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PointClouds: Distributing Points Uniformly on a Surface

The concept of a Point Cloud has played an increasingly important role in many areas of Engineering, Science, and Mathematics. Examples are: LIDAR, 3D-Printing, Data Analysis, Computer Graphics, Machine Learning, Mathematical Visualization, Numerical Analysis, and Monte Carlo Methods. Entering point cloud into Google returns nearly 3.5 million results! A point cloud for a finite volume manifold M is a finite subset or a sequence in M, with the essential feature that it is a representative sample of M. The definition of a point cloud varies with its use, particularly what constitutes being representative. Point clouds arise in many different ways: in LIDAR they are just 3D data captured by a scanning device, while in Monte Carlo applications they are constructed using highly complex algorithms developed over many years. In this article we outline a rigorous mathematical theory of point clouds, based on the classic Cauchy Crofton formula of Integral Geometry and its generalizations. We begin with point clouds on surfaces in R^3, which simplifies the exposition and makes our constructions easily visualizable. We proceed to hyper-surfaces and then sub-manifolds of arbitrary co-dimension in R^n, and finally, using an elegant result of Jurgen Moser to arbitrary smooth manifolds with a volume element.