1. Computer Animation Algorithms and Techniques
Integration

2. Integration
Given acceleration, compute velocity & position by integrating over time

3. Projectile
given initial velocity under gravity

4. Euler integration
For arbitrary function, f(t)
Truncated Taylor series

5. Integration – derivative field
For arbitrary function, f(t)
Truncated Taylor series

6. Step size

7. Numeric Integration Methods
(explicit or forward) Euler Integration
2nd order Runga Kutta Integration (Midpoint Method)
4th order Runga Kutta Integration
Implicit (backward) Euler Integration
Semi-implicit Euler Integration

8. Runge Kutta Integration: 2nd order
Aka Midpoint Method
For unknown function, f(t); known f ’(t)
Truncated Taylor series

9. Step size
Euler Integration
Midpoint Method

10. Runge Kutta Integration: 4th order
For unknown function, f(t); known f ’(t)
Truncated Taylor series

11. Implicit Euler Integration
For arbitrary function, f(t), find next point whose derivative updates last value to this value: required numeric method (e.g. Newton-Raphson)
Search for point such that negative of tangent points back at original point
Truncated Taylor series

12. Differential equation, initial boundary problem
(implicit/backward)
Euler method
(explicit/forward)
Euler method
Truncated Taylor series
e.g. linearize f’ and use Newton-Raphson

13. Semi-Implicit Euler Integration
Truncated Taylor series

14. Methods specific to update position from acceleration
Heun Method
Verlet Method
Leapfrog Method
Truncated Taylor series

15. Heun Method
Truncated Taylor series

16. Verlet Method
Truncated Taylor series

17. Leapfrog Method
Truncated Taylor series