Ellipsoidal projection mapping

Details Ellipsoidal projection mapping Ellipsoidal projection mapping

1998-09-30

2001-02-20

Powell, Mark R. (Department: 2779)

Computer graphics processing and selective visual display system

Computer graphics processing

Graph generating

C345S440000, C702S169000

Reexamination Certificate

active

06191795

ABSTRACT:

BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates generally to computer graphics, and more particularly to the generation of texture coordinates for a surface using an ellipsoidal projective body.
2. Related Art
In the field of computer graphics, texture mapping is a well known technique for imparting realism to rendered surfaces. A surface S is amenable to this technique if there is a defined mapping M from S to a rectangular region T known as texture space. The region T is typically identified with an image whose values modulate or define, via the mapping M, rendered attributes of the surface S.
For a parametric surface
S
(
u,v
)=(
X
(
u,v
),
Y
(
u,v
),
Z
(
u,v
))
such a mapping exists and is simply the inverse mapping S
−1
. However, for non-parametric surfaces (e.g., implicit or polygonal surfaces), such a mapping does not exist a priori. Mappings must be therefore be constructed for these non-parametric surfaces.
Planar, cylindrical or spherical projections are commonly used methods for constructing such mappings. For example, cylindrical projection mapping uses a cylinder of length L positioned so that it encloses the surface to be mapped. Each point on the cylinder can be coordinatized by (y, &thgr;) where y is the distance of the point from the base of the cylinder, and &thgr; is an angle that sweeps around the axis of the cylinder. Each point on the surface can be assigned the coordinate (y, &thgr;) of the nearest corresponding point on the cylinder, and in this way the surface can be mapped to the rectangle [0, L]×[0,2&pgr;]. By scaling, the surface can be mapped to an arbitrary texture rectangle T.
Conventional projection mapping embeds an unmapped surface in a curvilinear coordinate system defined by a given projective body in a given position and orientation. In general we can consider a curvilinear coordinate system (u
1
, u
2
, u
3
), where
x=x
(
u
1
, u
2
, u
3
),
y=y
(
u
1
, u
2
, u
3
),
z=z
(
u
1
, u
2
, u
3
)
u
1
=u
1
(
x,y,z
),
u
2
=u
2
(
x,y,z
),
u
3
=u
3
(
x,y,z
)
Examples of curvilinear coordinate systems are planar (also known as cartesian), cylindrical and spherical coordinate systems. A point (x,y,z) on an unmapped surface embedded in such a curvilinear coordinate system will map to curvilinear coordinates (u
1
, u
2
, u
3
). By dropping one of these coordinates (which is equivalent to the mathematical operation of projection), we can coordinatize the point (x,y,z) as a pair of coordinates (u
i
, u
j
), thereby mapping the surface onto a portion of the plane.
For example, consider using a sphere of unit radius centered at the origin as a projective body. The sphere can be parameterized by the angular coordinates (&phgr;, &thgr;), where &phgr; is the azimuth angle (around the vertical z axis) and &thgr; is the elevation angle (from the horizontal xy plane). Each point (x, y, z) on the unmapped surface maps to the 3D coordinates (r, &phgr;, &thgr;). Dropping r maps the 3D coordinates to the nearest point on the sphere (&phgr;, &thgr;), thus mapping the surface to the texture rectangle [0,2&pgr;]×[0, &pgr;].
Conventional mapping systems and methods require the user to choose a projective body (i. e., a rectangle, cylinder, or sphere) and then position and orient the body around the surface which is to be mapped. Manually positioning and orienting a projective body can be time consuming and, if not done correctly, can result in mappings that either make poor use of the texture rectangle or introduce a high degree of distortion. A need therefore exists for an improved system and method for positioning and orienting a projective body.
Furthermore, conventional projective bodies used for projection texture mapping often don't “fit” surfaces very well, because they have limited degrees of freedom within which they can be altered. For instance, a sphere can only be altered by its radius, a cylinder only by its length and radius. Requiring a user to select an appropriate projective body can also lead to mistakes and poor texture mappings. A need therefore also exists for a projection texture mapping which uses a projective body that can be modified to better fit surfaces, preferably where the dimensions of the projective body are chosen automatically.
SUMMARY OF THE INVENTION
Briefly stated, the present invention is directed to a system and method for aligning a projective body with a surface and generating texture coordinates for the surface.
A preferred embodiment of the present invention includes selecting an appropriate projective body for a given surface, orienting and positioning the surface with respect to the projective body, projecting the surface onto the projective body, and then mapping the projective body onto the texture rectangle. An inertial ellipsoid is preferably used to orient the surface and the projective body. The inertial ellipsoid is also preferably used as the projective body. Furthermore, ellipsoidal coordinates are preferably used to project the surface onto the projective body.
An advantage of the present invention is that a projective body and a surface are automatically positioned and oriented with respect to one another. This feature relieves the user of having to manually position and orient either the surface or the projective body.
An advantage of the present invention is that ellipsoidal projective bodies can better fit the surfaces that are to be projected. Besides position and orientation, ellipsoids have more degrees of freedom than a sphere or rectangle. Ellipsoids are defined by three axes, one along each of the x, y, and z axes, whereas a sphere has one degree of freedom (i.e., a radius) and a rectangle has two (i. e., a length and a width). This added degree of adjustability allows a user to modify a given ellipsoid to “tune” the projection.
Another advantage of the present invention is that ellipsoidal projective bodies generalize rectangles, cylinders and spheres. That is, the axes of an ellipsoid can be modified such that the resulting ellipsoid approximates a rectangle (one axis set to zero), a cylinder (one axis long in relation to the other two), or a sphere (all three axes approximately equal).
Another advantage of the present invention is that ellipsoidal projective bodies don't have “poles” like spherical projective bodies. For a spherical projection, the small areas around the poles map in a highly distorted manner across the top and bottom of the texture rectangle.
Another advantage of the present invention is the automatic generation of a projective body having an approximate fit to the surface. The inertial ellipsoid can be calculated automatically, i.e., without any input from the user, and used as a projective body. The inertial ellipsoid is a first order approximation of the shape and orientation of the surface.
Further features and advantages of the invention, as well as the structure and operation of various embodiments of the invention, are described in detail below with reference to the accompanying drawings. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. The drawing in which an element first appears is indicated by the leftmost digit(s) in the corresponding reference number.


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patent: 6016152 (2000-01-01), Dickie
Williamson et al.,Calculus of Vector Functions, 3rdEdition, Prentice-Hall, 1972, pp. 310-322 and 361-376.
Kiyoshi, I. (Ed.),Encyclopedic Dictionary of Mathematics, 2ndEdition, MIT Pres, 1993, pp. 334-337 and 1006-1007.
Eisenhart, L.P.,A Treatise on the Differential Geometry of Curves and Surfaces, Ginn and Co., 1909 (reprinted 1960), pp. 226-231.
Akhiezer, N.I.,Elements of the Theory of Elliptic Functions, American Mathematical Society, 1990, pp. 175-181.
Gray, A.,Modern Differential Geometry of Curves and S

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