Hyperbolic Angle Measure of PolygonsIs there a way to determine the angle measure of a regular polygon in hyperbolic space? I know that this depends on the length of the sides. A an example, I know that for an equilateral triangle with side length a and angle A, then sec ⁡ A = 1 + 2 e a e 2 a + 1 .Is there a similar formula for higher regular polygons?

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Hyperbolic Angle Measure of PolygonsIs there a way to determine the angle measure of a regular polygon in hyperbolic space? I know that this depends on the length of the sides. A an example, I know that for an equilateral triangle with side length a and angle A, then   sec  ⁡  A  =  1  +            2              e        a                            e                  2          a                    +      1      .Is there a similar formula for higher regular polygons?

Dwayne Hood · Accepted Answer

Step 1You can use the hyperbolic cosine rule for angles, namely   cos  ⁡      α    =  −  cos  ⁡      β    cos  ⁡      γ    +  sin  ⁡      β    sin  ⁡      γ    cosh  ⁡      a    , on a triangle with vertices at the centre of the polygon and two consecutive vertices of the polygon. Taking a as the side length, the central angle is obviously   2  π      /    n and the triangle is isosceles, so   β  =  γ, which gives  cos  ⁡                              2          π                n              =  −      cos    2    ⁡      β    +      sin    2    ⁡      β    cosh  ⁡      a    =  −      1    2    (  1  +  cos  ⁡      2    β    )  +      1    2    (  1  −  cos  ⁡      2    β    )  cosh  ⁡      a    .Step 2The internal angle A is then   2  β, so we find after some hyperbolic identities that   sec  ⁡      A    =      tanh    2    ⁡                    1        2              a    −  cos  ⁡                              2          π                n                  sech    2    ⁡                    1        2              a

Is there a way to determine the angle measure of a regular polygon in hyperbolic space?

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