Spherical Law of Cosines: POV-Ray Tutorial

The Math

The spherical law of cosines: [Illustration]

cos c = cos a cos b + sin a sin b cos C

where a, b, and c are the angular lengths of the sides of a spherical triangle; and C is the angle of the vertex opposite line c.

By definition, for a sphere of radius 1, the length of a side is equal to its angular length in radians.

Solve for a:

cos c = ±√(1 − sin2a) cos b + sin a sin b cos C

cos c − sin a sin b cos C = ±√(1 − sin2a) cos b

cos2c + sin2a sin2b cos2C − 2 cos c sin a sin b cos C = (1 − sin2a) cos2b

cos2c + sin2a sin2b cos2C − 2 cos c sin a sin b cos C = cos2b − sin2a cos2b

sin2a sin2b cos2C + sin2a cos2b − 2 cos c sin a sin b cos C + cos2c − cos2b = 0

(sin2b cos2C + cos2b) sin2a − (2 cos c sin b cos C) sin a + (cos2c − cos2b) = 0

Now solve the quadratic equation for sin a, yielding up to 2 unique intermediate solutions. Note that since the sine function does not have a unique inverse, there may be more than 2 unique final solutions.

Solve for C:

cos c − cos a cos b = sin a sin b cos C

(cos c − cos a cos b) ⁄ (sin a sin b) = cos C

cos−1[(cos c − cos a cos b) ⁄ (sin a sin b)] = C

Reminder

All POV-Ray trigonometric functions use radians. To take a function of an angle in degrees, use: 

  sin (radians (AngleInDegrees))
  cos (radians (AngleInDegrees))

or 

  #include "math.inc"
  sind (AngleInDegrees)
  cosd (AngleInDegrees)

To convert an inverse trigonometric function to degrees (e.g., for use in a rotate transformation) use: 

  degrees (asin (X))
  degrees (acos (X))

or 

  #include "math.inc"
  asind (X)
  acosd (X)

Links

Derivation | Spherical trigonometry | POV-Ray

Copyright © 2013 Richard Callwood III. Some rights reserved.
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