>>11968711Jesus christ this thread is awful. Only the schizo was right.
Here's your answer OP (and next time don't add a bait image if you want an answer):
"Hyperbolic space" is a catch-all term for spaces which always have negative curvature. This means that at each point of the space, a neighborhood of that point looks like a saddle (compare to positive curvature where neighborhoods look like domes/cups).
The curvature at a point is the product of the amount the saddle at that point curves upwards times the amount it curves downwards. In mathematical terms, it's the product of the max and the min of the second directional derivatives from the point.
When people say "hyperbolic space" they usually mean a space with constant negative curvature. But that curvature can be anything, -K.
The circumference of a circle in hyperbolic space is larger than 2pi*r, in fact, it's not even going to grow linearly with r anymore. But the circumference will grow faster the more negative curvature there is.
We can use the Poincare disk model to determine the circumference if we look up the hyperbolic metric on the Poincare disk for a fixed curvature, and then compute the point on the x axis which is distance r from 0, and then compute the length of the curve defined by a circle through that point with center at 0. This is easy to do presumably, not at my computer for the moment but I can give it a shot.
