###### Descriptions and Examples for the POV-Ray Raytracerby Friedrich A. Lohmüller     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
5.2: 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

# Speed Up and Slow Down (1)Realistic nonlinear movements with useful basic functions.

Realistic simulation of
Acceleration and Retardation

4 useful macros:
 ```//--------------------------- #macro Cos_01( X ) (0.5-0.5*cos( pi*X)) #end //--------------------------- #macro Cos_010( X ) (0.5-0.5*cos(2*pi*X)) #end //--------------------------- #macro Cos_10( X ) (1-(0.5-0.5*cos( pi*X))) #end //--------------------------- #macro Cos_101( X ) (1-(0.5-0.5*cos( 2*pi*X))) #end //--------------------------- ```

###### macro 'Cos_101( TIME )'Start smooth, return smooth, come back and end smooth.

Here some example how to use:
 ```#macro Cos_010( X ) //------------ (0.5-0.5*cos(2*pi*X)) #end //--------------------------- // hard #object{ Tower // left translate<2,0,1+3*clock>} // smooth #object{ Tower // right translate<3,0,1+3*Cos_01(clock)>} //-------------------------------- ```
###### Move hard (left) and smooth (right) Scene descriptions for POV-Ray: "Move_hard_n_smooth_1.ini" and "Move_hard_n_smooth_1.pov"
 ```#macro Cos_010( X ) //----- (0.5-0.5*cos(2*pi*X)) #end //-------------------- #declare Fold_Angle = -90*Cos_010(clock); //------------------------- ```

For complete scene file and more details see: Folding of a Cube.
###### folding without acceleration

Note:
Using exponential functions like f(X)= 1/(1+exp(-X*A+A/2)), with A >=10, meight be a mathematical more correct approximation, but the the problem her: the start and end points are not exact at y=0 resp. y=1!
So f(X)=(0.5-0.5*cos(2*pi*X)) is a better approximation!
Also f(X)= 3*X*X - 2*X*X*X has f(0)= 0, f(1)= 1 and f'(0)=f'(1)=0.
For movements with start and end at v=0 and a=0, a function with 2nd deviation = 0 at start and end, like
f(X)= 6*X5 -15*X4 +10*X3 = X*X*X*(10+X*(6*X-15)) is a more perfect approximation!
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