Bicubic surface rendering

Details Bicubic surface rendering Bicubic surface rendering

2000-12-11

2003-05-13

Mancuso, Joseph (Department: 2697)

Computer graphics processing and selective visual display system

Computer graphics processing

Shape generating

C345S423000

Reexamination Certificate

active

06563501

ABSTRACT:

FIELD OF THE INVENTION
The present invention relates to computer graphics and more specifically to a method and apparatus for rendering a bicubic surface on a computer system.
BACKGROUND OF THE INVENTION
Object models are often stored in computer systems in the form of surfaces. The process of displaying the object (corresponding to the object model) generally requires rendering, which usually refers to mapping the object model onto a two dimensional surface. At least when the surfaces are curved, the surfaces are generally subdivided or decomposed into triangles in the process of rendering the images.
A cubic parametric curve is defined by the positions and tangents at the curve's end points. A Bezier curve, for example, is defined by a geometry matrix of four points (P
1
-P
4
) that are defined by the intersections of the tangent vectors at the end points of the curve. Changing the locations of the points changes the shape of the curve.
Cubic curves may be generalized to bicubic surfaces by defining cubic equations of two parameters, s and t. In other words, bicubic surfaces are defined as parametric surfaces where the (x,y,z) coordinates in a space called “world coordinates” (WC) of each point of the surface are functions of s and t. Varying both parameters from 0 to 1 defines all points on a surface patch. If one parameter is assigned a constant value in the other parameters vary from 0 to 1, the result is a cubic curve, defined by a geometry matrix P comprising 16 control points (FIG.
4
).
While the parameters s and t describe a closed unidimensional interval (typically the interval [0,1]) the points (x,y,z) describe the surface:
x=f(s,t), y=g(s,t), z=h(s,t) s⊃[0,1], t⊃[0,1], where D represents an interval between the two coordinates in the parenthesis.
The space determined by s and t, the bidimensional interval [0,1]x[0,1] is called “parameter coordinates” (PC). Textures described in a space called “texture coordinates” (TC) that can be two or even three dimensional are described by sets of points of two ((u,v))or three coordinates ((u,v,q)). The process of attaching a texture to a surface is called “texture—object association” and consists of associating u, v and q with the parameters s and t via some function:
u=a(s,t) v=b(s,t) (and q=c(s,t))
FIGS. 1A and 1B
are diagrams illustrating a process for rendering bicubic surfaces. As shown in
FIG. 1A
, the principle used for rendering such a curved surface
10
is to subdivide it into smaller four sided surfaces or tiles
12
by subdividing the intervals that define the parameters s and t. The subdivision continues until the surfaces resulting from subdivision have a curvature, measured in WC space, that is below a predetermined threshold. The subdivision of the intervals defining s and t produces a set of numbers {si}i=1,n and {tj}j=1,m that determine a subdivision of the PC. This subdivision induces a subdivision of the TC, for each pair (si,tj) we obtain a pair (ui,j,vi,j) (or a triplet (ui,j,vi,j,qi,j) ). Here ui,j=a(si,tj), vi,j=b(si,tj), qi,j=c(si,tj). For each pair (si,tj) we also obtain a point (called “vertex”) in WC, Vi,j (f(si,tj),g(si,tj),h(si,tj)).
This process is executed off-line because the subdivision of the surfaces and the measurement of the resulting curvature are very time consuming. As shown in
FIG. 1B
when all resulting four sided surfaces (tiles)
12
is below a certain curvature threshold, each such resultant four-sided surface
12
is then divided into two triangles
14
(because they are easily rendered by dedicated hardware) and each triangle surface gets the normal to its surface calculated and each triangle vertex also gets its normal calculated. The normals are used later on for lighting calculations.
As shown in
FIG. 2
, bicubic surfaces
10
A and
10
B that share boundaries must share the same subdivision along the common boundary (i.e., the tile
12
boundaries match). This is due to the fact that the triangles resulting from subdivision must share the same vertices along the common surface boundary, otherwise cracks will appear between them.
The conventional process for subdividing a set of bicubic surfaces in pseudocode is as follows:
Step 1.
For each bicubic surface
Subdivide the s interval
Subdivide the t interval
Until each resultant four sided surface is below a certain
predetermined curvature range
Step 2
For all bicubic surfaces sharing a same parameter (either s or t)
boundary
Choose as the common subdivision the reunion of the
subdivisions in order to prevent cracks showing along the
common boundary
Step 3
For each bicubic surface
For each pair (si,tj)
Calculate (ui,j v,j qi,j Vi,j)
Generate triangles by connecting neighboring vertices
Step 4
For each vertex Vi,j
Calculate the normal Ni,j to that vertex
For each triangle
Calculate the normal to the triangle
The steps 1 through 4 are executed on general purpose computers and may take up to several hours to execute. The steps of rendering the set of bicubic surfaces that have been decomposed into triangles are as follows:
Step 5.
Transform the verices Vi,j and the normals Ni,j
Transform the normals to the triangles
Step 6.
For each vertex Vi,j
Calculate lighting
Step 7
For each triangle
Clip against the viewing viewport
Calculate lighting for the vertices produced by clipping
Step 8
Project all the vertices Vi,j into screen coordinates (SC)
Step 9
Render all the triangles produced after clipping and projection
Steps 5 through 9 are typically executed in real time with the assistance
of specialized hardware found in 3D graphics controllers.
The conventional process for rendering bicubic surfaces has several disadvantages. For example, the process is slow because the subdivision is so computationally intensive, and is therefore often executed off-line. In addition, as the subdivision of the tiles into triangles is done off-line, the partition is fixed, it may not account for the fact that more triangles are needed when the surface is closer to the viewer versus fewer triangles being needed when the surface is farther away. The process of adaptively subdividing a surface as a function of distance is called “automatic level of detail”.
Furthermore, each vertex or triangle plane normal needs to be transformed when the surface is transformed in response to a change of view of the surface, a computationally intensive process that may need dedicated hardware. Also, there is no accounting for the fact that the surfaces are actually rendered in a space called “screen coordinates” (SC) after a process called “projection” which distorts such surfaces to the point where we need to take into consideration the curvature in SC, not in WC.
Because the steps required for surface subdivision are so slow and limited, a method is needed for rendering a curved surface that minimizes the number of required computations, such that the images can potentially be rendered in real-time (as opposed to off-line). The present invention addresses such a need.
SUMMARY OF THE INVENTION
The present invention provides a method and system for rendering bicubic surfaces of an object on a computer system. Each bicubic surface is defined by sixteen control points and bounded by four boundary curves, and each boundary curve is formed by boundary box of line segments formed between four of the control points. The method and system of include transforming only the control points of the surface given a view of the object, rather than points across the entire bicubic surface. Next, a pair of orthogonal boundary curves to process is selected. After the boundary curves have been selected, each of the curves is iteratively subdivided, wherein two new curves are generated with each subdivision. The subdivision of each of the curves is terminated when the curves satisfy a flatness threshold expressed in screen coordinates, whereby the number of computations required to render the object is minimized.
According to the s

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