Geodesic Balls in the Nil-Space
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W. Heisenberg's real matrix group provides a noncommutative translation group of an affine 3-space. The Nil-geometry, which is one of the eight Thurston 3-geometries, can be derived from this group. It was proved by E. Molnár that the homogeneous 3-spaces have a unified interpretation in the projective 3-sphere 
). In this Demonstration we visualize the geodesic balls of the Nil-space with the origin as the center, radius in 
, and translated by 
.
Contributed by: Benedek Schultz, János Pallagi (April 2009)
Suggested by: Jenő Szirmai
Open content licensed under CC BY-NC-SA
Snapshots
Details
We get the geodesic ball by rotating the following curve about the 
axis (lying in the plane 
): 
, 
if 
; if 
: 
, 
.
If 
, then the curve is basically half of the intersection of the geodesic sphere with the [
, 
] plane and looks like this:
The coordinates of a point 
rotated by 
around the 
axis are (
, 
, 
).
Finally, we can translate the geodesic sphere with a vector (
, 
, 
) to get 
, 
, 
). This translation is defined by left multiplication with Heisenberg's matrix:
It is a good idea to zoom in for a better view as well as to rotate the image.
If the radius is less than 
, then the ball is convex in the affine-Euclidean sense of our model, but if the radius is in 
, then it is not convex. Also the geodesic sphere exists in Nil if and only if 
. For example if 
, then the curve used to rotate about the 
axis would be:
Reference:
J. Szirmai, "The Densest Geodesic Ball Packing by a Type of Nil Lattices," Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry), 48(2), 2007 pp. 383-397.
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