Algorithmic Animation & Particle Systems

Key Frame Animation Hand-drawn animation was straight-ahead (start at one position, and animate each small movement Computer Animation is usually based on key frames – a frame for the starting position and a frame for the end position

Key Frame Animation How do you calculate the frame positions in between key frames? Interpolation: Approximation of a function between two known points

Why interpolation? Attributes (position, size, color) change as a function over time Function often given by physics f = ma Why approximate a known function? Solving function at certain point requires solving differential equations

Interpolation Idea: Don’t want to solve equation at each point Approximate some points and connect them with Lines Arcs Other functions (trig)

Euler Integration Use first derivative to get the direction of the curve at a point. Draw lines in direction of first derivative Accuracy depends on the frequency with which you re-evaluate the derivative

Euler Integration

Linear Interpolation Use a line to connect 2 way points In C#: Mathf.Lerp(from, to, percent) Vector3.Lerp Quaternion.Lerp Color.Lerp

Example – Moving between way points private Vector3 start; private Vector3 end; private float startTime; private float speed; void Start() { startTime = Time.time; } void Update() { float journeyLength = Vector3.Distance(start, end); float distCovered = (Time.time - startTime)* speed; float fracJourney = distCovered / journeyLength; transform.position = Vector3.Lerp(start, end, fracJourney);

More on Approximation Collision Detection: Can’t always calculate every collision Isolate groups of things that may collide Approximate the result

Example: Hair

Example: Springs Hooke’s law: f = – kx k is “springiness” constant x is extension Also know f = ma Spring forces act on other elements(& other springs)

Particle Systems Create one simple object Objects follow rules of behavior Many instances of the simple object create an interesting overall effect

Examples: Small spheres Fireworks Waterfalls

Particle Systems: Springs Hair Cloth

Distributed Computation De-centralized control Aggregate behavior emerges Simple rules for the individual lead to complex behavior for the group Often very natural-looking

Distributed Computation Examples Braitenberg Vehicles Light lover Light fear Aggression Traffic Patterns

Distributed Computation Examples Boids: three rules Separation Alignment Cohesion

Distributed Computation Examples Fish Virtual Fishtank Xiaoyuan Tu

Artificial Life Ants Know “home” Look for food If they find food source, pick it up and head home Drop pheromones – chemicals “smelled” by antennae Result: All food gets taken home

Artificial Life Termites Rules Result: food in one place If you reach food and don’t have any, pick some up If you reach food and have some already, put it down Move randomly Result: food in one place

Artificial Life Creatures Random joints with degrees of freedom Try to move Learn to move from one place to the other