It is shown that two generic triangular surface patches with no base points and of parametric degree m and n respectively, intersect in a curve of degree m2n2 which is generally of genus 2m2n2 - 3 2m2n - 3 2n2m + 1. Similarly, two generic tensor product surface patches of parametric degree m1×m2 and n1×n2 respectively, intersect in a curve of degree 4m1m2n1n2 and generally of genus 8m1m2n1n2 - 2m1m2(n1 + n2) - 2n1n2(m1 + m2) + 1. For example, two general bicubic patches in general position intersect in a curve of degree 324 and of genus 433. The significance of this genus value lies in the fact that only curves of genus 0 can be expressed parametrically using rational polynomials. Genus and degree equations are also derived for intersection curves involving surface patches with simple base points. A class of surfaces is identified for which any plane section is a rational curve.
ASJC Scopus subject areas
- Modeling and Simulation
- Automotive Engineering
- Aerospace Engineering
- Computer Graphics and Computer-Aided Design