# Uniform distributed points on triangulated surfaces

Michael Lahanas

We have found a simple algorithm for the distribution of uniformly points on the surface of triangulated objects.

This is for example necessary if we want to determine the dose distribution on a surface obtained by parallel slices of an object.

A simple solution could be to unfold the triangulated polyhedron and produce the points in the unfolded plane.

Unfortunately the unfolding cannot be done always without intersections and is also too complicated.

We use the stochastic universal sampling where points inside a triangle are generated with a probability equal to its area fraction of the surface. Barycentric coordinates are used to generate points in each selected triangle. The example shows a tetrahedron. Three points have to be distributed on the tehrahedron. The circle is divided into 3 pieces by three markers. The triangles with markers are selected. Here 1 point is generated in triangle 1 and 2 in triangle 3.

Actual the distribution of points can be made more uniform if the points are virtually charged and they are allowed to move on the surface. A minimization procedure is then used to minimize the potential energy of the charged points.

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