1. Surface Modeling with Oriented Particle System Szeliski and Tonnesen Siggraph 1992

2. Overview Use particle systems to simulate deformable surface models Set up potential functions for internal forces The dynamics controlled by external forces, internal forces, gravity, and damping

3. Surface Modeling Freeform Surface Modeling

4. Particle System Oriented Particle System

5. Oriented Particles Pi: particle (global) position Ri: particles orientation; 3 rd column of Ri is the local normal vector Behavior of (oriented) particles is governed by external forces and desired potential functions. Equilibrium states rest at lowest energy state.

6. Intermolecular Potential Function Dynamics: long-range attraction force and short-range repulsion force pipi pjpj r ij, f ij

7. Expect Particles to be Part of a Flat Surface …

8. Weighting Function  (r) The weighting function  (r)is a monotone decreasing function used to limit the range of inter-particle interactions. Convert to local coordinate

9. Particle Dynamics Potential functions specify the “internal forces” Particle systems are under additional external forces and damping forces

10. Computation of Internal Forces

11. Misc. Numerical time integration –Euler method, Runge- Kutta, semi-implicit methods, … Controlling Complexity –Kd tree to subdivide the tree to efficiently find the neighbors within some radius Rendering –Axes, discs, triangulation (wireframe or shaded)

12. Modeling Operations Weld two surfaces together

13. Cutting a surfaces into two

14. Putting a crease into the surface

15. Particle Creation and 3D Interpolation

16. 3D Interpolation

19. Homework Oriented Particle: 2D version

20. Summary State of each particle: Design potential as in page 7 Weighting function

21. Operation Anchored at two end points; fix one of the normal (  ) Insert middle points Deform the curve by moving one middle points Etc.