Function Repository Resource:

# HyperbolicPoincareLine

Represent a hyperbolic line segment or polyline embedded in the Poincaré disk

Contributed by: Jan Mangaldan
 ResourceFunction["HyperbolicPoincareLine"][{p1,…,pn}] represents the hyperbolic line segments joining a sequence of points pi, embedded in the Poincaré disk.

## Details

ResourceFunction["HyperbolicPoincareLine"] can be used as a graphics primitive.
ResourceFunction["HyperbolicPoincareLine"] returns a JoinedCurve object.
The points pi must all have dimension 2, and must all lie within the unit disk.

## Examples

### Basic Examples (2)

A hyperbolic line segment:

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Show the hyperbolic line segment in the Poincaré disk:

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A hyperbolic polyline:

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Show the hyperbolic polyline in the Poincaré disk with styling:

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### Scope (3)

Construct a hyperbolic line segment of given length with one endpoint at the origin:

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A hyperbolic line segment with endpoints at infinity:

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Show the outline of a regular polygon centered at the origin:

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### Properties and Relations (1)

HyperbolicPoincareLine returns a JoinedCurve object:

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### Possible Issues (1)

All vertices of HyperbolicPoincareLine must lie within the unit disk:

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### Neat Examples (1)

Create a Manipulate of joined points on the Poincaré disk, where points can be added or deleted by Alt-Click:

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## Version History

• 1.0.0 – 16 March 2021

## Author Notes

HyperbolicPoincareLine relies the NURBS representation of a circle arc, which can also easily handle the case where a geodesic representing a segment passes through a diameter of the disk.