  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.
