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 AnimationAlgorithms 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

For arbitrary function, f(t)


Step size


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


Runge Kutta Integration: 2nd order

Aka Midpoint Method

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


Step size

Euler Integration

Midpoint Method


Runge Kutta Integration: 4th order

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


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


Differential equation, initial boundary problem

(implicit/backward)

Euler method

(explicit/forward)

Euler method

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


Semi-Implicit Euler Integration


Methods specific to update position from acceleration

Heun Method

Verlet Method

Leapfrog Method


Heun Method


Verlet Method


Leapfrog Method


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