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.