Nonlinear Tessellations

Construction

Step 3

Draw the circle with center A and passing through Q.

Let R denote the intersection of this circle and the segment OA.

Generating the arc RQ.

We show

1. The circle W with center A and passing through Q is a geodesic in the Poincare disc. Proof

2. The arc RQ bisects the angle OQP. Proof

3. The arc RQ is orthogonal to the segment AO. Proof

So we have shown

angle ROQ = angle RPQ = 30

angle OQR = angle PQR = 45

angle ORQ = angle PRQ = 90

Return to construction.