Geodesics · Drag points to build a hyperbolic polygon · Angle sums < Euclidean
Click inside the disk to add a point (max 6). Drag existing points to move them.
The Poincaré disk model represents the entire infinite hyperbolic plane inside a finite Euclidean unit disk. In this model, hyperbolic "straight lines" (called geodesics) are represented either as diameters of the disk, or as arcs of circles that meet the boundary circle at exactly 90° (perpendicular to it). Given two points inside the disk, the unique geodesic through them is found by constructing the circle through both points that crosses the unit circle at right angles — this simulator computes that circle directly using the standard inversion-geometry formula, then draws the correct arc, rather than approximating it with a straight line.
This model gives a concrete picture of Euclid's parallel postulate failing: given a geodesic and a point not on it, there are infinitely many distinct geodesics through that point that never meet the first one ("parallel" in the sense of never intersecting) — not just the single unique parallel line Euclidean geometry guarantees. Hyperbolic triangles built from three geodesic edges also have angle sums strictly less than 180°, with the "missing" amount (the angular defect) growing larger the bigger the triangle is, unlike Euclidean triangles where the angle sum is always exactly 180° regardless of size.
The Poincaré disk model was popularized by Henri Poincaré in the 1880s, but the underlying hyperbolic geometry dates to the independent 1820s–30s work of János Bolyai and Nikolai Lobachevsky, who showed that a fully self-consistent geometry exists in which Euclid's fifth postulate simply does not hold. The Dutch artist M. C. Escher used the Poincaré disk model (introduced to him by the mathematician H. S. M. Coxeter) as the basis for his famous "Circle Limit" woodcut series, tiling the disk with repeating, shrinking fish and angel/devil motifs that all represent congruent shapes in genuine hyperbolic space, despite looking smaller and smaller toward the disk's edge.
This simulator implements the Poincaré disk model of hyperbolic geometry. Given any two points you place inside the unit disk, it computes the real geodesic — the circular arc perpendicular to the boundary circle (or the diameter, in the degenerate case) — connecting them, using the standard construction from inversive geometry, and reports the resulting hyperbolic polygon's angle sum, which is always less than the Euclidean value for the same number of sides.
For two points inside the unit disk, the unique circle through them that meets the unit circle at a right angle is found by reflecting one point through the unit circle (circle inversion) and solving for the circle through both original points and the inverted point. That circle's arc between the two points is the true hyperbolic geodesic.
Click inside the disk to add up to 6 points (drag to move them); consecutive points are joined by geodesic arcs forming a hyperbolic polygon. Toggle Euclidean lines for comparison, and enable the tiling overlay to see an approximate {7,3} or {5,4} hyperbolic tessellation rendered as visual context, not an exact Möbius-transformation tiling.
M. C. Escher's famous "Circle Limit" woodcuts use exactly this model: the repeating fish and angel/devil figures shrink toward the disk's edge only in Euclidean appearance — in the actual hyperbolic geometry the model represents, every repeated figure is exactly the same size.
A geodesic is the hyperbolic equivalent of a straight line: the shortest path between two points in hyperbolic space. In the Poincaré disk model, geodesics are represented as arcs of circles that cross the boundary of the unit disk at a right angle, or as diameters through the centre in the special case where such a perpendicular circle degenerates to a straight line.
Given two points inside the disk, one point is reflected through the unit circle using circle inversion (mapping a point at distance r from the centre to distance 1/r along the same ray). The circle passing through the two original points and this inverted point automatically meets the unit circle at a right angle, and its arc between the two original points is the geodesic.
In hyperbolic geometry, the sum of a triangle's interior angles is always strictly less than 180 degrees, with the shortfall called the angular defect, and larger triangles have larger defects. This pattern extends to polygons: an n-sided hyperbolic polygon's angle sum is always less than the Euclidean value of (n-2) times 180 degrees, reflecting the negative curvature of the space.
Euclid's parallel postulate states that through a point not on a given line, exactly one line can be drawn parallel to it. In hyperbolic geometry, through a point not on a given geodesic, infinitely many distinct geodesics can be drawn that never intersect the original one, which is why hyperbolic geometry is called a non-Euclidean geometry.
It is a simplified, geometrically plausible approximation built by recursively placing scaled and positioned polygons following the {7,3} or {5,4} tiling's known combinatorial structure, rather than a fully exact recursive application of the hyperbolic reflection (Mobius transformation) group. It is intended to give an accurate visual sense of how such tilings shrink toward the boundary, not a precision mathematical construction.