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

  - 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
      Internal Tangents
      External Tangents
     
     
     
     
     
     
     
     
     
   - Geometric 3D Animations

                                               

Regular Tetrahedron
Some useful geometrical facts

In the following we write for the square root of a number the expression "sqrt(ZAHL)"
conforming to the syntax used in POV-Ray.

Dimensions
Length of tetrahedron edge: a.
The height of an regular tetrahedron:
h = sqrt (2/3) * a ;

The radius of circumsphere:
R = sqrt( 3/8 ) * a ;

The height of the incenter O or
the radius of the insphere:

r = 1/3 * R or   r = 1/sqrt(24) * a ;

The radius of midsphere (tangent to edges):
rm = 1/sqrt(8) * a ;

The angle between two faces: ~ 70,53 (yellow)
Angle(C,MAB,D) = degrees(atan(2*sqrt(2)));

The angle between an edge and a face: ~ 54,74 (green)
Angle(0,A,D) = degrees( atan(sqrt(2)));

The angle vertex-center-vertex: ~ 109.471 (violet)
Angle(A,0,D) = degrees( acos( -1/3 ));

Two regular tetrahedron in a cube
A regular tetrahedron

Folding of a regular tetrahedron

A tetrahedron of vectors
How to make it with POV-Ray
Animation in POV-Ray

Methan CH4
Tetrapode
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© Friedrich A. Lohmüller, 2013
http://www.f-lohmueller.de