```Newsgroups:     sci.math.research
From:           greg@dent.uchicago.edu (Greg Kuperberg)
Subject:        Re: Computing the classical voronoi diagram
Date:           Wed, 5 May 1993 21:37:30 GMT
Organization:   Dept. of Mathematics
```

```>I am looking for a fast (hopefully C) program to compute the "classical"
>voronoi diagram associated to a quadratic form.

This is a complicated problem.

>I am really only interested in the two-dimensional case at present.

Ah!  That changes everything.  The Voronoi cell is going to be a
hexagon in the general case and a parellelogram in special cases.  If
your lattice is just ordinary Z^2 and distance is given by a quadratic
form, i.e. a 2 x 2 symmetric, positive definite real matrix, the
following algorithm outputs the four or six Voronoi neighbors of the
origin, after doing a series of changes of basis.  It is then a matter
of simple linear algebra to find equations for the six sides.

Let Q = (a b) be your quadratic form, and let P = (1 -1) and N = (1  0)
(b c)                                     (0  1)         (0 -1)

be matrices that will be used for change of basis.  Recall that Q is
positive definite iff a,c, and the determinant, ac-b^2 are positive.
The determinant will not change throughout the computation.

The algorithm
-------------
In each step of the algorithm, update a,b, and c as necessary to
reflect changes in Q.

1. If b<0, replace Q <-- N Q N.
2. While a<b, replace Q <-- P^T Q P.
3. While c<b, replace Q <-- P Q P^T.
4. If Q changed in step 2 or 3, go back to step 2.
5. (+-1,0) and (0,+-1) are Voronoi neighbors, and so are (1,-1) and (-1,1)
provided that b > 0.

Geometric interpretation of the algorithm
-----------------------------------------
The set of positive def. quadratic forms with a given determinant,
say determinant 1, (equivalently the set of ellipses centered at the
origin with area pi) is a hyperbolic plane.  The set for which the
Voronoi cells of the lattice Z^2 are parallelograms is a collection
of geodesics that divide the hyperbolic plane into a tiling by
ideal triangles.  You can find a continuous path from any quadratic
form to any other in the same ideal triangle without degeneration
of the sides of the Voronoi region, which means that the origin has
the same Voronoi neighbors for all quadratic forms in the same triangle.

The goal of the algorithm is to find which ideal triangle contains a
given quadratic form.  Each ideal triangle divides the hyperbolic plane
into three complementary regions.  Therefore the adjacency graph of the
triangles is an infinite tree in which every vertex has three
neighbors.  The algorithm is a binary search using the fact that the
equation b = 0 determines an edge in the triangle tiling, and the fact
that an integral change of basis induces an isometry of the tiling.
Step 1 reflects the hyperbolic plane if the quadratic form Q is on the
wrong side of the geodesic b=0.  Step 2 checks if Q lies in one of the
triangles which is a left descendant of the "root" triangle T adjacent
to b=0, while step 3 does the same for the right descendants.  By the
time you get to step 4, Q lies in T itself.

Possible acceleration of the algorithm
--------------------------------------
In step 2, for example, you replace Q by P^T Q P repeatedly by a < b.
This has the effect of replacing b by b-a; a does not change.  You know
ahead of time that you are going to do this n = floor(b/a) times.  So
you can replace Q by (P^n)^T Q P^n and move to step 3.  P^n = (1 -n)
(0  1)
```