Greg Egan on Nostr: I went crazy last night trying to make sense of the claim that a horosphere comes ...

I went crazy last night trying to make sense of the claim that a horosphere comes from a vanishing dot product with a null vector … then finally realised that it’s nonsensical.

Suppose the null vector is (1,0,0,1). Its dot product with a generic vector (t,x,y,z) is:

d = –t+z

so the only vectors for which d=0 will take the form:

w = (t,x,y,t)

But the dot product of w with itself is:

x^2+y^2

So w is either null or spacelike! There is no intersection with the hyperboloid, which consists of timelike unit vectors!

[Edited to add: I don’t actually see the claim that the dot product *vanishes* in Wikipedia.

From the article on the horosphere,
https://en.wikipedia.org/wiki/Horosphere#Models

“In the hyperboloid model, a horosphere is represented by a plane whose normal lies in the asymptotic cone.”]