In this paper we propose a principled approach for shape comparison. Given two surfaces, one to one correspondences are determined using the Laplace equation. The distance between corresponding points is then used to define both global and local dissimilarity statistics between the surfaces. This technique provides a powerful method to compare shapes both locally and globally for the purpose of segmentation, registration or shape analysis. For improved accuracy, we propose a Boundary Element Method. Our approach is applicable to datasets of any dimension and offers subpixel resolution. We illustrate the usefulness of the technique for validation of segmentation, by defining global dissimilarity statistics and visualizing errors locally on color-coded surfaces. We also show how our technique can be applied to multiple shapes comparison.