An Approach for Randomly Distributing 3D Points with Even Density inside the Regular Octahedron

Abstract

In this paper, we propose a theory and method for random point distribution so that the generated 3-dimensional points are distributed with even density inside the regular octahedron without any point discarded. The approach for doing these is as follows; At first, the X- and Y-coordinates based functions for distributing random numbers with uniform density inside the regular octahedron are proposed in this paper. They make 3-dimensional points be distributed randomly within the regular octahedron in proportion to the area of its X-coordinate section and the length of Y-coordinate segment constituting that section. Secondly, the inverse functions are derived from those distribution functions and we present how to use them to generate random points with even density distribution. Finally, experimental results by those proposed distribution functions are compared with those by uniform distribution and normal distribution. As the results, it has been validated that our distribution functions randomly distribute 3-dimensional points into the inside of the regular octahedron with much more even density than these two distribution functions.