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

                                               

The Law of Cosines
Some useful geometrical facts on the sides and angles of triangles.

Note: The trigonometric functions sin(X), cos(X) and tan(X) in POV-Ray
need their arguments X in radians !!!   The symbole π = pi in POV-Ray.
According to this the reverse fuctions asin(x), acos(x) and atan(x) are giving the angles in radians.

 
For every tiangle ABC we have:
  c2 = a2 + b2 - 2*a*b*cos( γ )   (1)
  b2 = a2 + c2 - 2*a*c*cos( β )   (2)
  a2 = b2 + c2 - 2*b*c*cos( α )   (3)
For γ = 90° = pi/2 (right-angled triangle)
we have cos(γ) = 0 and for this by (1):
c2 = a2 + b2 (Pythagorean Theorem).
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For the angles of a triangle ABC
we get the following formulas:

γ = acos ( ( a2 + b2 - c2 )/(2*a*b) )   (4)
β = acos ( ( a2 + c2 - b2 )/ (2*a*c) )   (5)
α = acos ( ( b2 + c2 - a2 )/ (2*b*c) )   (6)
A triangle ABC in 2D.
A triangle ABC in 3D.
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© Friedrich A. Lohmüller, 2014
http://www.f-lohmueller.de