Surface Modeling with Oriented Particle System

Szeliski and Tonnesen

Siggraph 1992

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

Surface Modeling

Freeform Surface Modeling

Particle System

Oriented Particle System

Oriented Particles

Pi: particle (global) positionRi: particles orientation; 3rd 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.

Intermolecular Potential FunctionDynamics: long-range attraction force and short-range repulsion force

pi

pj

rij ,fij

Expect Particles to be Part of a Flat Surface …

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

Particle Dynamics

• Potential functions specify the “internal forces”

• Particle systems are under additional external forces and damping forces

i

ii

iii

vp

av

mfa

/

ii

ii

iii

q

I

1

Computation of Internal Forces

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)

Modeling Operations

Weld two surfaces together

Cutting a surfaces into two

Putting a crease into the surface

Particle Creation and 3D Interpolation

3D Interpolation

Homework

Oriented Particle: 2D version

Summary

• State of each particle:

• Design potential as in page 7

• Weighting function

iiiiii nwhereyx sin,cos,,

bab

y

a

xKyx

,

22exp,

2

2

2

2

Operation

• Anchored at two end points; fix one of the normal ()

• Insert middle points

• Deform the curve by moving one middle points

• Etc.