Particle systems

Particles are objects

• with position and velocity,
• without shape or orientation,
• that may have mass or may respond to forces.

Particle Systems consist of

• many particles, or
• many particle systems (hierarchically organized)

Basic particle systems:

• A collection of many minute particles that together form a fuzzy object
  • Points to define shapes rather than polygons
• Procedural/Statistical
• Stand alone

Particle lifetime

1. Seeding
2. Attribute assignment
3. Animation

Seeding particles

Creation/Deletion of particles.

Location:

• Randomly within volume/surface
• At point of event
• In regions sparsely populated

When:

• at start
• per frame
• on event (e.g. particle death)

Attributes

• Position
• Velocity
• Color
• Age
• Mass
• Radius
• Temperature
• Density
• and so on.

Animation

Types of rules that govern motion:

• Procedural (follow specification/algorithm)
• Statistical (to add variation)
• Physics-based collision/dynamics

Update cycle:

1. Clear forces
2. Calculate and accumulate forces
3. Solve for accelerations
4. Integrate velocity/position
5. Record state
6. Update other variables

Particle Physics

Newtonian Particle

Each particle is an ideal point mass.

Six degrees of freedom: 3 position coords, 3 velocity coords.

Newton’s second law: $F = ma$

• $f = mg$ for gravity
• $f = -k_d v$ for viscous drag

• $f = k_s (

  x

  -r ) - k_d v$ for Hooke’s Law

  • first term: stiffness
  • second term: damping factor (viscous drag)
  • Modelling particles as springs.
  • Virtual springs between particles

Forces: depend on position velocity time

Acceleration (2nd derivative) $x’’ = f(x, x’, t) / m$ can be:

• constant (e.g. gravity)
• position/time dependent (force fields/collisions)
• velocity dependent (drag/damping)
• n-ary (many springs: make more stiff/flexible)
  • mass spring models

Vector field of velocities/accelerations (depends on what this field represents)

Differential equations to solve initial value problems.

Euler’s method

Approximation. Break into discrete time steps. Choie of $\Delta t$ (time interval) is thus critical. Wrong choice can make movement inaccurate and unstable.

Velocity-time: $v_t = x’_t = f(x_t, t)$.

Position-time: $x_{t+1} = x_t + \Delta t v_t$.

Midpoint method is a modification of Euler’s method.

1. $\Delta x = \Delta t f(x, t)$
2. $v_\text{mid} = f(x + \frac{\Delta x}{2}, t + \frac{\Delta t}{2})$
3. $x_{t + 1} = x_t + \Delta t v_\text{mid}$

Use the midpoint velocity to derive the next step’s position, instead of the velocity at the start of the interval.

Advanced Particle Systems

1. Mass Spring systems
2. Smoothed particle hydrodynamics (SPH)
3. Crowd simulation (human crowds)

Mass spring systems

Hooke’s law: $f_s = k_s(x - r) - k_d v$

• 1D line of springs (hair)
• 2D (cloth)
  • Triangulated particles (see below)
• 3D (jello)
  • More sophisticated methods FEMs (Finite elemental methods)

Right layout, stiffness, integration.

Layout of 2D mass spring particles in cloth: