We present a closed form solution to the problem of registration and detection of dense 3-D point clouds undergoing unknown rigid deformations. The solution is obtained by adapting the general framework of the universal manifold embedding (UME) to the case where the deformations the object may undergo are rigid. The UME nonlinearly maps functions (e.g., images, 3D models) related by geometric transformations of coordinates to the same linear subspace of some Euclidean space. Therefore registration, matching and classification are solved as linear problems in a lower dimensional space. In this paper we extend the UME framework to the special case where it is a-priori known that the geometric transformations are rigid (e.g. pose change of a 3-D rigid object). We further demonstrate the applicability of the methodology for the registration of 3-D point clouds. In the case where point correspondences are unknown, the majority of existing methods for registering 3-D point clouds are based on iteratively finding a transformation which minimizes some distance between the object and a model. The method proposed in this paper is notably different as registration is performed using a closed form solution that employs the UME low dimensional representation of the shapes to be registered.