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:
sum of angles = o
Periodic image for a sum of exactly 180o.
Decoration of elliptic space for more than 180o.
(Stereographic projections of repeating patterns on a sphere.)
Periodic pattern in hyperbolic space for less than 180o.
(Zoom in at the border of the Poincaré disc representation.)
Rapid convergence of the iterated mapping:
The distance between points and the boundary of the Poincaré disc increases nearly exponentially with the number of circle inversions.