Computer animation algorithms and techniques
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Computer Animation Algorithms and Techniques. Integration. Integration. Given acceleration, compute velocity & position by integrating over time. Projectile. given initial velocity under gravity. Euler integration. For arbitrary function, f(t). Integration – derivative field.

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Computer Animation Algorithms and Techniques

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Computer animation algorithms and techniques

Computer AnimationAlgorithms and Techniques

Integration


Computer animation algorithms and techniques

Integration

Given acceleration, compute velocity & position by integrating over time


Computer animation algorithms and techniques

Projectile

given initial velocity under gravity


Computer animation algorithms and techniques

Euler integration

For arbitrary function, f(t)


Computer animation algorithms and techniques

Integration – derivative field

For arbitrary function, f(t)


Computer animation algorithms and techniques

Step size


Computer animation algorithms and techniques

Numeric Integration Methods

(explicit or forward) Euler Integration

2nd order Runga Kutta Integration (Midpoint Method)

4th order Runga Kutta Integration

Implicit (backward) Euler Integration

Semi-implicit Euler Integration


Computer animation algorithms and techniques

Runge Kutta Integration: 2nd order

Aka Midpoint Method

For unknown function, f(t); known f ’(t)


Computer animation algorithms and techniques

Step size

Euler Integration

Midpoint Method


Computer animation algorithms and techniques

Runge Kutta Integration: 4th order

For unknown function, f(t); known f ’(t)


Computer animation algorithms and techniques

Implicit Euler Integration

For arbitrary function, f(t), find next point whose derivative updates last value to this value: required numeric method (e.g. Newton-Raphson)

Search for point such that negative of tangent points back at original point


Computer animation algorithms and techniques

Differential equation, initial boundary problem

(implicit/backward)

Euler method

(explicit/forward)

Euler method

e.g. linearize f’ and use Newton-Raphson


Computer animation algorithms and techniques

Semi-Implicit Euler Integration


Computer animation algorithms and techniques

Methods specific to update position from acceleration

Heun Method

Verlet Method

Leapfrog Method


Computer animation algorithms and techniques

Heun Method


Computer animation algorithms and techniques

Verlet Method


Computer animation algorithms and techniques

Leapfrog Method


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