Shortest paths, distance formulas, and the exponential growth of circles
In Euclidean geometry, the shortest path between two points is always a straight line. In hyperbolic space, “straight” takes on a richer meaning: geodesics curve through space following the intrinsic metric, appearing as circular arcs in the Poincare disk, semicircles in the upper half-plane, chords in the Klein model, and great hyperbolas on the hyperboloid. Distances grow exponentially near the boundary, causing circles, triangles, and Voronoi cells to behave in ways that defy Euclidean intuition.
Click two points in the Poincare disk (top-left) to draw a geodesic. The same path is simultaneously displayed in the upper half-plane, Klein disk, and hyperboloid models. Color-coded segments mark equal hyperbolic-length intervals, revealing how the metric stretches differently in each representation.
Try it: Draw a geodesic close to the boundary of the Poincare disk. Watch how the segments bunch up at the edge of the Poincare and Klein disks but remain evenly spaced on the hyperboloid projection.
A circle of hyperbolic radius r centered at the origin fits inside the Poincare disk with Euclidean radius tanh(r/2). Its circumference C = 2π sinh(r) and area A = 2π(cosh(r)−1) both grow exponentially, dwarfing the Euclidean formulas at large radii. Use the slider to watch circles swell and compare the growth curves side by side.
Drag the three vertices of a hyperbolic triangle to explore side lengths, angles, and the angular deficit. The hyperbolic law of cosines cosh(c) = cosh(a) cosh(b) − sinh(a) sinh(b) cos(C) is verified numerically in real time. The area of any hyperbolic triangle equals π − (A+B+C), which is always positive and bounded by π.
| a (B→C) | 1.6778 |
| b (A→C) | 1.7567 |
| c (A→B) | 1.6807 |
| A | 0.7116 | (40.8°) |
| B | 0.7900 | (45.3°) |
| C | 0.7143 | (40.9°) |
| LHS cosh(c) | 2.777778 |
| RHS | 2.777778 |
| Error | 0.00e+0 |
Key insight: In hyperbolic geometry, the angle sum of every triangle is strictly less than π. The “missing” angle is exactly the triangle's area, a fact with no Euclidean analogue.
Click inside the disk to place seed points. Each pixel is colored by whichever seed is closest in the hyperbolic metric. Because the metric inflates near the boundary, cells near the edge look much larger in the Euclidean picture even though they span similar hyperbolic areas. Drag seeds to reshape cells in real time.