Computer graphics processing and selective visual display system – Computer graphics processing – Three-dimension
Reexamination Certificate
2001-03-02
2004-04-13
Nguyen, Phu C. (Department: 2671)
Computer graphics processing and selective visual display system
Computer graphics processing
Three-dimension
Reexamination Certificate
active
06720963
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates to a three-dimensional CAD (three-dimensional computer aided design: 3D-CAD) system, particularly to a three-dimensional CAD system applying a basic geometric theory called “topology CAD” (topology-applied CAD), and the recording medium utilized for the system.
DESCRIPTION OF THE RELATED ART
Currently, a system called a “three-dimensional CAD” is widely noticed, which enables the user to create a three-dimensional design using a computer. Most of the conventionally used 3D-CAD applies a method so-called a “solid CAD using Boolean expression”.
According to the method of solid CAD using Boolean expression, in order to define a solid graphic as shown in
FIG. 1
[
1
] where a cylinder CY penetrates above a square box BX, two basic graphics, the cylinder CY and the solid BX, are superposed, the intersection curve(trim line) TL of surface S
1
and surface S
2
is computed, and a “trimming process” is performed thereto where unnecessary portion of the surfaces are removed. Moreover, the Boolean operation tends to create an angular or boxy graphic, so it is necessary to provide thereto a process called a “fillet process” where the angular surfaces are rounded using a fillet surface FS. However, both the trim line TL and the fillet surface FS are approximate solutions, and accurately, the surfaces are not connected to each other.
Generally, the three-dimensional curved surface within a three-dimensional rectangular coordinate system having x, y, z axes representing real space is mathematically expressed as shown in the following equation (1):
f
(
x, y, z
)=0 (1)
However, according to conventional CAD system, parameters (&agr;, &bgr;) are commonly used to represent a point (x, y, z) within a rectangular coordinate system as shown in the following equation (2):
(
x, y, z
)
=S
(&agr;, &bgr;) (2)
wherein S is a vector function.
Here, we will refer to this surface (plane) defined by (&agr;, &bgr;) as a “parameter plane”, and the surface in real-space defined by a group of infinite points (x, y, z) corresponding to this parameter plane (&agr;, &bgr;) is referred to as “mapped plane”.
In this case, according to a normal CAD method, as shown in
FIG. 2
, the parameter plane (&agr;, &bgr;) is defined by a regular quadrilateral shape on a rectangular coordinate axis [vertex (0, 0), (1, 0), (1, 1), (0, 1)], the mapped plane is defined within this square region in the parameter plane, therefore a surface in real-space is expressed usually within the range of 0≦&agr;, &bgr;≦1.
FIG. 2
shows that within the parameter plane (&agr;, &bgr;), there are (m+1) points in direction &agr; (&agr;=0,&agr;=&agr;
1
,&agr;=&agr;
2
, . . . , &agr;=1), and (n+1) points in direction &bgr; (&bgr;=0, &bgr;=&bgr;
1
, &bgr;=&bgr;
2
, . . . , &bgr;=1); a total of (m+1)×(n+1) control points are set, and that the mapped plane (vertex Poo, Pmo, Pmn, Pon) is defined corresponding to these control points. The point P (x, y, z) on the mapped plane in real-space is the function of parameters &agr;, &bgr; using the rectangular coordinate system, and can be expressed by the following equations (3):
x=X
(&agr;, &bgr;)
y=Y
(&agr;, &bgr;)
z=Z
(&agr;, &bgr;) (3)
Most of the free curved surface used by the currently-applied CAD system is called a NURBS (non-uniform rational B-spline) surface, and is expressed by the following formula (4):
S
(
α
,
β
)
=
∑
i
=
0
m
∑
j
=
0
n
Nik
(
α
)
Njk
(
β
)
WijPij
∑
i
=
0
m
∑
j
=
0
n
Nik
(
α
)
Njk
(
β
)
Wij
(
4
)
wherein:
Nik (&agr;), Njk (&bgr;); so-called a B-spline function
Wij; weight function,
Pij; surface control point (position vector),
m+1, n+1: the number of control points in directions &agr; and &bgr;, the total number of control points being (m+1)×(n+1).
According to the prior art CAD method, as shown in FIG.
2
, a rectangular coordinate system is used for the parameter plane (&agr;, &bgr;), and when defining a surface having a complex local shape, the number n×m of the control points are increased for example to 10×10, . . . , or 50×100 according to the level of complexity.
Moreover, the straight lines drawn parallel to the vertical axis and the horizontal axis in the parameter plane (&agr;, &bgr;) [that is, lines &agr;=0, &agr;=1, &bgr;=0 or &bgr;=1 (expressing the boundary of the surface in real-space), or lines expressed by a fixed value of &agr; or a fixed value of &bgr;] is expressed by a higher function of parameters &agr;, &bgr; in mapping space (real-space). For example, according to methods such as the cubic Bezier patch or Coons patch, these lines in the parameter plane (&agr;, &bgr;) are expressed by a cubic Bezier curve: cubic parametric curve. However, even according to this method, the mapped curve in real-space corresponding to a straight line (u=&agr;=&bgr;) that diagonally cuts the parameter plane (&agr;, &bgr;) expressed by &agr;=&bgr; becomes a sextic function expressed by the sextic equation of the parameter (u).
It is possible to continuously define a plural number of surface to the horizontal and vertical directions using the curved surface expressed by such curved method, but it is not possible to bond plural surface diagonally. In other words, the cubic Bezier curved surface defined by a group of infinite cubic Bezier curved lines may look like a group of sextic Bezier curved lines, and it is impossible to share the cubic curve and the sextic curve as the boundary between surfaces. Therefore, such conditions provide an extremely strict restriction to the creation of graphics.
Generally, the definition of the interior of a surface is not clear from a surface expressed with boundary lines. There are cases where a cross-sectional line (diagram) cutting the surface with a plane is used to evaluate the shape of the surface. However, coloring (shading) is a natural method to confirm the shape of a surface visually. Moreover, the virtual surface within the computer is realized by machining along the mathematically defined surface using an NC machine tool.
It is impossible to perform a consistent logical operation process according to the solid CAD method using various surfaces or by the conventional free curved surfaces defined by a complex formula. Therefore, as shown in
FIG. 3
, in the final step, the curved surface is resolved into small surface segments expressed by the minimum degree of freedom, or in other words, microplanes [normally, very small triangular surface elements] PT. All the curved surfaces of a solid is resolved into microplanes PT, and the operation process is carried out using a linear discrete process approximating this group of microplanes [called “polygon”] PP. The result of the operation is approximated by planes and lines. As explained, the currently applied CAD method utilizes a polygon PP where triangular surfaces PT formed by cutting portions of a plane defined by infinite expansion are connected so as to approximately express the curved surface, and a surface expression representing a wide space (such as NURBS surface) is used to control the alignment of the triangular microplanes PT.
Moreover, when shading is to be performed according to the conventional CAD method or CG (computer graphics), the color (shadow level) at each vertex of the polygon is computed, and continuous gradation process (glow shade, phon shade) is performed to the rest of the area. According to the conventional computer technology, it is not possible to evaluate the continuity of the surfaces visually. As explained, the linear-discrete approximation according to such CAD technology also has various problems for application to an NC process where a machine tool is controlled to cut o
Shiiba Eiji
Yoshida Yasuhiko
CITEC Corporation
Nguyen Phu C.
Westerman Hattori Daniels & Adrian LLP
LandOfFree
Three-dimensional CAD system and recording medium for... does not yet have a rating. At this time, there are no reviews or comments for this patent.
If you have personal experience with Three-dimensional CAD system and recording medium for..., we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Three-dimensional CAD system and recording medium for... will most certainly appreciate the feedback.
Profile ID: LFUS-PAI-O-3196163