This thesis investigates three related topics on the geometry of the space of planar shapes.
In the first part, we setup the EPDiff (Euler-Poincaré equation on the group of diffeomorphisms) and investigate its singular solutions, Teichons, analogues of solitons. A symplectic integration scheme has been implemented to solve a Hamiltonian problem that arises in this part.
In the second part, the formula for the sectional curvature of the Weil-Petersson metric has been derived. It has been shown that the formula coincides with Arnold’s formula in the case of the metric without a null space.
The third part concerns numerical implementation of optimization problem of finding geodesics by minimizing the Weil-Petersson energy. The comparison of methods is shown alongside with the examples of geodesics.
|School Location:||United States -- Rhode Island|
|Source:||DAI-B 71/11, Dissertation Abstracts International|
|Subjects:||Applied Mathematics, Mathematics|
|Keywords:||Planar shapes, Sectional curvature, Space, Weil-Petersson metric|
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