The intersection of two ringed surfaces and some related problems

Hee Seok Heo, Sung Je Hong, Jun Kyung Seong, Myung Soo Kim, Gershon Elber

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪uCu generated by a moving circle. Given two ringed surfaces ∪uC1 u and ∪vC2 v, we formulate the condition C1 u∩C2 v Ø ≠(i.e., that the intersection of the two circles C1 u and C2 v is nonempty) as a bivariate equation λ(u, v) = 0 of relatively low degree. Except for redundant solutions and degenerate cases, there is a rational map from each solution of λ(u, v) = 0 to the intersection point C1 u ∩ C2 v. Thus it is trivial to construct the.intersection curve once we have computed the zero-set of λ(u, v)= 0. We also analyze exceptional cases and consider how to construct the corresponding intersection curves. A similar approach produces an efficient algorithm for the intersection of a ringed surface and a ruled surface, which can play an important role in accelerating the ray-tracing of ringed surfaces. Surfaces of linear extrusion and surfaces of revolution reduce their respective intersection algorithms to simpler forms than those for ringed surfaces and ruled surfaces. In particular, the bivariate equation λ(u, v) = 0 is reduced to a decomposable form, f(u) = g(v) or ∥f(u) - g(v)∥ = |r(u)|, which can be solved more efficiently than the general case.

Original languageEnglish
Pages (from-to)228-244
Number of pages17
JournalGraphical Models
Volume63
Issue number4
DOIs
Publication statusPublished - 2001 Jul 1
Externally publishedYes

Fingerprint

Intersection
Ruled Surface
Curve
Circle
Efficient Algorithms
Surface of revolution
Rational Maps
Zero set
Extrusion
Robust Algorithm
Ray Tracing
Sweep
Decomposable
Ray tracing
Trivial
Form

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Geometry and Topology
  • Modelling and Simulation

Cite this

The intersection of two ringed surfaces and some related problems. / Heo, Hee Seok; Je Hong, Sung; Seong, Jun Kyung; Kim, Myung Soo; Elber, Gershon.

In: Graphical Models, Vol. 63, No. 4, 01.07.2001, p. 228-244.

Research output: Contribution to journalArticle

Heo, Hee Seok ; Je Hong, Sung ; Seong, Jun Kyung ; Kim, Myung Soo ; Elber, Gershon. / The intersection of two ringed surfaces and some related problems. In: Graphical Models. 2001 ; Vol. 63, No. 4. pp. 228-244.
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