Hyperbolic geometry in 3D Minkowski space and the projective Klein disk
The Poincare disk and upper half-plane are conformal models — they preserve angles but distort distances. Two other models offer complementary advantages. The hyperboloid model embeds the hyperbolic plane as a surface in 3D Minkowski space, making isometries manifest as Lorentz transformations. The Klein disk model (also called the Beltrami–Klein model) is a projective model: geodesics appear as straight chords of the disk, but angles are distorted.
Together these models reveal the deep connection between hyperbolic geometry and special relativity. The hyperboloid x² + y² − t² = −1 is the locus of unit timelike vectors in Minkowski space, and a Lorentz boost — the relativistic change of reference frame — acts as a hyperbolic translation on this surface.
The upper sheet of the two-sheeted hyperboloid x² + y² − t² = −1 carries the hyperbolic metric inherited from Minkowski space. Geodesics on this surface are the intersections of planes through the origin with the hyperboloid itself. Rotate the view to see how the surface curves away from the vertex at (0, 0, 1).
Drag to rotate, scroll to zoom. Colored lines are geodesics on the hyperboloid surface (intersections of planes through the origin).
Try it: Drag to orbit the hyperboloid. The glowing lines on the surface are geodesics — shortest paths in the Minkowski metric. Each one lies in a plane through the origin.
In the Klein disk, geodesics are ordinary straight chords — no arcs needed. But this simplicity comes at a cost: angles are distorted. Click two points in the Klein disk (left) to draw a geodesic chord, and see the corresponding circular arc appear simultaneously in the Poincare disk (right).
Key insight: The Klein model is projective (geodesics are straight lines) but not conformal (angles are distorted). The Poincare model is conformal (angles are preserved) but not projective (geodesics are curved arcs). No single model can be both.
All four models — Poincare disk, upper half-plane, Klein disk, and hyperboloid — describe the same hyperbolic plane. Drag the point in the Poincare disk quadrant and watch the corresponding coordinates update in real time across all four representations, together with the conversion formulas.
k = 2p / (1 + |p|²)
w = i(1 + z) / (1 − z)
(x, y, t) = (2p, 1+|p|²) / (1−|p|²)
A Lorentz boost in special relativity is a hyperbolic rotation in Minkowski space. On the hyperboloid, this corresponds to a hyperbolic translation — points slide along the surface, but the surface itself is invariant. Use the rapidity slider to apply a boost along the x-axis and watch the marked points flow along geodesics.
The transparent yellow cones represent the light cone at the origin. Colored dots are points on the hyperboloid; their trails show the path traced by the Lorentz boost as rapidity increases from 0. The hyperboloid surface is invariant under this transformation.
Physics connection: The rapidity parameter is the hyperbolic angle of the boost. Velocities add via the hyperbolic tangent of rapidities, giving the relativistic velocity addition formula directly from hyperbolic geometry.