# A Marching Method for Computing Intersection Curves of Two Subdivision Solids

• Xu-Ping Zhu
• Shi-Min Hu
• Chiew-Lan Tai
• Ralph R. Martin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)

## Abstract

This paper presents a marching method for computing intersection curves between two solids represented by subdivision surfaces of Catmull-Clark or Loop type. It can be used in trimming and boolean operations for subdivision surfaces. The main idea is to apply a marching method with geometric interpretation to trace the intersection curves. We first determine all intersecting regions, then find pairs of initial intersection points, and trace the intersection curves from the initial intersection points. Various examples are given to demonstrate the robustness and efficiency of our algorithm.

## Keywords

Intersection Point Intersection Curve Subdivision Surface Control Mesh Control Vertex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Barnhill, R.E., Farin, G., Jordan, M., Piper, B.R.: Surface/surface intersection. Computer Aided Geometric Design 4, 3–6 (1987)
2. 2.
Barnhill, R.E., Kersey, S.N.: A marching method for parametric surface/surface intersection. Computer Aided Geometric Design 17, 257–280 (1990)
3. 3.
Biermann, H., Kristjansson, D., Zorin, D.: Approximate Boolean operations for subdivision surfaces. In: Proc. SIGGRAPH 2001, pp. 185–194 (2001)Google Scholar
4. 4.
Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design 10(6), 350–355 (1978)
5. 5.
Doo, D., Sabin, M.: Analysis of the behavior of recursive division surfaces near extraordinary points. Computer Aided Design 10(6), 257–268 (1978)
6. 6.
Grinspun, E., Schröder, P.: Normal bounds for subdivision-surface interference detection. In: Proc. IEEE Scientific Visualization (2001)Google Scholar
7. 7.
Hohmeyer, M.: A surface intersection algorithm based on loop detection. International Journal of Computational Geometry and Applications 1(4), 473–490 (1991)
8. 8.
Hu, S.M., Sun, J.G., Jin, T.G., Wang, G.Z.: Computing the parameters of points on NURBS curves and surfaces via moving affine frame method. Journal of Software 11(1), 49–53 (2000)Google Scholar
9. 9.
Kobbelt, L.: $$\sqrt{3}$$ subdivision. In: SIGGRAPH 2000 proceedings, pp. 103–112 (2000)Google Scholar
10. 10.
Krishnan, S., Manocha, D.: An efficient surface intersection algorithm based on lower dimensional formulation. ACM transactions on Graphics 16(1) (1997)Google Scholar
11. 11.
Litke, N., Levin, A., Schröder, P.: Trimming for Subdivision Surfaces. Technical report, Caltech (2000)Google Scholar
12. 12.
Loop, C.T.: Smooth Subdivision Surfaces Based on Triangles. M.S. Thesis, Department of Mathematics, University of Utah (1987)Google Scholar
13. 13.
Nasri, A.H.: Polyhedral Subdivision Methods for Free-form Surfaces. ACM Trans. Graphics 6(1), 29–73 (1987)
14. 14.
Seidel, R.: The nature and meaning of perturbations in geometric computing. Discrete and Computational Geometry 19(1), 1–17 (1998)
15. 15.
Stam, J.: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In: Proc. SIGGRAPH 1998, pp. 395–404 (1998)Google Scholar
16. 16.
Stam, J.: Evaluation of Loop subdivision surfaces. In: SIGGRAPH 1999 Course Notes (1999)Google Scholar
17. 17.
Zorin, D., Schröder, P.: Subdivision for modeling and animation. In: SIGGRAPH 2000 Course Notes (2000)Google Scholar
18. 18.
Zorin, D., Kristjansson, D.: Evaluation of piecewise smooth subdivision surfaces. Visual Computer (2001)Google Scholar

## Authors and Affiliations

• Xu-Ping Zhu
• 1
• Shi-Min Hu
• 1
• Chiew-Lan Tai
• 2
• Ralph R. Martin
• 3
1. 1.Department of Computer Science and TechnologyTsinghua UniversityBeijingChina
2. 2.Department of Computer ScienceHong Kong University of Science and TechnologyHong KongChina
3. 3.School of Computer ScienceCardiff UniversityCardiff