Here is the interactive and theoretical part of the subject..
The Hyperbolic Machine
The basic construction consists of the hyperbolic triangle with the angles α, β and γ given by natural numbers k, n, m as fractions of 180°.
α = 180°/k, β = 180°/n and
γ = 180°/m with the condition
1/k + 1/m + 1/n < 1
The traingle has two straight lines AB and AC and an arc side BC. You controll the situation of the points A (yellow), B (orange) by moving the small circles in the drawing area. The machine then completes the construction. The point C is calculated under the condition of the given angles. α is measured normally and the angles β and γ are measured between the straight lines and the circle line tangents. The sum of the three angle values has to be less than 180°. As a second condition the circle line of BC with center M (white) outside has to cut the circle with centre A vertically.
Therefor the angle δ and the centre M of the construction circle is given. The blue fixed circle cuts the construction circle around M vertically. If the angles of the hyperbolic triangle are given by natural numbers k, m, n, so that the conditions are like noticed left then we have some interesting results:
The machine is easy to use. Chose the natural numbers k, n und m without hurting the condition above. If the image looks too "dotty", let draw again or increase the values for dots and iterations.