Place lights based on honeycomb

[roice3]

Now that we've added a larger class of honeycombs, the standard light positions don't always work well, often being partially submerged into walls.

We could probably use the geometry of the fundamental simplex to place them well for all cases.

[henryseg]

Put lights at centers of the faces of the fundamental simplex? Or the point on the face that a geodesic to the fourth vertex meets the face at right angles? (Does this work if the vertex is hyperideal?)

What is the correct way to canonically choose a point in the center of a face?

[roice3]

I've been thinking about this, and a good canonical choice is to take the insphere of the fundamental simplex, then use the 4 kissing points of contact with the faces as the light locations.

These points are also the edge midpoints of the omnitruncated form, and will always be finite. A geodesic starting at one of these points and perpendicular to the face will go through the simplex incenter. The geodesics don't meet the simplex vertices in general, so a bonus is we shouldn't need the vertices for the calculation.

A 2D analogue to think about this is here.

See also this page on triangle centers, which shows scenarios when dropping geodesics from vertices to be at right-angles to the opposing face (an orthocenter calculation) - that won't work well in all cases.