Computer Animation Particle systems

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Computer Animation Particle systems

• Some slides courtesy of Jovan Popovic Ronen
  Barzel

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Animation

• How do we describe and generate motion of object
  in the scene?

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Conventional Animation

• Draw each frame of the animation
• great control
• tedious
• Reduce burden with cel animation
• Layer, keyframe, inbetween,
• Example cel panoramas (Disneys Pinocchio)

4
Keyframing

• automate the inbetweening
• use spline curves
• good control
• less tedious
• creating a good animation still requires
  considerable skill and talent

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Procedural animation

• describes the motion algorithmically
• express animation as a function
• of small number of parameteres
• Example a clock with second, minute and hour
  hands
• hands should rotate together
• express the clock motions in terms of a seconds
  variable
• the clock is animated by varying the seconds
  parameter

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Physically Based Animation

• Assign physical properties to objects (masses,
  forces, inertial properties)
• Simulate physics by solving equations
• Realistic but difficult to control

v0
m
g
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Motion Capture

• Usually uses optical markers and multiple
  high-speed cameras
• Triangulate to get marker 3D position
• Faces or joints
• Captures style, subtle nuances and realism
• You must observe someone do something

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See also

• http//www.pixar.com/howwedoit/index.html

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Questions?
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Keyframing

• Describe motion of objects as a function of time
  from a set of key object positions. In short,
  compute the inbetween frames.

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What values are keyframed?

• Camera positions and orientations
• Hierarchical controls
• Smile control, eye blinking, etc.
• Keyframes for these higher-level controls
• Skeletal joint angles
• Inverse kinematics
• Endpoint positions
• Skinning
• Muscle parameters, skin parameters,
• Building 3D models and associated controls is a
  major componenet of every animation pipeline.

Images from the Maya tutorial
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Example Interpolating Positions

• Given positions
• find curve such that

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Linear Interpolation

• Simple problem linear interpolation between
  first two points assuming
• The x-coordinate for the complete curve in the
  figure

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Polynomial Interpolation

• An n-degree polynomial can interpolate any n1
  points. The Lagrange formula gives the n1
  coefficients of an n-degree polynomial that
  interpolates n1 points. The resulting
  interpolating polynomials are called Lagrange
  polynomials. On the previous slide, we saw the
  Lagrange formula for n 1.

parabola
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Spline Interpolation

• Lagrange polynomials of small degree are fine but
  high degree polynomials are too wiggly. Spline
  (piecewise cubic polynomial) interpolation
  produces nicer interpolation.

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Spline Interpolation

• Reuse tools developed for geometric modeling
  Bezier, Hermite, B-Spline,
• The parameter u of curve q(u) becomes the current
  animation time.

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Interpolating Key Frames
Interpolation is not fool proof. The splines may
undershoot and cause interpenetration. The
animator must also keep an eye out for these
types of side-effects.
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Questions?
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Traditional Animation Principles

• The in-betweening, was once a job for apprentice
  animators. Splines accomplish these tasks
  automatically. However, the animator still has to
  draw the key frames. This is an art form and
  precisely why the experienced animators were
  spared the in-betweening work even before
  automatic techniques.
• The classical paper on animation by John Lasseter
  from Pixar surveys some the standard animation
  techniques
• "Principles of Traditional Animation Applied to
  3D Computer Graphics, SIGGRAPH'87, pp. 35-44.
• See also

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Squash and stretch

• Squash flatten an object or character by
  pressure or by its own power
• Stretch used to increase the sense of speed and
  emphasize the squash by contrast

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Timing

• Timing affects weight
• Light object move quickly
• Heavier objects move slower
• Timing completely changes the interpretation of
  the motion. Because the timing is critical, the
  animators used the draw a time scale next to the
  keyframe to indicate how to generate the
  in-between frames.

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Anticipation

• An action breaks down into
• Anticipation
• Action
• Reaction
• Anatomical motivation a muscle must extend
  before it can contract. Prepares audience for
  action so they know what to expect. Directs
  audiences attention. Amount of anticipation can
  affect perception of speed and weight.

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Questions?
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Now
Dynamics
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Types of dynamics

• Point
• Rigid body
• Deformable body (include clothes, fluids, smoke,
  etc.)

Animation by Mark Carlson
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Today we focus on point dynamics

• But we use tons of points
• Particles systems

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Overview

• Generate tons of particles
• Describe the external forces with a force field
• Integrate the laws of mechanics
• Lots of differential equations -(
• Each particle is described by its state
• Position, velocity, color, mass, lifetime, shape,
  etc.
• More advanced versions exist flocks, crowds

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What is a particle system?

• Collection of many small simple particles
• Particle motion influenced by force fields
• Particles created by generators
• Particles often have lifetimes
• Used for, e.g
• sand, dust, smoke, sparks, flame, water,

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Videos

• Demos from http//users.rcn.com/mba.dnai/software/
  flow/
• flow is a particle  animation application under
  development by Mark  B. Allan and
  theReptileLabourProject. 

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Questions?
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Particle motion

• mass m, position x, velocity v
• equations of motion
• Ordinary Differential Equation

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What is an ODE?

• Ordinary Differential Equation
• Relates value of a function to its derivatives
• Ordinary function of one variable
• Partial Differential Equation (PDE) more
  variables

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Standard ODE

• Generic form for first-order ODE
• Note
• typically t is time
• sometimes use Y instead of X, sometimes x
  instead of t
• names sometimes confusing often

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Why do we care?

• Differential equations describe (almost)
  everything
• in the world
• physics
• chemistry
• engineering
• ecology
• economy
• weather
• Also useful for animation!
• ODEs are fundamental. PDEs build on ODEs

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Solving differential equations

• Analytic solutions to differential equations
• Many standard forms, e.g.
• E.g. x kx !
• But most cant be solved analytically
• 3-body problem

x(t)x0ekt
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3 body problem

• 3 punctual masses under the laws of Newtonian
  mechanics (gravity)
• e.g. the Sun and 2 planets
• No, I do not exclude Pluto
• Cannot be solved analytically
• (related to the geeky term 2-body problem)

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Numerical solutions to ODEs

• Given a function f(X,t) compute X(t)
• Typically, initial value problems
• Given values X(t0)X0
• Find values X(t) for t gt t0
• Also, boundary value problems, constrained
  problems,

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Solving ODEs for animation

• For animation, want a series of values
• samples of the continuous function X(t)
• i.e., frames of an animation

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Questions?
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Path through a field

• f(X,t) is a vector field defined everywhere
• E.g. a velocity field
• it may change based on t

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Path through a field

• f(X,t) is a vector field defined everywhere
• E.g. a velocity field
• X(t) is a path through the field

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Higher order ODEs

• E.g., Mechanics has 2nd order ODE
• Express as 1st order ODE

by defining v(t)
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E.g., for a 3D particle

• We have a 6 dimension ODE problem

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For a collection of 3D particles
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Still, a path through a field

• X(t) path in multidimensional state space
• X(t) is an array/vector of numbers.

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Questions?
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Intuitive solution
take steps

• Current state X
• Examine f(X,t) at (or near) current state
• Take a step to new value of X
• Most solvers do some form of this

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Eulers method

• Simplest and most intuitive.
• Define step size h
• Given X0X(t0), take step
• Piecewise-linear approximation to the curve

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Effect of step size

• Step size controls accuracy
• Smaller steps more closely follow curve
• For animation, may need to take many small steps
  per frame

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Eulers method inaccurate

• Moves along tangent can leave curve, e.g.
• Exact solution is circle
• Eulers spirals outward
• no matter how small h is
• will just diverge more slowly

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Eulers method unstable

• Exact solution is decaying exponential
• Limited step size
• If k is big, h must be small

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Analysis Taylor series

• Expand exact solution X(t)
• Eulers method approximates
• First-order method Accuracy varies with h
• To get 100x better accuracy need 100x more steps

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Questions?
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Can we do better?

• Problem
• Idea look at f at the arrival of the step and
  compensate for variation

f has varied along the step
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2nd order methods

• Let
• Then
• This is the trapeziod method,
• AKA improved Eulers method
• Analysis omitted (see 6.839)

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Can we do better?

• Problem f has varied along the step
• Idea look at f at the arrival of the step and
  compensate for variation

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2nd-order methods continued

• Could also have chosen
• then rearrange the same way, let
• and get
• This is the midpoint method

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Comparison
a

• Midpoint
• ½ Euler step
• evaluate fm
• full step using fm
• Trapezoid
• Euler step (a)
• evaluate f1
• full step using f1 (b)
• average (a) and (b)
• Not exactly same result
• Same order of accuracy

f1
fm
b
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Can we do even better

• You bet!
• You will implement Runge Kutta for assignment 4
• See the Physically-Base Modeling Siggraph course
  notes, e.g. at http//www.cs.berkeley.edu/daf/gam
  es/webpage/SimList/Papers/physicallyBasedModeling-
  sg2002.pdfsearch22physically-based20modeling2
  2
• the reference for anythign related to physics and
  computer animation

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Questions?
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Particle Animation

• AnimateParticles(n, y0, t0, tf)
• y y0
• t t0
• DrawParticles(n, y)
• while(t ! tf)
• f ComputeForces(y, t)
• dydt AssembleDerivative(y, f)
• //there could be multiple force fields
• y, t ODESolverStep(6n, y, dy/dt)
• DrawParticles(n, y)

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What is a force?

• Forces can depend on location, time, velocity
• Implementation
• Force a class
• Computes force function for each particle p
• Adds computed force to total in p.f
• There can be multiple force sources

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Forces gravity on Earth

• depends only on particle mass
• f(X,t) constant
• for smoke, flame make gravity point up!

v0
mi
G
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Forces gravity for N-body problem

• Depends on all other particles
• Opposite for pairs of particles
• Force in the direction of pipj with magnitude
  inversely proportional to square distance
• Fij

G mimj / r2
Pi
Pj
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Forces
damping

• force on particle i depends only on velocity of I
• force opposes motion
• removes energy, so system can settle
• small amount of damping can stabilize solver
• too much damping makes motion like in glue

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Forces spatial fields

• force on particle i depends only on position of i
• arbitrary functions
• wind
• attractors
• repulsers
• vortexes
• can depend on time
• note these add energy, may need damping

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Forces spatial interaction

• e.g., approximate fluid Lennard-Jones force
• Repulsive attractive force
• O(N2) to test all pairs
• usually only local
• Use buckets to optimize. Cf. 6.839

force
distance
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Questions?
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Where do particles come from?

• Often created by generators (or emitters)
• can be attached to objects in the model
• Given rate of creation particles/second
• record tlast of last particle created
• create n particles. update tlast if n gt 0
• Create with (random) distribution of initial x
  and v
• if creating n gt 1 particles at once, spread out
  on path

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Particle lifetimes

• Record time of birth for each particle
• Specify lifetime
• Use particle age to
• remove particles from system when too old
• often, change color
• often, change transparency (old particles fade)
• Sometimes also remove particles that are
  offscreen

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Rendering and motion blur

• Particles are usually not shaded (just emission)
• Often, they dont contribute to the z-buffer
  (rendered last with z-buffer disabled)
• Draw a line for motion blur
• (x, xvdt)
• Sometimes use texture maps (fire, clouds)

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Questions?
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Particle Animation Reeves et al. 1983
Start Trek, The Wrath of Kahn

• Star Trek, The Wrath of Kahn Reeves et al. 1983

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How they did it?

• One big particle system at impact
• Secondary systems for rings of fire.

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Particle Modeling Reeves et al. 1983

• The grass is made of particles

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Other uses of particles

• Water
• E.g. splashes in lord of the ring river scene
• Explosions in games
• Grass modeling
• Snow, rain
• Screen savers

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Questions?
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Collisions

• Detection
• Response
• Overshooting problem (when we enter the solid)

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Detecting collisions

• Easy with implicit equations of surfaces
• H(x,y,z)0 at surface
• H(x,y,z)lt0 inside surface
• So just compute H and you know that youre inside
  if its negative
• More complex with other surface definitions

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Collision response
N
v

• tangential velocity vb unchanged
• normal velocity vn reflects
• coefficient of restitution
• change of velocity -(1?)v
• change of momentum Impulse -m(1?)v
• Remember mirror reflection? Can be seen as
  photon particles

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Collisions - overshooting

• Usually, we detect collision when its too late
  were already inside
• Solutions back up
• Compute intersection point
• Ray-object intersection!
• Compute response there
• Advance for remaining fractional time step
• Other solutionQuick and dirty fixup
• Just project back to object closest point

backtracking
fixing
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Questions?
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Implementation Particle

• Particles p have attributes that specify mass
  p.m, position p.x, and velocity p.v.
• Other attributes such as color, particle age, and
  other can also be included for more advanced
  effects.
• For implementation convenience, each particle
  will also have a force accumulator p.f that sums
  up all forces acting on the particle.

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Implementation Particle Systems

• Particle system is an array of particles.
• It will support numerical integration with the
  following methods
• Compute dimension of the particle system
• Set current state (positions and velocities)
• Get current state (positions and velocities)
• Compute accelerations f(X, t)
• Integration uses only these four methods to
  simulate evolution of a particle system

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Implementation Accelerations

• / compute accelerations and place in f /
• int ComputeAccelerations(ParticleSystem ps,
  double f)
• int i / particle counter /
• int j / force dimension counter /
• ClearForces(ps) / zero force accumulators
  /
• ComputeForces(ps) / accumulate forces /
• for(i0, j0 i lt ps.n i)
• fj ps.pi.v0 / xdot v /
• fj ps.pi.v1
• fj ps.pi.v2
• fj ps.pi.f0/m / vdot f/m /
• fj ps.pi.f1/m
• fj ps.pi.f2/m

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Implementation Get State

• / gather state from the particles into X /
• int GetState(ParticleSystem ps, double X)
• int i, j
• for(i0, j0 i lt ps.n i)
• Xj ps.pi.x0
• Xj ps.pi.x1
• Xj ps.pi.x2
• Xj ps.pi.v0
• Xj ps.pi.v1
• Xj ps.pi.v2

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Implementation Set State

• / scatter state from X into particles /
• int SetState(ParticleSystem ps, double X)
• int i, j
• for(i0, j0 i lt ps.n i)
• ps.pi.x0 Xj
• ps.pi.x1 Xj
• ps.pi.x2 Xj
• ps.pi.v0 Xj
• ps.pi.v1 Xj
• ps.pi.v2 Xj

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Implementation Euler Integration

• void EulerStep(ParticleSystem ps, double dt)
• ComputeAccelerations(ps, f)
• GetState(ps, X0)
• X1 X0 dt f / Euler step /
• SetState(ps, X1)
• ps.t dt / advance time /

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Questions?
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More advanced version

• Flocking birds, fish school
• http//www.red3d.com/cwr/boids/
• Crowds

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Flocks

• From Craig Reynolds
• Each bird modeled as a complex particle (boid)
• A set of forces control its behavior
• Based on location of other birds and control
  forces

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Flocks

• From Craig Reynolds
• Boid" was an abbreviation of "birdoid", as his
  rules applied equally to simulated flocking
  birds, and schooling fish.

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Questions?
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Additional references

• http//www.cse.ohio-state.edu/parent/book/outline
  .html
• http//www.pixar.com/companyinfo/research/pbm2001/
  
• http//www.cs.unc.edu/davemc/Particle/