Module for working with hyperbolic geometry. Uses the Beltrami–Klein model.

Ideal points have Beltrami coordinates with length 1.

Hyper ideal points have Beltrami coordinates with length > 1.

The origin is the point in the center.

An ideal point is a point at infinity, uniquely defined by an angle.

Returns if two point can be considered equal.

Get the distance between two points

Distance to the origin

If you want to construct a tiling in the hyperbolic space, then the length of the polygon specifies how many polygons fit around a point.

If you look up tilings, you will usually find a Schläfli symbol {p,q}. In that case p = vertices, q = polygonsAroundAPoint.

The resulting length is for BeltramiCoords

We define a line by two ideal points.

An ideal point is a point at infinity and can be represented by an angle

A hyper ideal point is a point that only exists in the euclidean projection of the hyperbolic plane into the Beltrami–Klein model.

A line segment is defined by two points

Any line except lines going through the origin, can be associated by a unique hyper ideal point.

Constructs a line from two ideal points

lineFromIdealPoints : IdealPoint -> IdealPoint -> Line
lineFromIdealPoints =
    Tuple.pair

Constructs a line segment to a point

lineSectionTo : BeltramiPoint -> BeltramiPoint -> LineSegment
lineSectionTo =
    Tuple.pair

constructs a line from two points

Convert a line segment into a line

Construct the midpoint of a line

Get the midpoint of a line segment

Construct a line using an hyper ideal point

Point as to lie on the line

Two lines may intersect at exactly one point

Construct a bisector through a point

reflect a point by a line

Gyrovector in Poincare-Model

Creates a vector from two points. The length of the vector

Negate a vector

Using the Möbius Addition to add two vectors in the poincare disc

Return the length of a vector

Addition for Gyrovectors in the Beltrami Disc Model.

Rotate the vector clockwise

Rotate a point clockwise around the origin

scale a vector by a factor

Constructs a given amount of point along a line

Constructs a given amount of point along a line segment

Constructs a point using polar coordinates.

Projects points from euclidean space into hyperbolic space. However, you can not expect the proportions to stay the same.

Points further out experience more distortion.

Project Hyperbolic points to the Beltrami klein disc

Project Hyperbolic points to the Poincare disc

Converts Beltrami Coordinates into Axial coordinates.

Converts Axial coordinates into Beltrami Coordinates.