Poincaré Disk Tutorial - Hyperbolic Geometry

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.

The distance between points increases exponentially as you approach the boundary. This means you can never reach the edge of the disk in hyperbolic space.

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 P

Click anywhere within the disk to create a new hyperbolic point. Points are the fundamental building blocks for all other geometric entities.

Add segments S

Click two points to create a finite hyperbolic segment between them. The segment follows the unique geodesic path between the points.

Add lines L

Click two points to create an infinite hyperbolic line (geodesic) passing through them and extending to the boundary of the disk.

Add circles C

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 T

Click and drag inside the disk to apply a complex hyperbolic isometry (translation and rotation) to the entire scene.

Rotation mode R

Click and drag to rotate the entire space around the center of the disk. This preserves all hyperbolic distances.

Center point O

Select a point to automatically transform the space such that the selected point moves to the origin (center) of the disk.

Drag & Drop point M

Precisely move individual points. All geometric objects attached to these points (lines, segments, circles) will update in real-time.

Measurement tools

Measure distance I

Select two points sequentially to calculate the hyperbolic distance between them. The result is displayed in the info box.

Measure angle A

Select three points sequentially (with the vertex as the second point) to calculate the hyperbolic angle. Press Esc to reset the selection.

Delete objects

Delete selection D

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 Delete

Clear the entire workspace, confirmation is required. Use this to start a new construction from scratch.

3. Measurement Tools

Precise quantitative analysis is key to understanding hyperbolic space. The measurement tools allow you to extract geometric data from your constructions. The results are displayed just above the tool description card in the sidebar.

Measure Distance

Select two points sequentially. The hyperbolic distance between them will be displayed in the sidebar. This distance remains invariant under isometries.

Measure Angle

Select three points (A, B, C). Point B is treated as the vertex. The hyperbolic angle ∠ABC is calculated and displayed. Note that hyperbolic triangles always have an angle sum less than 180°.

4. Zoom & Navigation

The visualization of hyperbolic space can be adjusted to explore precise details or to get an overview of the Poincaré disk.

Zoom Buttons

Each view has + and - buttons in its header to adjust the zoom level. This visually enlarges objects without changing their coordinates in the model.

Mouse Wheel

Use Ctrl + Scroll with the mouse over a canvas to quickly zoom in or out. The zoom is centered on the current cursor position.

Navigation

When you zoom in, scrollbars appear if the view exceeds the card size. You can then navigate within the Poincaré disk while maintaining a high level of detail.

Important Note: Zooming is purely visual (View Space). Geometric calculations and isometry transformations (like "Transform" mode) always apply to the actual coordinates of the Poincaré disk, regardless of the zoom level.

5. 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.

5.1 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.

Visual Analogy: Imagine looking through a extreme fish-eye lens. Objects in the center look "normal," but as they move toward the edges, they appear to curve and compress. In hyperbolic space, this "distortion" is actually the reality of the geometry.

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.

5.2 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°.

Visual Analogy: Think of a triangle drawn on a saddle-shaped surface. The sides bow inward, making the corners sharper and the total angle sum smaller than on a flat sheet of paper.

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°.

5.3 Hyperbolic Circles

In the Poincaré disk, a hyperbolic circle is the set of all points at a constant hyperbolic distance (the radius \( r \)) from a central point. Visually, these appear as Euclidean circles, but they are "shifted" toward the boundary: their Euclidean center does not coincide with their hyperbolic center.

Area Formula: The area \( A \) of a hyperbolic circle with radius \( r \) is given by: \[ A = 4\pi \sinh^2\left(\frac{r}{2}\right) \] where \( \sinh(x) = \frac{e^x - e^{-x}}{2} \).

Exponential Growth

Unlike Euclidean geometry where area grows quadratically (\( A = \pi r^2 \)), hyperbolic area grows exponentially with the radius. For large \( r \), the area is approximately \( \pi e^r \). This reflects how "more space" is available as you move away from any given point.

Model Comparison

The visual representation of a hyperbolic circle depends on the chosen model:

Poincaré Disk: Represented as a Euclidean circle, but with an offset center.
Beltrami-Klein Disk: Represented as a Euclidean ellipse. This model preserves straight lines as Euclidean segments but distorts the circular shape of equidistant points.

Key Properties

Boundary Proximity: As a circle moves closer to the boundary, it appears increasingly compressed from a Euclidean perspective.
Maximum Radius: While a hyperbolic circle can have an infinitely large radius, its Euclidean representation always remains strictly within the disk. An "infinite circle" is called an horocycle.

6. 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.