### 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_{1}
^{u} and ∪_{v}C_{2}
^{v}, we formulate the condition C_{1}
^{u}∩C_{2}
^{v} Ø ≠(i.e., that the intersection of the two circles C_{1}
^{u} and C_{2}
^{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 C_{1}
^{u} ∩ C_{2}
^{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 language | English |
---|---|

Pages (from-to) | 228-244 |

Number of pages | 17 |

Journal | Graphical Models |

Volume | 63 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2001 Jul 1 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Graphical Models*,

*63*(4), 228-244. https://doi.org/10.1006/gmod.2001.0553

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

Research output: Contribution to journal › Article

*Graphical Models*, vol. 63, no. 4, pp. 228-244. https://doi.org/10.1006/gmod.2001.0553

}

TY - JOUR

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

AU - Heo, Hee Seok

AU - Je Hong, Sung

AU - Seong, Jun Kyung

AU - Kim, Myung Soo

AU - Elber, Gershon

PY - 2001/7/1

Y1 - 2001/7/1

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 ∪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.

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 ∪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.

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 -