Descriptions and Examples for the POV-Ray Raytracer by Friedrich A. Lohmüller
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    3D Animations with POV-Ray
        Some basics and examples on animations.
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  3D Animation Tutorial
   Index of Content
  0. Basic Knowledge
     1. Basic Example
     2. Example 2
     3. Images to Animated Gif
     4. From Images to Video
     5. Basic Terms
     6. Animation Commands  
  I. Cyclic Animations
     1. Rotating Objects
     1.2. Planets in Orbit
     1.3. Clock Animation
     2. Rotating Camera
     2.1. Straight Moving Camera
     3. Western Wheel
         Problem
  > 3.1. Rolling Wheels
     4. Gears
     4.1. Roller Chain
     4.2. Bike Chain
     5. Swinging Pendulum
     5.1. Newton's Cradle
     5.2: Rock the Rocker
     6. Spiral Pendulum
     7. Coupling Rods
     7.1. Connecting Rods
     8. Psychedelic + Op-Art
     9. Counters + Countdowns
    10. Folding of a Cube
  II. Non-linear Movements
     1.0 Speed Up/Slow Down 1
     1.1 Speed Up/Slow Down 2
     2. Fall + Bounce
     3. Acceleration by
          physical Formulas
     4. Speed Controll by
          Spline Functions
  III. Animation Paths
      with Spline Curves
     1. Spline Curves
     2. Closed Splines
     3. Animation Paths
                                                       
The Rolling of the Wheels without slipping or spinning.


For correct rolling wheels we have
to know this basic on circles:

Circumference(C) = 2*pi*Radius(R) ;    

or shorter:   C = 2*pi*R
The Greek sign for Pi ~ 3.14159...
in POV-Ray is simply "pi" (lower case!).


If a wheel rotates once around the axis,
then it will move foreward by the length
of it's circonference C.

If a wheel rotates by an angle Alpha,
the wheel will move forward by
  S = C * Alpha/360 or
S = 2*pi*R*Alpha/360.

On the other hand:
If we want a wheel to move forward by S,
this wheel will rotate during this
movement by an angle of
  Alpha = S*360/C or
  Alpha = S*360/(2*pi*R).

Note: Here all angles were messured in degrees!
We should use capital letters for our variables
to avoid any collision with predefined POV-Ray variables!

Rolling wheel

Rolling wheel

Rolling wheel: rolled off circonference (orange) and angle of rotation (green)

Rolling Wheels with Different Radii

If we want to roll a vehicle with wheels with different radii R1 and R2
the wheels will rotatate with different speed.
 
While the first wheel with R1 is turning by an angle of Alpha1 degrees,
the second wheel with R2 has to be turned by an angle of Alpha2 = Alpha1*R1/R2 degrees.

wheels with different radii

Rolling wheels with different radii.
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© Friedrich A. Lohmüller, 2013
http://www.f-lohmueller.de