Title
The intersection of two ringed surfaces and some related problems
Abstract
We present an efficient and robust algorithm to compute the intersection curve of two ringed surfaces, each being the sweep ∪ u C u generated by a moving circle. Given two ringed surfaces ∪ u C u 1 and ∪ v C v 2 , we formulate the condition C u 1 ∩ C v 2 ≠ ∅ (i.e., that the intersection of the two circles C u 1 and C v 2 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 C u 1 ∩ C v 2 . 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.
Year
DOI
Venue
2001
10.1006/gmod.2001.0553
Graphical Models
Keywords
Field
DocType
ringed surface,related problem,ruled surface,surface of revolution,ray tracing
Degenerate energy levels,Combinatorics,Surface of revolution,Bivariate analysis,Intersection curve,Mathematics,Ruled surface
Journal
Volume
Issue
ISSN
63
4
Graphical Models
Citations 
PageRank 
References 
12
0.69
15
Authors
5
Name
Order
Citations
PageRank
Hee-Seok Heo1342.58
Sung Je Hong226728.92
Joon-Kyung Seong324818.55
Myung-soo Kim4118292.56
Gershon Elber51924182.15