Physically Based Animation and Modeling

CSE 3541Matt Boggus

Overview

• Newton’s three laws of physics

• Integrating acceleration to find position

• Particle Systems

• Common forces in physically based animation

Mass and Momentum

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• Associate a mass with an object. We assume that the mass is constant

• Define a vector quantity called momentum (p), which is the product of mass and velocity

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Newton’s First Law

• A body in motion will remain in motion• A body at rest will remain at rest, unless acted upon

by some force

• Without a force acting on it, a moving object travels in a straight line

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Newton’s Second Law

• Newton’s Second Law says:

• This relates the kinematic quantity of acceleration to the physical quantity of force

(Kinematics – the branch of mechanics concerned with the motion of objects without reference to the forces that cause the motion)

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Newton’s Third Law• Newton’s 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

• Every action has an equal and opposite reaction– Do we really want this for games and animation?

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Integration

Given acceleration, compute velocity & position by integrating over time

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Physics review, equations for:Zero accelerationConstant acceleration

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Constant accelerationNo acceleration

Pseudocode for motion within an animation loop (Euler method)

To update an object at point x with velocity v:

a = (sum all forces acting on x) / m [ ∑vectors scalar: m ]

v = v + a * dt [ vectors: v, a scalar: dt ]

x = x + v * dt [ vectors: x, v scalar: dt ]

Pseudocode for motion within an animation loop (Euler 2)

To update an object at point x with velocity v:

a = (sum all forces acting on x) / m [ ∑vectors scalar: m ]

endv = v + a * dt [vectors: endv, v, a scalar: dt]

x = x + [vectors: x, endv, v scalars: 2, dt]

v = endv [vectors: endv, v]

Comparison of methods

See spreadsheet example

Particle Systems• A collection of a large number of point-like elements• Model “fuzzy” or “fluid” things– Fire, explosions, smoke, water, sparks, leaves, clouds, fog,

snow, dust, galaxies, special effects

• Model strands– Fur, hair, grass

• Star Trek 2 – genesis sequence (1982)– The making of the scene

• More examples

Lots of small particles - local rules of behaviorCreate ‘emergent’ elementCommon rules for particle motion:

Do collide with the environmentDo not collide with other particles

Common rules for particle rendering:Do not cast shadows on other particlesMight cast shadows on environmentDo not reflect light - usually emit it

Particle Systems

Particle Example

source Particle’s birth: constrained and time with initial color and shading (also randomized)

Particle’s demise, based on constrained and randomized life span

Collides with environment but not other particles

Particle’s midlife with modified color and shading

Particle system implementation

Update Steps1. for each particle

1. if dead, reallocate and assign new attributes2. animate particle, modify attributes

2. render particles

Use constrained randomization to keep control of the simulation while adding interest to the visuals

Constrained randomization

particleX = x + random(-1,1)particleY = y + random(-1,1)

particleX = xparticleY = y

Particle (partial example in C#)class Particle {

Vector3 position; // Updates frame to frameVector3 velocity; // Updates frame to frameVector3 force; // Reset and recomputed each frameGameObject geom;// other variables for mass, lifetime, …

public:void Update(float deltaTime);void ApplyForce(Vector3 &f) { force.Add(f); }void ResetForce() { force = Vector3.zero; }// other methods…

};

Particle emitter (partial example in C#)

using System.Collections.Generic;using System.Collections;

class ParticleEmitter {ArrayList Particles = new ArrayList();

public:void Update(deltaTime);

};

Particle Emitter Update()

Update(float deltaTime) {

foreach (Particle p in Particles) {// …add up all forces acting on p…

}

foreach (Particle p in Particles){p.Update(deltaTime);p.ResetForce();

}}

Creating GameObjects

for(int i = 0; i < numberOfAsteroids; i++){ GameObject aSphere = GameObject.CreatePrimitive(PrimitiveType.Sphere); aSphere.transform.parent = transform; aSphere.name = "sphere" + i.ToString();

aSphere.transform.position = new Vector3(Random.Range(-10.0f, 10.0f), Random.Range(-10.0f, 10.0f),

Random.Range(-10.0f, 10.0f));

aSphere.transform.localScale = new Vector3(Random.Range(0.0f, 1.0f), Random.Range(0.0f, 1.0f),

Random.Range(0.0f, 1.0f));}

Deleting GameObjects

GameObject myParticle;

// …create, animate, etc. …

Destroy(myParticle); Note: this affects the associated GameObject; it does not delete the variable myParticle

Lab3

• Implement a particle system where each particle is a GameObject

• Restrictions– No RigidBodies– No Colliders– Minimal credit if you use these for lab3

Forces – gravity

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Forces

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Nkk fkf Kinetic friction

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for small objectsNo turbulence

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Forces• Aerodynamic drag is complex and difficult to model accurately

• A reasonable simplification it to describe the total aerodynamic drag force on an object using:

• Where ρ is the density of the air (or water, mud, etc.), cd is the coefficient of drag for the object, a is the cross sectional area of the object, and e is a unit vector in the opposite direction of the velocity– In short – create a scaled vector in the opposite direction of velocity

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Forces – spring-damper

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Hooke’s Law

Damping example

Animation from http://www.acs.psu.edu/drussell/Demos/SHO/damp.html

Spring-mass-damper system

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Springs

• At rest length l, the force f is zero

• Points are located at r1 and r2

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[direction of displacement]

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Spring-mass system

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Example – Jello cube http://www.youtube.com/watch?v=b_8ci0ZW4vI

Spring mesh – properties for cloth

Each edge is a spring-damper

Angular springs connect every other mass point

Each vertex is a point mass

Global forces: gravity, wind

Diagonal springs for rigidity

Example http://www.youtube.com/watch?v=ib1vmRDs8Vw

Virtual springs – soft constraints