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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