1. Numerical integration for physically based animation
CSE 3541 / 5541
Matt Boggus

2. Recording motion
First, save a moving object’s position over time. Then, given time, look up position ; y = f(time)
Plot roughly based on dropping a non very bouncy ball

3. Sampling A fixed amount of time passes between frames,
approximate the continuous position curve with discrete samples.

4. Sampling
Low sampling rate (large dt)

5. Sampling
High sampling rate (smaller dt)

6. Kinematics terms Position (x,y,z) Velocity (x,y,z)
Point with respect to the origin
Velocity (x,y,z)
Speed (vector magnitude)
Direction
Acceleration (x,y,z)
Rate of change of velocity
Magnitude and direction r
Kinematics - the branch of mechanics concerned with the motion of objects without reference to the forces that cause the motion.

7. Problem statement
Compute an unknown function f(time), using its known derivative f’(time)
Known current position and velocity
i.e. values at start of Update()
Unknown “next” position and velocity
i.e. what should values be at end of Update()
Known forces acting on object
Integrate to compute “next” velocity and position

8. Example Initial conditions:
acceleration
Initial conditions:
p = 0, v = 5
If we have the function for acceleration, we can integrate it and use initial conditions to solve for the velocity and position functions
velocity
position

9. Step in the direction of the derivative
Euler integration
For arbitrary function f (ti) with known derivative
Also draw locations of new point if doubling dt or negating dt
Euler pronunciation: like “oiler”
Step in the direction of the derivative

10. Example of inaccuracy during integration
For arbitrary function, f(t)
Ex: wind, springs
The force acting on a point may vary in space, i.e in most cases

11. Integration and step size
Here x is the same thing as time or t in the previous slides

12. Inaccuracy and instability

13. Runge Kutta Integration: 2nd order aka Midpoint Method
Compute a “full” Euler step
Evaluate f’ at midpoint
Take a step from the original point using the midpoint f’ value

14. Runge Kutta Integration: 2nd order
aka Midpoint Method
For unknown function, f(t); known f ’(t)

15. Step size
Euler Integration
Midpoint Method

16. Integration comparison
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17. Integration comparison
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18. Integration comparison
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19. Additional slides

20. Other integration techniques
Implicit Euler
Huen
Verlat
Leapfrog

21. OSU CSE course: Numerical Methods
5361
541 (from quarter system)

22. List of 5361/541 topics
Mathematical Preliminaries: Derivatives, Taylor Series
Representation of Numbers: Accuracy, Precision
Root Finding
Polynomial Interpolation
Numerical Differentiation
Numerical Integration
Random Numbers and Monte-Carlo Techniques
Linear Systems and Gaussian Elimination