We present a Riemannian geometric framework for variational approaches to geometric design. Optimal surface design is regarded as a special case of the more general problem of finding a minimum distortion mapping between Riemannian manifolds. This geometric approach emphasizes the coordinate-invariant aspects of the problem, and engineering constraints are naturally embedded by selecting a suitable metric in the physical space. In this context we also present an engineering application of the theory of harmonic maps.

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