TY - JOUR

T1 - The intersection of two ringed surfaces and some related problems

AU - Heo, Hee Seok

AU - Je Hong, Sung

AU - Seong, Joon Kyung

AU - Kim, Myung Soo

AU - Elber, Gershon

N1 - Funding Information:
The authors thank the anonymous referees for their comments which were very useful in improving the presentation of this paper. All the algorithms and f gures presented in this paper were implemented and generated using the IRIT solid modeling system [1], developed at the Technion, Israel. This research was supported in part by the Korean Ministry of Information and Communication under the University Basic Research Program of the year 1999, in part by the Korean Ministry of Science and Technology under the National Research Lab Project, and also in part by the Korean Ministry of Education under the Brain Korea 21 Project.

PY - 2001/7

Y1 - 2001/7

N2 - 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 ∪uC1u and ∪vC2v, we formulate the condition C1u∩C2v Ø ≠(i.e., that the intersection of the two circles C1u and C2v 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 C1u ∩ C2v. 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.

AB - 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 ∪uC1u and ∪vC2v, we formulate the condition C1u∩C2v Ø ≠(i.e., that the intersection of the two circles C1u and C2v 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 C1u ∩ C2v. 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.

UR - http://www.scopus.com/inward/record.url?scp=0035410673&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035410673&partnerID=8YFLogxK

U2 - 10.1006/gmod.2001.0553

DO - 10.1006/gmod.2001.0553

M3 - Article

AN - SCOPUS:0035410673

VL - 63

SP - 228

EP - 244

JO - Graphical Models

JF - Graphical Models

SN - 1524-0703

IS - 4

ER -