This paper reports a new method for 3D shape classification. Given a 3D shape M, we first define a spectral function at every point on M that is a weighted summation of the geodesics from the point to a set of curvature-sensitive feature points on M. Based on this spectral field, a real-valued square matrix is defined that correlates the topology (the spectral field) with the geometry (the maximum geodesic) of M, and the eigenvalues of this matrix are then taken as the fingerprint of M. This fingerprint enjoys several favorable characteristics desired for 3D shape classification, such as high sensitivity to intrinsic features on M (because of the feature points and the correlation) and good immunity to geometric noise on M (because of the novel design of the weights and the overall integration of geodesics). As an integral part of the work, we finally apply the classical multidimensional scaling method to the fingerprints of the 3D shapes to be classified. In all, our classification algorithm maps 3D shapes into clusters in a Euclidean plane that possess high fidelity to intrinsic features—in both geometry and topology—of the original shapes. We demonstrate the versatility of our approach through various classification examples.