This books gives a detailed overview on the mathematical background of the Laplace Beltrami operator (LBO) for a Riemannian manifold. A few analytical computations are presented for special domains in 2D and 3D. Furthermore the numerical computation of the spectra of the LBO are described using a special method for planar domains and the FEM method with up to cubic form functions for the general setup in 2D and 3D. Since the spectrum of the LBO is an isometry invariant, it is invariant under translation, rotation, and invariant under a change of parametrization. It is therefore possible to construct complex objects by gluing several parameter spaces to each other. Some examples are presented, also employing the medial axis as a tool to guide the parametrization of any 2D domain in a way that ensures good meshes without any error at the boundary.

Furthermore this book describes the extraction of geometric data from the spectrum. For this purpose the rapid convergence of the Heat trace is shown. Finally several applications and examples are presented to employ the spectrum in the field of shape recognition. Examples of isospectral 3D solids are shown, that can be distinguished by the spectrum of their boundary surface. The robustness is demonstrated when objects on different mesh resolutions are successfully compared. It is shown how the Neumann boundary condition can be used instead of the Dirichlet condition in cases where the boundary is not supposed to play an important role. finally several complex objects are matched successfully employing their Laplace-Beltrami spectrum.