Hyperbolic Geometry

Hyperbolic geometry is one of the three classical non-Euclidean geometries, discovered independently in the 19th century by mathematicians including Bolyai, Lobachevsky, and Gauss. Unlike Euclidean geometry where parallel lines remain equidistant, hyperbolic space has a constant negative curvature, causing parallel lines to diverge exponentially. This geometry describes the intrinsic structure of saddle-shaped surfaces and appears naturally in many areas of mathematics and physics.

The Poincaré Disk

The Poincaré disk is a model of hyperbolic geometry where the entire hyperbolic plane is represented inside a circle. In this fascinating non-Euclidean geometry, the parallel postulate doesn't hold — through a point not on a line, infinitely many parallel lines can be drawn.

Drag the dots to explore how hyperbolic shapes behave. Notice how "straight lines" (geodesics) appear as circular arcs that meet the boundary at $90^\circ$. As you drag points toward the edge, watch how the angles shrink toward zero.

Mathematical Insight

In the Poincaré disk model, straight lines (geodesics) are represented as circular arcs that intersect the boundary circle at right angles. This model is conformal, meaning angles are preserved — if two curves meet at a certain angle in hyperbolic space, they appear to meet at the same angle in the disk model.

However, distances are distorted. As you move toward the boundary, distances grow exponentially. The boundary itself represents points at infinite hyperbolic distance from the center. The hyperbolic distance from the origin to a point at radius $r$ is approximately $2\text{atanh}(r)$.

Triangles, squares, and pentagons in hyperbolic geometry have angle sums less than their Euclidean counterparts. For instance, a hyperbolic triangle always has an angle sum less than $180^\circ$, with the deficit proportional to its area.