Poincaré Disk Tutorial
A guide to understanding hyperbolic geometry through the Poincaré disk model.
This tutorial explains the fundamental concepts of hyperbolic geometry using the Poincaré disk model. You can use the interactive tools to create objects and observe their properties in real-time.
1. Introduction to the Model
The Poincaré disk model maps the infinite hyperbolic plane onto a finite disk. The boundary of this disk represents "infinity." As objects move closer to the boundary, they appear smaller from a Euclidean perspective, even though their hyperbolic size remains constant.
2. Toolbar Interface
The floating toolbar at the bottom provides all necessary tools for interacting with the hyperbolic plane. It is organized into functional groups:
Adding objects
Add points
Click anywhere within the disk to create a new hyperbolic point. Points are the fundamental building blocks for all other geometric entities.
Add segments
Click two points to create a finite hyperbolic segment between them. The segment follows the unique geodesic path between the points.
Add lines
Click two points to create an infinite hyperbolic line (geodesic) passing through them and extending to the boundary of the disk.
Add circles
Click a center point, then click a second point to define the hyperbolic radius. Note that hyperbolic circles appear Euclidean but their centers are offset from their Euclidean centers.
Transform the space
Transform mode
Click and drag inside the disk to apply a complex hyperbolic isometry (translation and rotation) to the entire scene.
Rotation mode
Click and drag to rotate the entire space around the center of the disk. This preserves all hyperbolic distances.
Center point
Select a point to automatically transform the space such that the selected point moves to the origin (center) of the disk.
Drag & Drop point
Precisely move individual points. All geometric objects attached to these points (lines, segments, circles) will update in real-time.
Delete objects
Delete selection
Select this tool and click on a point to remove it. Any lines, segments, or circles that depend on that point will also be deleted.
Delete Everything
Instantly clear the entire workspace. Use this to start a new construction from scratch.
3. Hyperbolic Geometry
Building on the model's structure, we can explore how familiar geometric concepts transform in hyperbolic space. This section covers the fundamental behavior of lines, parallelism, and triangles.
3.1 Hyperbolic Lines
In Euclidean geometry, a line is the shortest path between two points. In the Poincaré disk, these "shortest paths" (geodesics) are circular arcs that intersect the disk's boundary at perfect right angles. If a line passes through the exact center of the disk, it appears as a straight Euclidean diameter.
The Parallel Axiom
The defining difference between Euclidean and hyperbolic geometry lies in parallelism. In a flat plane, given a line and a point not on it, exactly one parallel line can be drawn. In hyperbolic space, this axiom fails: infinitely many lines can pass through that same point without ever intersecting the original line.
Measuring Distance
Distances in hyperbolic space are not measured with a Euclidean ruler. As you move toward the boundary, space "expands." A segment that looks short near the edge actually represents a massive hyperbolic distance. This ensures that the boundary is truly at infinity; you can travel forever and never reach it.
3.2 Sum of Angles in a Triangle
Because hyperbolic lines curve "away" from each other, triangles in this space look "thin" or "pinched" compared to Euclidean ones. Consequently, the sum of the interior angles of a hyperbolic triangle is always less than 180°.
The "defect" (how much the sum falls short of 180°) is directly proportional to the triangle's area. A tiny triangle behaves almost like a Euclidean one, while a massive triangle with vertices near the boundary will have an angle sum approaching 0°.
3.3 Hyperbolic Circles
Content for hyperbolic circles will be added in a future update.
4. Tilings and Generation
Procedural generation in hyperbolic geometry allows for the creation of complex, symmetrical patterns that fill the entire space. Unlike Euclidean space, hyperbolic space expands exponentially, allowing for intricate tilings and branching structures that would overlap or distort in a flat plane.
The following buttons allow you to instantly generate predefined hyperbolic structures. Each action will clear the current laboratory and apply the new parameters.