Nonlinear Tessellations

Construction

Step 2, proof

Showinf angle OPM = 30 deg.

In the Euclidean triangle ABO, angle AOB + angle OAB + angle ABO = 180.

The segment OB is perpendicular to AB, so angle AOB + angle OAB = 90.

The segment MN is tangent to U at P, so angle APB + angle APN = 90.

The segments AB and PB are radii of the circle U, so triangle ABP is isosceles.

Consequently, angle PAB = angle APB, so angle APB + angle APN = 90 implies angle PAB + angle APN = 90.

Combining angle PAB + angle APN = 90 and angle PAB + angle AOB = 90, we see angle APN = angle AOB.

But angle APN = angle OPM, so angle OPM = angle AOB = 30.

Finally, since MN is tangent to U at P, angle OPM is the hyperbolic angle OPQ.

We have constructed the hyperbolic triangle OPQ with angles 30, 30, and 90.

Return to Step 2.