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Physically Based Animation and Modeling
CSE 3541 Matt Boggus
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Overview Newton’s three laws of physics
Integrating acceleration to find position Particle Systems Common forces in physically based animation
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Mass and Momentum 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) Force = change in momentum over time = mass * acceleration
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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 acceleration Constant acceleration
No acceleration Constant acceleration f a m Similarly, you’d need a different equation to handle cases where acceleration is linear, quadratic, or some other higher order. a v’ v vave
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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 ]
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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 + 𝑒𝑛𝑑𝑣+𝑣 2 ∗dt [vectors: x, endv, v scalars: 2, dt] v = endv [vectors: endv, v]
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See spreadsheet example
Comparison of methods See spreadsheet example
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Particle Systems Star Trek 2 – genesis sequence (1982) More examples
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
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Particle Systems Lots of small particles - local rules of behavior
Create ‘emergent’ element Common rules for particle motion: Do collide with the environment Do not collide with other particles Common rules for particle rendering: Do not cast shadows on other particles Might cast shadows on environment Do not reflect light - usually emit it
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Particle Example source
Collides with environment but not other particles Particle’s midlife with modified color and shading Particle’s demise, based on constrained and randomized life span source Particle’s birth: constrained and time with initial color and shading (also randomized)
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Particle system implementation
Update Steps for each particle if dead, reallocate and assign new attributes animate particle, modify attributes render particles Use constrained randomization to keep control of the simulation while adding interest to the visuals
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Constrained randomization
particleX = x particleY = y particleX = x + random(-1,1) particleY = y + random(-1,1)
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Particle (partial example in C#)
class Particle { Vector3 position; // Updates frame to frame Vector3 velocity; // Updates frame to frame Vector3 force; // Reset and recomputed each frame GameObject 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… };
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Particle emitter (partial example in C#)
using System.Collections.Generic; using System.Collections; class ParticleEmitter { ArrayList Particles = new ArrayList(); public: void Update(deltaTime); };
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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();
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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)); }
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Deleting GameObjects GameObject myParticle; // …create, animate, etc. … Destroy(myParticle); Note: this affects the associated GameObject; it does not delete the variable myParticle
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Lab3 Implement a particle system where each particle is a GameObject
Restrictions No RigidBodies No Colliders Minimal credit if you use these for lab3
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Forces – gravity
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Forces Static friction Kinetic friction Viscosity for small objects
No turbulence For sphere
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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
Hooke’s Law
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Animation from http://www.acs.psu.edu/drussell/Demos/SHO/damp.html
Damping example Animation from
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Spring-mass-damper system
f -f
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Springs At rest length l, the force f is zero
Points are located at r1 and r2 [scalar displacement] [direction of displacement]
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Example – Jello cube http://www.youtube.com/watch?v=b_8ci0ZW4vI
Spring-mass system V3 E23 E31 V2 V1 E12 Example – Jello cube
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Spring mesh – properties for cloth
Each vertex is a point mass Each edge is a spring-damper Diagonal springs for rigidity Angular springs connect every other mass point Global forces: gravity, wind Example
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Virtual springs – soft constraints
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