Can one hear shape?

Abstract

The question “Can one hear the shape of a drum” has been asked in several contexts before (e.g., by Bers and Kac). It is a pictorial way of asking if the eigenvalues of the Laplacian on a given domain completely characterize its shape, in other words, if the spectrum is a complete shape descriptor (which it is not in general). Since the spectrum contains geometrical information and since it is an isometry invariant and therefore independent of the object's representation, parametrization, spatial position, and optionally of its size, it is well suited to be used as a fingerprint (Shape-DNA) in contemporary computer graphics applications like database retrieval, quality assessment, and shape matching in fields like CAD, engineering or medicine. We will explain why isometry invariance is so important and point out future directions of research. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)