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Current point cloud processing algorithms do not have the capability to automatically extract semantic information from the observed scenes, except in very specialized cases. Furthermore, existing mesh analysis paradigms cannot be directly employed to automatically perform typical shape analysis tasks directly on point cloud models. We present a potent framework for shape analysis, similarity, and segmentation of noisy point cloud models for real objects of engineering interest, models that may be incomplete. The proposed framework relies on spectral methods and the heat diffusion kernel to construct compact shape signatures, and we show that the framework supports a variety of clustering techniques that have traditionally been applied only on mesh models. We developed and implemented one practical and convergent estimate of the Laplace-Beltrami operator for point clouds as well as a number of clustering techniques adapted to work directly on point clouds to produce geometric features of engineering interest. The key advantage of this framework is that it supports practical shape analysis capabilities that operate directly on point cloud models of objects without requiring surface reconstruction or global meshing. We show that the proposed technique is robust against typical noise present in possibly incomplete point clouds, and segment point clouds scanned by depth cameras (e.g. Kinect) into semantically-meaningful sub-shapes., Comment: 21 pages, 20 figures. To appear in The Journal of Computer Aided Geometric Design's Special Issue on Heat Diffusion Equation and Optimal Transport in Geometry Processing and Computer Graphics