We have a triangle of two reflecting straight lines and a reflecting circle arc.

**How do we map a point into this triangle ?**

The straight lines generate a dihedral group.

This group generates images of the circle arc.

They make a polygon around the center of the image.

Inside the polygon the dihedral symmetry maps any point into the triangle.

**How do we map a point into this polygon ?**

Use inversions at the circles.

**
At which circle ?
**

Use "ray tracing": Draw a straight line from the center to the point and take the circle that intersects first with this line.

**
Repeat until the point is inside the polygon.
**

**Choose the rotational symmetries:**

center | |

left | |

right |

showing:

sum of angles = ^{o}

**Results:**

Periodic image for a sum of exactly 180^{o}.

Decoration of *elliptic space* for more than 180^{o}.

(Stereographic projections of repeating patterns on a sphere.)

Periodic pattern in *hyperbolic space* for less than 180^{o}.

(Zoom in at the border of the PoincarĂ© disc representation.)

**Rapid convergence of the iterated mapping:**

The distance between points and the boundary of the PoincarĂ© disc increases nearly exponentially with the number of circle inversions.