Maths and Technologies for GamesAnimation: Interpolation

CO3303

Week 2

Interpolation for AnimationInterpolation for Animation

• Given two transforms for a model, we can calculate the transforms in between the two:– This is called interpolation

• Animation is stored as a sequence of key frames (transforms)

• To get frames between the key frames, we use interpolation

• For example a rotating arm has 2 key frames stored– One at the start of rotation (T1), one at the end (T2)

– Calculate (don’t store) frames in between with interpolation

Previous InterpolationPrevious Interpolation

• We have interpolated between elements before:– Alpha blending – source and destination colours– Skinning – the effects of multiple bones

• We used Linear Interpolation each time:– Alpha blending:

Dest.R = Source.A * Source.R + (1-Source.A) * Dest.R

– Skinning (with two bones)

– Do you see a pattern?

• N.B. Details for this section in Van Verth

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• General linear interpolation between two mathematical elements P0 and P1 takes the form:

• Typically t is in the range [0,1] - note that:

• The parameter t selects a element P(t) in between the start and end elements P0 and P1

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tPtPtP 10 )1()(

Linear Interpolation (Lerp)Linear Interpolation (Lerp)

• If P0 & P1 are points, then P(t) will be on the line between them:

• Hence linear interpolation

Linear Interpolation of TransformsLinear Interpolation of Transforms

• A transformation is typically made up of a position, a rotation and a scaling component– Can be stored in several ways, e.g.

• Position: vector / scalars (x,y,z) or a matrix• Rotation: Euler angles, a matrix or a quaternion• Scaling: vector / scalars or a matrix• Can have a single combined matrix for everything

• We will consider the effect of linear interpolation on each of the components separately

• Then consider how this affects our storage choice

Linear Interpolation of PositionLinear Interpolation of Position

• Linear interpolation applied to key-frame positions will give sensible in-between positions:

• With several key-frames we get piecewise linear interpolation

• Motion between points has sharp changes in direction:

• May prefer smoother motion and we can use higher order interpolations to do this– But linear interpolation works and will

suffice for nowPiecewise Linear Interpolation

Higher Order Interpolation

Linear Interpolation of ScalingLinear Interpolation of Scaling

• Scaling can also be interpolated linearly:

• Works fine for one pair of key-frames• Again, linear interpolation over many frames will cause

scaling rate to change sharply at each key-frame:– The speed of shrinking/expanding changes suddenly– Generally a less noticeable effect

• Again, possible to calculate smoother result with higher order interpolation

Linear Interpolation of RotationLinear Interpolation of Rotation

• Can apply linear interpolation to rotations if they are in matrix or quaternion form:– Ineffective with Euler angles

• Unlike position and scale, the result is not correct– Even with just one pair of

key-frames

• The tips of the rotated vectors are interpolated

– Resulting in an unwanted scaling– Incorrect angles – want quarter steps, but here α ≠ β

Normalised Lerp (Nlerp)Normalised Lerp (Nlerp)

• Can normalise the rotations to remove the scaling effect

• Still have inaccurate angles– Normalising quaternions is trivial

– Gram-Schmidt orthogonalisation used to normalise matrices (see Van Verth early chapter)

• Allows us to use linear interpolation of rotations– If the overall rotation is small enough

• Otherwise we need linear interpolation of the angles

Spherical Lerp (Slerp)Spherical Lerp (Slerp)

• Linear interpolation of angles is same as linear interpolation of an arc on a sphere

ttslerp )(),,( 1QPPQP

• So this technique is called spherical linear interpolation or slerp

• The formula is different from linear interpolation:

• As before – take a start and end rotation (P, Q), and a value t from [0,1]

• Formula calculates the correct interpolated rotation

Slerp for MatricesSlerp for Matrices

• We can apply the slerp formula if P, Q are matrices

• Matrix multiplication / inverse are understood• But how to calculate a matrix raised to a power?

– i.e. Mt

• Need to convert the matrix to axis-angle format, then calculate θ

t, then convert back• Very expensive

– More than 100 operations

tt,slerp )(),( 1QPPQP

Slerp for QuaternionsSlerp for Quaternions

• Can show that slerp for quaternions is:

• Expensive, mainly because of the sin functions– But sin θ can be precalculated

• It’s constant for any pair of key frames

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ttt,slerp

• Has potential accuracy problems for small θ

• However, more usable than the matrix version

Interpolation SummaryInterpolation Summary

• We want to interpolate between two key frames

• Can use simple linear interpolation (lerp) for the position and scaling

• For rotation, consider angle between the frames:– Can often use lerp on rotations & normalise the result– For large angles, use slerp

• In any case store rotations as quaternions if you intend to interpolate them– Matrices expensive to interpolate, Eulaer angles ineffective

• Note: there are some newer methods to approximate or avoid slerp