On 2D solid α-complexes of Poisson Disc Samplings

This paper studies triangulations obtained from Poisson disc sampling. The Poisson disc sampling strategy with radius parameter α can be used to obtain a set of points in the interior or on the boundary of a given solid object in the plane. Using this sampling, one can construct a triangulation from its α-complex. In this work we give two contributions: the quality of this triangulation in terms of triangles’ aspect ratio and a sampling condition with topological guarantee. If we choose the same parameter α for the Poisson disc sampling and for the solid α-complex, we prove that all the triangles in the complex have aspect ratio less or equal than 4 √ 3, which is only three times the aspect ratio of an equilateral triangle. Moreover, we prove that this bound is tight. We also establish a condition on radius parameter α that allows to recover, with topological guarantees, the original solid object from the α-complex representation.