Function Repository Resource:

# HyperbolicPoincarePolygon

Represent a hyperbolic polygon embedded in the Poincaré disk

Contributed by: Jan Mangaldan
 ResourceFunction["HyperbolicPoincarePolygon"][{p1,…,pn}] represents a filled hyperbolic polygon with points pi, embedded in the Poincaré disk.

## Details

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

## Examples

### Basic Examples (2)

A random hyperbolic triangle:

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Show the hyperbolic triangle in the Poincaré disk:

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### Scope (4)

Use directives to style the polygon:

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A concave polygon:

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A triangle with a side that goes through the origin:

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A triangle with points at infinity:

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### Properties and Relations (1)

HyperbolicPoincarePolygon returns a FilledCurve object:

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### Possible Issues (1)

All vertices of HyperbolicPoincarePolygon must lie within the unit disk:

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### Neat Examples (1)

Show a regular hyperbolic polygon in the Poincaré disk:

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## Version History

• 1.0.0 – 08 February 2021

## Author Notes

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