U. Clarenz, N. Litke, and M. Rumpf.
Axioms and variational problems in surface parameterization.
Comput. Aided Geom. Design, 21(8):727-749, 2004.
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For a surface patch on a smooth, two-dimensional surface in IR3, low-distortion parameterizations are described in terms of minimizers of suitable energy functionals. Appropriate distortion measures are derived from principles of rational mechanics, closely related to the theory of non-linear elasticity. The parameterization can be optimized with respect to the varying importance of conformality, length preservation and area preservation. A finite element discretization is introduced and a constrained Newton method is used to minimize a corresponding discrete energy. Results of the new approach are compared with other recent parameterization methods.