schematic diagram showing a procedure for cutting out a triangular unit area, and FIG. 3 is a schematic diagram showing another application example. , FIG. 4 is a flow chart showing the shade display processing procedure, and FIG. 5 is a schematic diagram, FIG. 6 and FIG. 6 used for explaining the confirmation of the result when the triangular cut-out processing is performed on the divided shared boundary with different parameters. FIG. 7 is a schematic diagram used for explaining a conventional shading processing method, FIG. 8 is a schematic diagram showing a procedure for cutting out the triangular unit area, and FIGS. 9 and 10 are shading processing methods for other free-form surfaces. Schematic diagram, FIG. 11 and FIG.
FIG. 2 is a schematic diagram showing a method of cutting out a triangular unit area from a patch. S _{(u, v) 1} …… First patch, S _{(u, v) 11} , S _{(u, v) 12} …… Crop area, S _{(u, v) 21} , S _{(u, v) 22} … … Split patches,
UA, UA11, UA12, UA21, UA22 ... Triangular unit area, COM1, COM11, COM12, C
OM4, COM5 ... Sharing boundaries.

Claims (1)

[Claims] 1. A freedom generated by forming a large number of frame spaces surrounded by boundary curves by a frame process and applying a patch represented by a vector function having a parameter representing a position to each frame space. With respect to the curved surface, a triangle unit area is cut out from the patches respectively set in the framework space, and based on the brightness information of the three vertex positions of the triangle unit area, the brightness information about the pixels included in the triangle unit area is obtained. In the shading display method in which the free curved surface is shaded by interpolation, in a triangle unit from a first patch connected so as to be adjacent to a plurality of second patches via a boundary curve on one side. Prior to cutting out the area, the first patch is divided into the plurality of second patches.
Is divided into a plurality of cutout areas corresponding to each patch, and the triangle unit areas are cut out using the parameters used in the corresponding patch of the plurality of second patches in each of the divided cutout areas. A shading display method characterized in that