Greg Egan on Nostr: I just want to add something about the horosphere and ideal points. I was a bit ...

I just want to add something about the horosphere and ideal points. I was a bit confused about that, but now I think I understand what’s going on.

A given horosphere will have a single limit point on the boundary of the Poincaré ball model. In fact, *all* the horospheres that are defined via all the affine planes that share the same normal null ray will have the same limit point in the ball model.

For example, all the horospheres defined by planes normal to (1,0,0,1) will have a unique limit point of (0,0,1) in the ball model.

But this does *not* mean that the defining equation of the horosphere:

g((1,0,0,1),(t,x,y,z)) = α

α < 0

has a unique null vector solution.

That equation is solved by a whole 2-parameter family of null vectors:

(t, x, ±√(–x^2–α(2t+α)), t+α)

But what matters, in terms of the ideal point, is the limit as t→∞ of the projection into the Poincaré ball:

(x, ±√(–x^2–α(2t+α)), t+α)) / (1+t)

which is (0,0,1).