Voronoi manuscripts of
These are five unpublished manuscripts I wrote in the mid 1980's when
I was temporarily infatuated with generating random Voronoi tessellations
on my new personal computer and the wonders of TeX. My attention was
soon absorbed by my Surface Evolver project, but word of my Voronoi
manuscripts got circulated somehow, and there have been requests for
copies sporadically. Therefore I have scanned the manuscripts (the
original TeX source being long lost), used OCR to convert them to LaTeX,
and then to PDF. For the OCR, I used
InftyReader, which does a very nice job of scanning math formulas
and converting them to LaTeX.
Some of the figures had to be regenerated, and I fixed
a few misprints, but these are pretty close to the originals. Beware
that the scanning process may have misread digits in the various
numerical tables; I found many such and fixed them, but there may
Statistics of Random Plane Voronoi Tessellations
Abstract: If S is a discrete set of points in a space,
and each point of the space is associated with the
nearest point of S, then the resulting partition is
called a Voronoi tessellation. This paper derives
a general scheme for setting up integrals for
statistics for tessellations generated from a Poisson
point process. For the case of the plane, the integrals
are evaluated to find the variances of
cell area, edge length, perimeter, and number of sides.
The distributions of several parameters,
including edge length, are also found.
200,000,000 Random Voronoi Polygons
Abstract: The results of computer simulation of random Voronoi
tessellations of the plane are presented. Statistics tallied include
frequencies, area, and perimeter of n-sided cells, and the
frequencies of n- and m-sided cells abutting.
Statistics of Three Dimensional Random Voronoi Tessellations
Abstract: This paper derives some integral formulas for first and
second order statistics of 3D Poisson Voronoi tessellations.
Statistics of Non-Poisson Point Processes in Several
The Poisson point process is the most commonly
studied random point process, but there
are others. It is not always justifiable to
assume that a random point process will have the
same statistics as a Poisson point process.
This paper derives the relationship between Poisson
processes and some other random processes for
some types of geometric statisics.
Random Voronoi Tessellations in Arbitrary Dimension
Abstract: Voronoi tessellations generated by Poisson
point processes in n-dimensional Euclidean space
are studied. Formulas for the expected measure of
the k-dimensional skeleton of the tessellation
are developed, along with formulas for q-dimensional
cross sections. As n goes to infinity with
q fixed, there is a limiting tessellation process,
which is intuitively a finite dimensional cross
section of an infinite dimensional tessellation.