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Implicit representation of parametric curves and surfaces

1984; Academic Press; Volume: 28; Issue: 1 Linguagem: Inglês

10.1016/0734-189x(84)90140-3

1557-895X

Thomas W. Sederberg, D. C. Anderson, Ron Goldman,

Advanced Theoretical and Applied Studies in Material Sciences and Geometry

The following two problems are shown to have closed-form solutions requiring only the arithmetic operations of addition, subtraction, multiplication and division: (1) Given a curve or surface defined parametrically in terms of rational polynomials, find an implicit polynomial equation which defines the same curve or surface. (2) Given the Cartesian coordinates of a point on such a curve or surface, find the parameter(s) corresponding to that point. It is shown that a two-dimensional curve defined parametrically in terms of rational degree n polynomials in t can be expressed implicitly as a degree n polynomial in z and y. It is also demonstrated that a "bi-m-ic" parametric surface (where e.g., m = 3 for bicubic) can be expressed implicitly as a polynomial in x, y, z of degree 2m2. The degree of a rational bi-m-ic surface is also shown to be 2m2. The application of these results to finding curve and surface intersections is discussed.