Ray tracing parametric patches
SIGGRAPH '82 Proceedings of the 9th annual conference on Computer graphics and interactive techniques
A unified approach to resultant matrices for Bernstein basis polynomials
Computer Aided Geometric Design
Shifting planes always implicitize a surface of revolution
Computer Aided Geometric Design
Exact, efficient, and complete arrangement computation for cubic curves
Computational Geometry: Theory and Applications
On the problem of proper reparametrization for rational curves and surfaces
Computer Aided Geometric Design
Bernstein Bezoutians and application to intersection problems
Computer Aided Geometric Design
The Bernstein polynomial basis: A centennial retrospective
Computer Aided Geometric Design
Using polynomial interpolation for implicitizing algebraic curves
Computer Aided Geometric Design
A partial solution to the problem of proper reparametrization for rational surfaces
Computer Aided Geometric Design
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In this paper vector techniques and elimination methods are combined to help resolve some classical problems in computer aided geometric design. Vector techniques are applied to derive the Bezout resultant for two polynomials in one variable. This resultant is then used to solve the following two geometric problems: Given a planar parametric rational polynomial curve, (a) find the implicit polynomial equation of the curve (implicitization); (b) find the parameter value(s) corresponding to the coordinates of a point known to lie on the curve (inversion). The solutions to these two problems are closed form and, in general, require only the arithmetic operations of addition, subtraction, multiplication, and division. These closed form solutions lead to a simple, non-iterative, analytic algorithm for computing the intersection points of two planar parametric rational polynomial curves. Extensions of these techniques to planar rational Bezier curves are also discussed.