Geodesic Cone in Nil-Geometry
W. Heisenberg's real matrix group provides a noncommutative translation group of an affine three-space. The Nil-geometry, which is one of the eight Thurston three-geometries, can be derived from this group. E. Molnár proved that the homogeneous three-spaces have a unified interpretation in the projective three-sphere ). In this Demonstration a geodesic line rotated around the axis, (a "geodesic cone") is visualized.
You get a "geodesic cone" by rotating a geodesic curve around the axis. The geodesic curves of the Nil-geometry are generally defined as having locally minimal arc length between any two (near enough) points. The system of equations of a parametrized geodesic curve is (where , ):
, , ,
if , and
, , ,
Here and are the parameters of a geodesic curve (, ); in this Demonstration you can adjust these values.
The following is an example of a geodesic curve with parameters , :
A point rotated through has the following coordinates: (, , ).
As you can see, a geodesic curve returns periodically to the axis. We get the "geodesic cone" by rotating the part of the geodesic curve between the origin and the first return to the axis around the axis.
If we rotated the whole curve, then it would look like this (with , ):
In this Demonstration you can adjust the and parameters, and according to this, the parameter (the arc length parameter) has a value .
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.