Chebyshev
Polynomials for
Boxing and
Intersections of
Parametric
Curves and
Surfaces
Alain
Fournier and
John
Buchanan
Appeared in Eurographics '94 (submitted March 1994)
TR-94-36
We propose and demonstrate the use of Chebyshev basis functions to speed up the computation of the intersections between rays and parametric curves or surfaces. The properties of Chebyshev polynomials result in the computation of better and tighter enclosing boxes. For surfaces they provide a better termination criterion to decide on the limits of subdivision, and allow the use of bilinear surfaces for the computation of the intersection when needed.
The efficiency of the techniques used depends on the relative magnitude of the coefficients of the Chebyshev basis functions. We show from a statistical analysis of the characteristics of several thousands surfaces of different origin that these techniques will result most of the time in significant improvement in speed and accuracy over other boxing and subdivision technqiues.