JPS638881A - Shade displaying method - Google Patents

Abstract

PURPOSE:To stick a shade so that a curved line sense held inherently with a patch cannot be lost by selecting the number of the division at a triangle unit area segmented from one patch in accordance with the necessity. CONSTITUTION:Many triangle unit areas UA are segmented from one patch S(u,v) and concerning respective triangle unit areas UA, a shade processing is executed. The number of the division of the triangle unit areas UA to be cut out is selected to the value so that an error delta between border curved lines BOD1-BOD4 of the patch S(u,v) and one side of the triangle unit areas UA segmented along the border curved lines BOD1-BOD4 can be smaller than the prescribed value. Thus, the shade, in which the curved line sense held inherently by the border curved lines BOD1-BOD4 practically is not lose, can be stuck to the patch S(u,v).

Description

DETAILED DESCRIPTION OF THE INVENTION A. Field of Industrial Application The present invention relates to a method for displaying shading, for example, CAD (com
putter aided design) or CA
M
This method is suitable for applying shading to a free-form surface generated in a process such as URING.

B. Summary of the Invention The present invention provides a shading display method in which a triangular unit area is cut out from one patch and shading processing is performed, by making it possible to select the number of divisions of the triangular unit area as necessary. , you can add shading to the patch without losing its original curved feel.

C Conventional technology For example, when designing the shape of an object with a free-form surface using a CAD method (gioa+etric mod
eling), the designer in IQ is 3 that the curved surface should pass through.
By specifying multiple points in a dimensional space (these are called nodes) and having a computer calculate a boundary curve network connecting the specified multiple nodes using a predetermined vector function, it can be expressed in a so-called wire frame. Create a curved surface. In this way, a large number of framework spaces surrounded by boundary curves can be formed (such processing is called framework processing). The boundary curve network formed by such framework processing itself has the rough shape that the designer intends to design, and an interpolation operation is performed to create a curved surface that can be expressed by a predetermined vector function using the boundary curves surrounding each framework space. If it is possible to do so, it is possible to generate a free-form surface (which cannot be defined by a quadratic function) that is designed by the designer as a whole. Here, the curved surfaces stretched across each framework space form basic elements constituting the entire curved surface, and these are called patches.

Conventionally, in this type of CAD system, easy-to-calculate vector functions representing boundary curve networks, such as the Bezier equation and B-spline, have been used.
A third-order tensor product formed by the equation (ne) is used, and is considered to be optimal for mathematically expressing a free-form surface that does not have any special features in terms of shape, for example.

In other words, a free-form surface that has no special features in shape is
When points given in space are projected onto the xy plane, the projected points are often arranged regularly in a matrix, and when the number of projected points is expressed as m x n, the The framework space can be easily spanned using quadrilateral patches expressed by cubic Bezier equations.

However, when trying to create a smooth free-form surface on a curved surface with a characteristic shape (for example, a curved surface with a greatly distorted shape), it is difficult to connect the patches, and advanced mathematical calculations are required. Because of the need to do so, conventionally there was a problem that the calculation processing by the computer was complicated and enormous, and the calculation time was also long.

As a way to solve this problem, we find internal control points that satisfy the condition of tangent plane continuity for the shared boundaries of adjacent framework spaces, and use a vector function that represents the free-form surface determined by the internal control points to A method of applying patches made of curved surfaces has been proposed (Japanese Patent Application No. 60-277)
No. 448, Patent Application No. 1984-290849, Patent Application No. 1983-
No. 298638, Japanese Patent Application No. 1983-33412, Japanese Patent Application No. 1983
1-59790, Japanese Patent Application No. 61-64560, Japanese Patent Application No. 61-69368, Japanese Patent Application No. 61-69385).

By the way, if it is possible to add shading to a free-form surface represented by free-form surface data generated by such a method by applying shading processing, it is possible to display the curved surface three-dimensionally on a display, thereby creating a graphic image. It is believed that high-quality images can be provided as images, and a method has been proposed in which such shading processing is realized by linear interpolation (Japanese Patent Application No. 1983-370).
No. 77).

This shading processing method, for example, as shown in FIGS. 3 and 4, uses a quadrilateral patch and a triangle patch
), the U and V directions representing the coordinates of the patch S (un v) are divided into a predetermined number of divisions (for example, 4×4), and a triangular unit area UA is cut out for each divided area, and the triangle is Obtain data representing the brightness of the free-form surface for the three vertices of the unit area UA, and calculate the brightness of all pixels included in the triangular unit area UA based on the brightness plane stretched over the triangular unit area UA using the three data. is designed to perform linear interpolation calculations.

In this way, the shading process can be executed in a much shorter time than when the brightness is calculated one by one for every pixel included in the triangular unit area UA.

D Problems to be Solved by the Invention However, the shape of the framework space processed by this method is, for example, as shown in FIG. 4iABOD1 to BoD4, by dividing the boundary curves BOD1 and BOD5, which form long sides facing each other, into four, for example, one boundary curve BOD
1 is divided by three dividing points QI0, QgoSQ3 (1 into four triangular unit areas UAI i UAI 2,
Similarly, the other boundary curve BOD5 is divided into four triangular unit areas UA31 . When extracting UA32, UA33, and UA34, the following problems arise.

That is, triangular unit areas UAII-UA14, UA3
When shading processing is performed by cutting out 1 to UA34, the boundary curve BOD 1 curves over a relatively long range.
With respect to BOD5 and GUP1, gaps GUP1 and GUP2 between adjacent dividing points and straight lines forming one side of the cut out triangular unit region have portions where shading cannot be performed.

While the boundary curves BOD1 and BOD5 are naturally curved and have a curved feel, these gaps CUP 1 and GUP 2 are shaded parts because shading is performed using the curve connecting the dividing points as a boundary. This results in a straight-line feel, and the shadow processing may create an unnatural impression.

The present invention has been made in consideration of the above points, and provides a shading display method that does not lose the curved feel of a free-form surface when cutting out a triangular unit area from a patch forming a free-form surface for shading processing. This is what I am trying to propose.

Means for Solving Problem E To solve this problem, in the present invention, a large number of triangular unit areas U
In a shading display method in which a patch S (1
The number of divisions of the triangular unit area UA to be cut out from 1+ vl is determined by the boundary curve BODl of patch S (un Vl
A value is selected such that the error between ~BoD4 and one side of the triangular unit area UA cut out along the boundary curve BODI-BOD4 is smaller than a predetermined value.

Since one side of the triangular unit area UA cut out along the boundary curve of the F-action patch S iu+ vl is a straight line, the boundary curve BODI to BOD4 of the patch S (un v)
In principle - don't lose.

However, if the number of divisions of the triangular unit area UA is selected so as to make the error δ smaller than a predetermined value, the patch S (un It can be attached to vl.

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

In Fig. 1, when considering two points Q and a line segment Qt on the boundary curve BOD of one patch S (, 1vl), if the distance between the points Q and Ql is infinitesimal, then
The boundary curve BOD can be considered as a curve on one plane,
And the curvature radii ρ1 and ρ8 of points Q and Q2 can be considered to be equal values ρ as shown in the following equation 2 % (1) with respect to the center of curvature 0,
Therefore, the curve between points Q and Q is a circular arc (this can be called an arc Ql
−Qt). In addition,(
In equation 1), ρi is the radius of curvature of the midpoint Q of the arc Q + −Q t.

Under this assumption, if the straight line drawn between points Q and 02 (represented by straight lines Q, -Q) is one side of the cut out triangular unit area, then the arcs Q, -Q, The straight line Q, -0 connecting the midpoint QI4 and the center of curvature 0, and the straight line Q,
-Q, is a right angle, and if the angle between straight line Q, -0 and straight line Qg-0 is θ, then the angle between straight line QM-0 and straight line Q1-0 is also θ. .

and the midpoint Q of direct vAQ, -Qt. . and the distance between the centers of curvature 0 is d, and the straight line fiQi. −Length of Q2 is L/
2, then the straight line Q. The length of -Q is also L/2.

From this relationship, the error δ between the distance from the midpoint QM on the boundary curve BOD to the center of curvature O and the length d of the straight line Q, -0 can be expressed as the following formula % Formula % (2) , and the straight line Q. - Length L/2 of Ql and straight line Q. . -The length L/2 of Q2 is □-ρ
sinθ・・・・・・(3)
It can be expressed as.

This error δ is determined by the boundary curve BOD described above with respect to FIG.
1 or BOD5 and the cut out triangular unit area UAI
It represents the width of the gap GUP1 and GUP2 between I to UA14 or UA31 to UA34 on one opposing side.

From equation (2), cos θ becomes sin θ=□ (5) 2ρ. Here, sin” θ + cos θ = 1 Old... (6
), so by substituting equations (4) and (5) into equation (6), we get (7), which can be calculated as the distance between points Q and 02 (that is, the straight line Q , -Q! length)L''-4δ(
2ρ−δ) . . . The relationship (8) is obtained.

When error δ occurs in equation (8), δ≧O・
・・・・・・(9) 2ρ−δ≧O・・・・・・(10) The following relationship holds true, because of the boundary surface! BOD is point Q1 and 0
Since the arcs Q and -Qt are considered to be formed with respect to the center of curvature 0 based on the fact that the distance between In the boundary curve BOD, since the error δ is smaller than the radius of curvature ρ, equation (10) holds true.

Therefore, from the relationship in equation (8), the straight lines Q, -Q! measure the length of It can be found as

In this way, when looking at the arc Q + -Q t from the center of curvature 0, we have made it possible to express the length of the straight line Q I - Q z representing its chord by the radius of curvature ρ and the error δ, so When the triangular unit area UA used for shading processing is cut out for the patch S (u+V) stretched in the framework space (Fig. 5), there are gaps GUPI and G.U.P.
2 occurs, the gap GUPI and GU
Assuming that the maximum width of P2 corresponds to the error δ in equation (11), the straight line 11J is set so that the error δ is below an allowable value.
Select the length between I Q l and Q m.

When cutting out a triangular unit area UA and performing shading processing using a linear interpolation method for a curved boundary curve, in practice, the gap between one side of the triangular unit area UA and the boundary curve It is possible to effectively avoid the possibility of creating shading that would cause loss of impression.

Such division processing of the triangular unit area UA can be realized by executing the processing procedure shown in FIG. 2 by a computer.

In FIG. 2, when the division processing program is started at step SP1, the computer starts at step SP2.
Load the patch data and display it as a wireframe. This patch data is obtained by separately framing a boundary curve network in a three-dimensional space when a designer designs a free-form surface.

In this display state, the operator checks in step SP whether there are any defects including inappropriate patches when performing shading processing on the free-form surface displayed in wireframe.
In step 3, the patch is visually checked and when it is determined that there is a defect, the process moves to step SP4, where the patch with the problem is divided into a plurality of areas and triangular unit areas are cut out, and then the process moves to step SP5.

In the processing procedure of steps SP3 and SF3, if the method of dividing the patches is not appropriate, when the shading process is performed for each patch, the position of the boundary curve of the adjacent patch is
Since there is a risk that the portions that cannot be shaded will remain as holes, the formation of holes between the patches is prevented.

If no defective patch is found in step SP3, the process immediately moves to step SP5.

This step SP5 is a step in which the operator specifies the value of the error δ between the shared boundary BOD and one side along the shared boundary BOD of the triangular unit area UA to be cut out, as described above regarding equation (11). After specifying this error δ, the computer moves to step SP6, determines the number of divisions of the triangular unit area UA corresponding to the error δ, and executes the division process.

As described above with reference to FIG. 5, the processing in steps SP5 and SF3 is performed to redivide the patch so as not to lose the curvilinear feeling that the patch originally has as a boundary curve when cutting out a triangular unit area from one patch. In the process, the computer confirms the processing result in step SP7, and if there are still gaps GUP 1 and GUP2 in practice, a positive result is obtained in step SP7, so the computer returns to step SP5 described above and calculates the value of the error δ. The designation and division of the triangular unit area UA corresponding to the error δ are repeated.

In this way, the gap GUP loses the sense of curve in practical use.
By setting the error δ such that I and GUP2 do not occur, the number of divisions of the corresponding triangular unit area UA is limited, and by the number of divisions, it is possible to prevent gaps CUP1 and GUP2 from occurring from patch S (an v) in practice. Conditions can be set such that the triangular unit area UA can be divided.

When such processing is completed, the computer moves to step SP8 and ends the division processing program.

According to the above-described embodiment, in order to perform linear interpolation in shading processing, when cutting out a triangular unit area UA from one patch, the number of divisions for the boundary curve BOD is reset as necessary. By doing this, it is possible to add shading without losing the curvaceous feeling of the boundary curve that the patch originally has.

H Effects of the Invention As described above, according to the present invention, when a triangular unit area is cut out from one patch and subjected to shading processing, the number of divisions of the triangular unit area with respect to the boundary curve can be selected as necessary. By doing this, it is possible to easily add shading to the boundary curve of the patch without losing its original curved feel.

[Brief explanation of the drawing]

FIG. 1 is a route map showing an embodiment of the shading display method according to the present invention, FIG. 2 is a flowchart showing the processing procedure, and FIGS. 3 and 4 are explanations of the process of cutting out a triangular unit area from one patch. FIG. 5 is a route map used to explain the parts that cannot be shaded. S (un vl "" = "patch, BOD 1~B
OD4...Boundary curve, cupt, GUP 2.
·····gap.

Claims (1)

[Scope of Claim] In a shading display method in which a large number of triangular unit areas are cut out from one patch and shading processing is performed on each of the triangular unit areas, the number of divisions of the triangular unit area to be cut out from the patch is determined by the number of divisions of the triangular unit area to be cut out from the patch. A shadow display device characterized in that a value is selected such that an error between a boundary curve and one side of a triangular unit area cut out along the boundary curve is smaller than a predetermined value.