Descriptions and Examples for the POV-Ray Raytracer by Friedrich A. Lohmüller
    Elementary Geometry for Raytracing
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  - Geometrical Basics
    for Raytracing

    Right-angled Triangle
    Pythagorean Theorem
    Trigonometry Basics
    Law of cosines
    Equilateral Triangle
    Regular Polygon
    Polyhedron
      Tetrahedron
      Octahedron
      Cube & Cuboid
      Dodecahedron
      Icosahedron
      Cuboctahedron
      Truncated Octahedron
      Rhombicuboctahedron
      Truncated Icosahedron
    Circles
      Tangent circles
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Regular Polygon

A regular polygon is a polygon with the properties, that all angles are equal and all sides have the same length.
The equilateral triangle (3), the sqare (4), the regular pentagon (5), the regular hexagon (6), the heptagon (7),
the octagon (8), the nonagon or enneagon (9), the decagon (10), the hendecagon (11), the dodecagon (12), ... ,
the hectagon (100), the chiliagon (1000), the miriagon (10000), the megagon (1000000),
the gigagon (1000 000 000), the teragon (1000 000 000 000 000 000),
the petagon (1000 000 000 000 000 000 000), ... , the circle (∞).

Note: The trigonometric functions sin(X), cos(X) and tan(X) in POV-Ray
need their arguments X in radians !!!   The symbole π = pi en POV-Ray.

In a regolar polygon with N corners or N sides:
The sum of the interior angles = 180*(N-2);
The interior angle = 180-360/N;
 equilateral triangle: 180-360/3 = 180 - 120 =  60;
 sqare:                180-360/4 = 180 -  90 =  90;
 regular pentagon:     180-360/5 = 180 -  72 = 108;
 regular hexagon:      180-360/6 = 180 -  60 = 120;
 regolar heptagon:     180-360/7 = 180-51,43 = 128,57;
 regolar octagon:      180-360/8 = 180 -  45 = 135;
 regolar enneagon:     180-360/9 = 180 -  40 = 140;
 regolar decagon:      180-360/10= 180 -  36 = 144;
 regolar hendecagon:   180-360/11= 180 -32,7 = 147,3;
 regolar dodecagon:    180-360/12= 180 -  30 = 150;
A regolar polygon with N corners and
with the radius of the circumcircle R,
or with the radius of the incircle Ri:

The length of side a:
a = 2* R * sin( radians( 180/N ) ); or
a = 2*R *sin( pi/N );
a = 2* Ri * tan( radians( 180/N ) ); or
a = 2*Ri*tan( pi/N );
With the length of the side a of the regolar polygon :
The radius of the circumcircle:
R = 1/2 * a/sin( radians( 180/N ) ) ); o
R = 1/2 * a/sin( pi/N );
The radius of incircle:
Ri = 1/2 * a/tan( radians( 180/N ) ) ); o
Ri = 1/2 * a/tan( pi/N );
A regolar polygon - angles, sides, radii.
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© Friedrich A. Lohmüller, 2013
http://www.f-lohmueller.de