Particle Systems

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Particle Systems
Lecture 8

• Abdennour El Rhalibi

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Particle Systems

• Particle systems have been used extensively in
  computer animation and special effects since
  their introduction to the industry in the early
  1980s
• The rules governing the behavior of an individual
  particle can be relatively simple, and the
  complexity comes from having lots of particles
• Usually, particles will follow some combination
  of physical and non-physical rules, depending on
  the exact situation

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Whats a Particle System?

• A collection of many small particles that
  together represent a fuzzy object
• Use points to define shapes rather than polygons

Star Trek II The Wrath of Khan
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Some fuzzy objects

• Grass, Smoke, fire, clouds, water
• Fireworks, explosions
• Fluid flow
• Physical simulations
• Flocking - Bird migration, schools of fish, riots

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20,000 particles in a galaxy-forming gravitation
simulation
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Physics
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Kinematics of Particles

• We will define an individual particles 3D
  position over time as r(t)
• By definition, the velocity is the first
  derivative of position, and acceleration is the
  second
• To render a particle, we need to know its
  position r.

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Uniform Acceleration

• How does a particle move when subjected to a
  constant acceleration?

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Uniform Acceleration

• This shows us that the particles motion will
  follow a parabola
• Keep in mind, that this is a 3D vector equation,
  and that there is potentially a parabolic
  equation in each dimension. Together, they form a
  2D parabola oriented in 3D space
• We also see that we need two additional vectors
  r0 and v0 in order to fully specify the equation.
  These represent the initial position and velocity
  at time t0

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Mass and Momentum

• We can associate a mass m with each particle. We
  will assume that the mass is constant
• We will also define a vector quantity called
  momentum (p), which is the product of mass and
  velocity

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Newtons First Law

• Newtons First Law states that a body in motion
  will remain in motion and a body at rest will
  remain at rest- unless acted upon by some force
• This implies that a free particle moving out in
  space will just travel in a straight line

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Force

• Force is defined as the rate of change of
  momentum
• We can expand this out

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Newtons Second Law

• Newtons Second Law says
• This relates the kinematic quantity of
  acceleration to the physical quantity of force

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Newtons Third Law

• Newtons Third Law says that any force that body
  A applies to body B will be met by an equal and
  opposite force from B to A
• Put another way every action has an equal and
  opposite reaction
• This is very important when combined with the
  second law, as the two together imply the
  conservation of momentum

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Conservation of Momentum

• Any gain of momentum by a particle must be met by
  an equal and opposite loss of momentum by another
  particle. Therefore, the total momentum in a
  closed system will remain constant
• We will not always explicitly obey this law, but
  we will implicitly obey it
• In other words, we may occasionally apply forces
  without strictly applying an equal and opposite
  force to anything.

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Energy

• The quantity of energy is very important
  throughout physics, and the motion of particle
  can also be formulated in terms of energy
• Energy is another important quantity that is
  conserved in real physical interactions
• However, we will mostly use the simple Newtonian
  formulations using momentum

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Forces on a Particle

• Usually, a particle will be subjected to several
  simultaneous vector forces from different sources
• All of these forces simply add up to a single
  total force acting on the particle

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Particle Simulation

• Basic kinematics allows us to relate a particles
  acceleration to its resulting motion
• Newtons laws allow us to relate acceleration to
  force, which is important because force is
  conserved in a system and makes a useful quantity
  for describing interactions
• This gives us a general scheme for simulating
  particles (and more complex things)

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Particle Simulation

• 1. Compute all forces acting within the system
  (making sure to obey Newtons third law)
• 2. Compute the resulting acceleration for each
  particle (af/m) and integrate to get positions
• - Repeat
• This describes the standard Newtonian approach
  to simulation. It can be extended to rigid
  bodies, deformable bodies, fluids, vehicles, and
  more

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Particle Example

• class Particle
• float Mass // Constant
• Vector3 Position // Evolves frame to frame
• Vector3 Velocity // Evolves frame to frame
• Vector3 Force // Reset and re-computed each
  frame
• public
• void Update()
• void Draw()
• void ApplyForce(Vector3 f) Force.Add(f)

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Particle Example

• class ParticleSystem
• int NumParticles
• Particle P
• public
• void Update()
• void Draw()

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Particle Example

• ParticleSystemUpdate(float time)
• // Compute forces
• Vector3 gravity(0,-9.8,0)
• for(i0iltNumParticlesi)
• Vector3 forcegravityParticlei.Mass // fmg
• Particlei.ApplyForce(force)
• // Integrate
• for(i0iltNumParticlesi)
• Particlei.Update(time)

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Particle Example

• ParticleUpdate(float time)
• // Compute acceleration (Newtons second law)
• Vector3 Accel(1.0/Mass) Force
• // Compute new position velocity
• VelocityAcceltime
• PositionVelocitytime
• // Zero out Force vector
• Force.Zero()

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Particle Example

• With this particle system, each particle keeps
  track of the total force being applied to it
• This value can accumulate from various sources,
  both internal and external to the particle system
• The example just used a simple gravity force, but
  it could easily be extended to have all kinds of
  other possible forces
• The integration scheme used is called forward
  Euler integration and is about the simplest
  method possible

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Integration
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Integration

• Computing positions and velocities from
  accelerations is just integration
• If the accelerations are defined by very simple
  equations (like the uniform acceleration), then
  we can compute an analytical integral and
  evaluate the exact position at any value of t
• In practice, the forces will be complex and
  impossible to integrate analytically, which is
  why we automatically resort to a numerical scheme
  in practice
• The ParticleUpdate() function described earlier
  computes one iteration of the numerical
  integration. In particular, it uses the forward
  Euler scheme

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Forward Euler Integration

• Forward Euler integration is about the simplest
  possible way to do numerical integration
• It works by treating the linear slope of the
  derivative at a particular value as an
  approximation to the function at some nearby
  value
• The algorithm we discussed for the roller-coaster
  used Euler integration

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Forward Euler Integration

• For particles, we are actually integrating twice
  to get the position
• which expands to

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Forward Euler Integration

• Note that this
• is very similar to the result we would get if we
  just assumed that the particle is under a uniform
  acceleration for the duration of one frame

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Forward Euler Integration

• Actually, it will work either way
• Both methods make assumptions about what happens
  in the finite time step between two instants, and
  both are just numerical approximations to reality
• As ?t approaches 0, the two methods become
  equivalent
• At finite ?t, however, they may have significant
  differences in their behavior, particularly in
  terms of accuracy over time and energy
  conservation
• As a rule, the forward Euler method works better
• In fact, there are lots of other ways we could
  approximate the integration to improve accuracy,
  stability, and efficiency

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Forward Euler Integration

• The forward Euler method is very simple to
  implement and if it provides adequate results,
  then it can be very useful
• It will be good enough for lots of particle
  systems used in computer animation, but its
  accuracy is not really good enough for
  engineering applications
• It may also behave very poorly in situations
  where forces change rapidly, as the linear
  approximation to the acceleration is no longer
  valid in those circumstances

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Forward Euler Integration

• One area where the forward Euler method fails is
  when one has very tight springs
• A small motion will result in a large force
• Attempting to integrate this using large time
  steps may result in the system diverging (or
  blowing up)
• Therefore, we must use lots of smaller time steps
  in order for our linear approximation to be
  accurate enough
• This resorting to many small time steps is where
  the computationally simple Euler integration can
  actually be slower than a more complex
  integration scheme that costs more per iteration
  but requires fewer iterations
• We will look at more sophisticated integration
  schemes in future lectures

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Modeling and Animation

• Lecture 8

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Physics Based Animation in 3DS Max

• Simulation User specifies objects, forces and
  processes to govern object and/or particle
  motion.
• Dynamic Space Warp Mechanism for specifying
  forces or constraints on motion of objects and/or
  particles.
• Path Follow, (PUS)dynaflect, (PUS)omniflect,
  (SU)deflector.
• Gravity, Wind, Push, Motor, Field, Pbomb, etc.
• Particle System Process that generates a
  collection of particles, with specified,
  randomized motion, over a region of space and a
  period of time
• (Super) Spray, Snow, Blizzard, PCloud, PArray.

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Example Max Files

• Dynamics
• A sphere falls under the influence of gravity and
  bounces off the floor bouncing-sphere.max
• One block falls and bounces while dangling from a
  damped spring attached to another block
  boxes-and-spring.max
• Particle Systems
• A stream of particles is emitted at a point and
  follows a spline path smoke-stack.max
• A stream of particles is emitted at a point, is
  blown by the wind, bounces off a wall and falls
  under the influence of gravity fountain.max

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Bouncing Sphere

• Create a small sphere in the ground plane.
• Go into Front View and pan vertically so you have
  room to watch the object fall downward.
• On the Command Panel, use Create, Space Warps,
  Particles and Dynamics to create a Gravity space
  warp in Top View.
• Check Front View to see that the arrow indicates
  a downward direction for gravity.

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Creating the Simulation

• Got to the Utilities tab on the Command Panel and
  click on Dynamics.
• Press New to create a new simulation, called
  Dynamics00.
• Press Edit Object List, select Sphere01 and use
  the gt button to include it in the simulation.
  Press OK to confirm.
• Press Edit Object and Assign Object Effects
  and use the gt button to include Gravity01 as an
  effect acting on Sphere01.

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Solve the Simulation

• Go to the Dynamics rollout on the Command Panel.
• Check the Update Display w/ Solve box.
• Press the Solve button The sphere accelerates
  downward, falling off the bottom of the window.
• Open Track View and look a the spheres Position
  track function curves.
• Notice that the curve is a parabola, described by
  lots of keys.
• Press Edit Time use the mouse to select the time
  range 0,100 and press Reduce Keys to reduce
  the number of keys.
• You may now edit this track as you wish.

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Adding Collisions

• Create a Box of size (W,L,H)(100,100,1) and
  location (x,y,z)(0,0,-100).
• Use Edit Object List on the Dynamics rollout on
  the Command Panel, to include Box01 in the
  Dynamics00 simulation.
• Press Edit Object, select Sphere01, press Assign
  Object Collisions and add Box01 to the
  collisions set for the sphere.
• Press Edit Object, select Box01 and check the box
  marked This object is unyielding under Dynamic
  Controls to make sure the box wont move.
• Press the Solve button and observe that the ball
  bounces.
• Go to track view to observe the function curves
  and reduce the number of keys, as before.

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Specification of Particle Systems

• Region of space over which particles are emitted.
  
• Period of time over which particles are emitted.
• Initial linear and rotational velocities of
  particles.
• Forces and/or process governing particles
  motion.
• Shape and material properties of particles.
• Lifetimes of particles.
• Probability distributions to specify randomness
  in any or all of the above.

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Fountain

• Use Create, Particle-Systems to create a Super
  Spray particle emitter in Perspective View at the
  center of the ground plane.
• Pull the Time Slide and see the particles that
  are emitted. Notice that no particles are emitted
  after the 30th frame.
• Use the Particle Generation rollout to arrange
  that particles are generated at a rage of 100 per
  second, over 100 frames and have a lifetime of
  100 frames.
• Increase the two Spread parameters in the Basic
  Parameters rollout to 45 degrees and 180.
• Pull the Time Slider again and observe the
  particle stream.

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Adding Wind

• Use Create, Space Warps, and Particles Dynamics
  on the Command Menu to create a Wind space warp
  in Left View.
• Use Bind Space Warp on the Main Toolbar to bind
  the Wind space warp to SuperSpray01.
• Notice that the particles are blown by the wind.

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Adding Gravity

• Use Create, Space Warps, and Particles Dynamics
  on the Command Menu to create a Gravity space
  warp in Left View.
• Set the Strength parameter of Gravity to 0.5
  units.
• Use Bind Space Warp on the Main Toolbar to bind
  the Gravity space warp to SuperSpray01.
• Notice that the particles are drawn downward by
  the Gravity.

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Adding a Deflector

• Use Create, Space Warps, and Particles Only on
  the Command Menu to create a POmniFlect space
  warp in Left View.
• Use Bind Space Warp on the Main Toolbar to bind
  the POmniFlect space warp to SuperSpray01.
• Rotate the POmniFlect space warp around the Y
  axis to deflect the particles upward.

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