Hyperbolic geometry or hypercycle, then the triangle has no circumscribed circle. As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent. A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides. In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line. However, in hyperbolic geometry, a regular apeirogon has sides of
Vibrations of a circular membrane equation with zero boundary conditions, Due to the circular geometry of formula_1, it will be convenient to use cylindrical coordinates, formula_13 Then, the above equations are written as Here, formula_16 is a positive constant, which gives the speed at which transverse vibration waves propagate in the membrane. In terms of the physical parameters, the wave speed, c, is given by where formula_18, is the radial membrane resultant at the membrane boundary (formula_19), formula_20, is the membrane thickness, and formula_21 is the membrane density. If the membrane has uniform tension, the uniform tension force at a given radius, formula_22 may be