For applications to mathematical form recognition, the author considers the shape space \(\mathrm{S}\) consisting of all closed surfaces in \(\mathbb{R}^3\), bounding a simply connected body. The preshape space \(\widetilde{\text S}\) is the space of all smooth embeddings \(f:S^2\to\mathbb{R}^3\) of the 2-sphere that are aligned so that the center of gravity and the first and second moments have prescribed values. \(\widetilde{\text S}\) is acted on by the group \(\mathrm{Diff}^+(S^2)\) of all orientation-preserving diffeomorphisms (reparametrizations) of \(S^2\), called the gauge group here, and \(\mathrm{S}\) is the factor space of \(\widetilde{\text S}\) under this action. As a subspace of \(C^\infty(S^2,\mathbb{R}^3)\), \(\widetilde{\text S}\) inherits the structure of a Fréchet manifold. It carries a Riemannian structure, called elastic metric, defined at a point \(f\in\widetilde{\text S}\) by considering variations of \(f\) in terms of the corresponding variations of the induced metrics and the corresponding normal vector fields. The tangent space at \(f\) decomposes into the direct sum of the tangent space \(T_f\mathcal{O}\) of the \(\mathrm{Diff}^+(S^2)\)-orbit \(\mathcal{O}\) and the orthogonal complement \(\mathrm{Nor}_f\) consisting of all vectors normal to \(\mathcal{O}\) at \(f\).
The elastic metric is invariant under the action of \(\mathrm{Diff}^+(S^2)\) and therefore it induces a metric on \(\mathrm{S}\) leading to the notion of distance between shapes. A disadventage however is that two deformations of a parametrized shape into another one might be of different length in \(\mathrm{S}\). In order to be able to measure the length of a metamorphosis between two surfaces in a way independent of the parametrization, a degenerate metric is considered in \(\widetilde{\text S}\) that is zero on \(T_f\mathcal{O}\) and equals some \(\mathrm{Diff}^+(S^2)\)-invariant metric on \(\mathrm{Nor}_f\), for instance the restriction of the elastic metric.
The article has survey character. Proofs are omitted. Instead, visual ideas are used to make methods of differential geometry and global analysis accessible to non-experts. We mention sequences of pictures of a jumping cat or a running horse visualizing paths in \(\mathrm{S}\).