1. Numerical Integration for physically based animation
CSE 3541
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. 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.

4. Problem statement 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()
Current forces or acceleration is known
Integrate to compute “next” velocity and position
Compute an unknown function f(time), using its known derivative f’(time)

5. 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

6. Step in the direction of the derivative
Euler integration
For arbitrary function f (ti) with known derivative
Step in the direction of the derivative

7. Integration – derivative field
For arbitrary function, f(t)
Ex: wind, springs
The force acting on a point may vary in space, i.e in most cases

8. Sampling A fixed amount of time passes between frames.
Approximate the continuous position curve with discrete samples.

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

10. Inaccuracy and instability

11. 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

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

13. Step size
Euler Integration
Midpoint Method

14. Integration comparison
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15. Integration comparison
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16. Integration comparison
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17. Additional slides

18. Other integration techniques
Implicit Euler
Huen
Verlat
Leapfrog

19. CSE course Numerical Methods
(5361) -
(541) from quarter system -

20. 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