Up: Models of Hyperbolic Geometry

This page constructed by John Hartman.

The Minkowski Model

If you aren't familiar with the concept of spacetime, please visit that page first.

The Minkowski Model of hyperbolic space is a bit different from the other models in that it needs to be embedded in a space of dimension one greater than its own. So, to view the 2-dimensional case, we need to step into a 3-dimensional space. Consider the set of points of Minkowski length i, with -- half a pseudospere. Then , so the resulting object is half a two-sheeted hyperboloid:

Points in the Minkowski model correspond to our usual notion of points on the hyperboloid.

Lines are the intersection of the hyperboloid with planes passing through the origin (of 3-space, not hyperbolic space). They appear as hyperbolae in our model. Also, they are the geodesics of the hyperboloid, so the shortest curve (with the standard Euclidean metric) between two points on the hyperboloid follows these lines.

Lengths simply correspond to the standard definition of lengths and angles in spacetime - we integrate the Minkowski metric along the curve.

To find the angle between two curves, take the normals to the planes they lie in. Then the angle phi is given by the formula:

where * represents the Minkowski dot product, and the norms are also those of a Minkowski space.

Up: Models of Hyperbolic Geometry

Created: Jul 15 1996 --- Last modified: Wed Aug 7 23:42:08 1996