Implicit representation of parametric curves and surfaces

✍ Scribed by T.W Sederberg; D.C Anderson; R.N Goldman

• Publisher: Elsevier Science
• Year: 1984
• Weight: 834 KB
• Volume: 28
• Category: Article
• ISSN: 0734-189X
• DOI: 10.1016/0734-189x(84)90140-3

✦ Synopsis

The following two problems arc 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 2m*. The degree of a rational bi-m-ic surface is also shown to be 2m'. The application of these results to finding curve and surface intersections is discussed.