Motion of Curves Constrained on Surfaces Using a Level-Set Approach

Author： Cheng L-T. Burchard P. Merriman B. Osher S.

ISSN： 0021-9991

Source： Journal of Computational Physics, Vol.175, Iss.2, 2002-01, pp. : 604-644

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Abstract

The level-set method has been successfully applied to a variety of problems that deal with curves in R² or surfaces in R³. We present here a combination of these two cases, creating a level-set representation for curves constrained to lie on surfaces. We study primarily geometrically based motions of these curves on stationary surfaces while allowing topological changes in the curves (i.e., merging and breaking) to occur. Applications include finding geodesic curves and shortest paths and curve shortening on surfaces. Further applications can be arrived at by extending those for curves moving in R² to surfaces. The problem of moving curves on surfaces can also be viewed as a simple constraint problem and may be useful in studying more difficult versions. Results show that our representation can accurately handle many geometrically based motions of curves on a wide variety of surfaces while automatically enforcing topological changes in the curves when they occur and automatically fixing the curves to lie on the surfaces. The method can also be easily extended to higher dimensions. © 2002 Elsevier Science (USA).