Surface approximations in geometric modeling

Jung-Hong Chuang, Purdue University

Abstract

One of the major research efforts in the field of solid modeling focuses on extending the geometric coverage of modeling systems and on incorporating complex free-form surfaces. Some major obstacles to this goal include computing and representing intersection curves of two general surfaces, and computing and rendering very complex surfaces, including offset, Voronoi, and blending surfaces. We present local and global approximation schemes that are expected to be of practical value in overcoming the above problems. For parametric curves and surfaces, we present a method for computing an implicit approximant of low degree that approximates the curves or surface locally and achieves an order of contact that can be prescribed in advance. In principle, the method is capable of exact implicitization. Several surfaces, including offsets, blends, and Voronoi surfaces can be defined as the natural projections to R$\sp3$ of 2-surfaces in R$\sp{n}$, $n\/>$ 3. The 2-surface in R$\sp{n}$ is the zero set of a system of nonlinear equations in $n$ variables. We present algorithms that compute the normal, tangent vectors, and normal curvatures of the projected surface directly from the nonlinear system without variable elimination. Methods are presented as well that compute the explicit and parametric approximations of the projected surface locally. Finally, for a given 2-surface in R$\sp{n}$, $n\/>$ 3, an algorithm is given that computes the piecewise linear approximation of the projected surface globally with all major computations performed in 3-space.

Degree

Ph.D.

Advisors

Hoffmann, Purdue University.

Subject Area

Computer science

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