Any visualization software for the intrinsic metric of a convex polyhedron?

Question

I'd like to find a visual simulation of what it would be like to 'live' in a polyhedron with the intrinsic, piecewise-Euclidean length metric. Of course, to make it easier to visualize, I'd prefer to see a simulation of the polyhedron crossed with the real numbers or the circle.

I've seen Jeffrey Weeks Geometry Games, but I did not find any manifolds with cone points there. I've also watched 'Not Knot' several times, but while it shows what sight-lines to a cone point would look like, it doesn't show what the eye would 'see'.

I've sketched out several potential images myself, but software often reveals hidden features; for instance, A Slower Speed of Light showed me some very unusual features due to special relativity.

So, has anyone created an image of what someone living in a (potentially thickened) convex polyhedron with the intrinsic metric would see?

Answer 1

I believe David Glickenstein's GEOCAM is exactly what you're interested in.

See his preprint "A bug's eye view: the Riemannian exponential map on polyhedral surfaces".

We explore the perspective of a bug living on the two-dimensional surface of a polyhedron. Images of various kinds of effects like lensing and cloaking are shown via color pictures of three viewpoints: the first person perspective of the bug, a map of the bug's viewpoint, and a look at the bug on the embedded polyhedron from a three-dimensional exterior viewer. The pictures were constructed by computing the exponential map of a polyhedron by cutting and rotating faces into the tangent plane of the bug.

Here's one representative image, but there are several more in the preprint, illustrating the phenomena of cloaking, fracturing, apexing and lensing.