Analyzing surfaces: the truth about elimination theory

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Analyzing surfaces: the truth about elimination theory

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Annual Symposium on Computational Geometry archive
Proceedings of the third annual symposium on Computational geometry table of contents

Waterloo, Ontario, Canada

Pages: 348 - 353

Year of Publication: 1987

ISBN:0-89791-231-4

Authors

A. Schwartz	Department of Mathematics, The University of Michigan Ann Arbor, Michigan
C. M. Stanton	Department of Mathematics, The University of Michigan Ann Arbor, Michigan

Sponsors

SIGACT: ACM Special Interest Group on Algorithms and Computation Theory
SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

Publisher

ACM New York, NY, USA

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ABSTRACT

In this paper we discuss the relationship between parametric and implicit representations of algebraic surfaces. This problem is fundamental in computer graphics and computer aided design. In addition to presenting our results, we are trying to expose a number of basic mathematical ideas and techniques that are useful in the theory of algebraic surfaces from the point of view of graphics. We also hope that this paper can serve as a useful, although necessarily only partial, introduction to the study of algebraic surfaces. We use extended versions of Sylvester's resultant to present a precise and accessible proof that a surface in C3 with a rational parameterization of degree n, is defined by a polynomial equation F(x1,x2,x3) = 0 of degree N ≤ n2. We make no claim that this result is original, although we have not found it anywhere. In addition to the calculus of resultants we use some field theory and Bezout's Theorem. We discuss some examples of surfaces from this point of view, and we prove that the real torus has no parameterization of degree n < 4.

REFERENCES

Note: OCR errors may be found in this Reference List extracted from the full text article. ACM has opted to expose the complete List rather than only correct and linked references.

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