Shortest paths, distance formulas, and the exponential growth of circles
In hyperbolic space the shortest path bends, and distances grow exponentially. Circles, triangles, and neighborhoods all behave in ways a flat-world eye does not expect.
Click two points in the Poincare disk (top-left) to draw a geodesic in all four models. Draw one near the edge: the equal-length segments bunch up in the disks but stay even on the hyperboloid.
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.
Geodesics across models — The same shortest path appears as a circular arc (Poincare), semicircle (half-plane), chord (Klein), or intersection curve (hyperboloid), but the hyperbolic length is identical in every model
Use the radius slider to watch a circle swell, and compare its growth curves with the Euclidean ones side by side.
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 and watch the side lengths, angles, and deficit update. The law of cosines is checked live as you move.
| 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 |
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.
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 seeds, then drag them. Cells near the edge look huge but cover about the same hyperbolic area.
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.
Spot a mistake or have an idea?