1. Integration for physically based animation CSE 3541 Matt Boggus

2. What is motion? Observe an object’s position over time We could say y = f(time)

3. Equations of motion – terms Position (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

4. Equations of motion graph examples 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 acceleration velocity position

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

6. Integration – derivative field For arbitrary function, f(t) The force acting on a point may vary in space

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

8. Integration and step size

9. Inaccuracy and instability

10. Runge Kutta Integration: 2 nd 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

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

12. Step size Euler Integration Midpoint Method

13. Integration comparison Image from http://www.physics.drexel.edu/students/courses/Comp_Phys/Integrators/simple.html