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Geometric Curve Modeling with Sobolev Gradients

GeometricCurve ModelingwithSobolevGradientsRobertJ. ComputerScience& EngineeringUniversity of NorthTexasDenton,TexasJuly20,2006 AbstractTheSobolevgradient method is a powerfultool treattheproblemof constructingfaircurves by minimizinga fair-nessmeasuresubjectto includecurve length,curvature,torsion,and/orvariation of may includespeci edvalues,tangent vectors, may alsorequireperiodicity in thecaseof closedcurves, representedby discreteverticesanddivideddi erenceapproximationsto derivatives withrespectto Sobolevgradient method isthenparticularlye ective IntroductionOur rstapplicationof theSobolevgradient method to a geometricmodelingprobleminvolvedtheconst ructionof a surfacewithminimalsurfaceareaandconstrai nedto passthrougha spacecurve ([2]).

ing parametric space curves that minimize variation of curvature and take on pre-speci ed values, tangent vectors, and curvature vectors. The problem of constructing elastic curves constrained to lie in a regular surface is treated in [5], and [1] is addressed to the construction of periodic closed geodesics in a regular surface.

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