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Distortion minimal morphing: the theory for stretching. (English) Zbl 1167.68055

The authors develop a mathematical approach to the study of distortion minimal morphings over a continuous family of states in the context of morphs between \(n\)-dimensional oriented compact connected smooth manifolds without boundary embedded in \(\mathbb R^{n+1}.\) They also suppose that all the intermediate states are equipped with the volume forms induced by the standard volume form on \(\mathbb R^{n+1}.\) By definition, a morph is a transformation between two shapes through a set of intermediate shapes [cf. G. Wolberg, “Image morphing: a survey”, The Visual Computer 14, No. 8/9, 360–372 (1998)], while a minimal morph [cf. G. Yu, N. M. Patrikalakis and T. Maekawa, “Optimal development of doubly curved surfaces”, Comput. Aided Geom. Des. 17, No. 6, 545–577 (2000; Zbl 0945.68175)] is such a transformation that minimizes distortion; the distortion involves bending and stretching.
The main result is the proof of the existence of a distortion minimal morph (with respect to stretching) between every pair of isotopic submanifolds in \(\mathbb R^{n+1}.\) The proof is based on the analysis of the natural cost functional (for stretching) that measures the total relative change of volume with respect to a family of diffeomorphisms defining the morph. In particular, it turns out that the extremals of this functional are global minimizers.

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
49Q10 Optimization of shapes other than minimal surfaces
58E99 Variational problems in infinite-dimensional spaces
68U07 Computer science aspects of computer-aided design

Citations:

Zbl 0945.68175
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References:

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