Emerging Technologies for Games Animation: Interpolation

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Emerging Technologies for GamesAnimation
Interpolation

• CO3303
• Week 3

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Interpolation 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
• One at the start of rotation, one at the end
• Generate frames in between with interpolation

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Previous 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)
• N.B. Refer to Van Verth 10.6 for this section

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Linear Interpolation or Lerp

• 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

• If P0 P1 are points, then P(t) defines the line
  between them
• Hence linear interpolation

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Linear 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

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Linear 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 can use higher
  order interpolations to do this
• But linear interpolation works and will suffice
  for now

Piecewise Interpolation
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Linear Interpolation of Scaling

• Scaling can also be interpolated linearly

• Again, piecewise linear interpolation will cause
  sharp changes
• This time in the speed of shrinking/expanding
• Less noticeable effect
• Again, possible to calculate smoother result with
  higher order interpolation

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Linear 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
  incorrect
• Even with just one pair of key-frames
• The tips of the rotated vectors are interpolated

• Resulting in an unwanted scaling
• Inconsistent angles quarter steps here, but a ?
  ß

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Improved Linear Interpolation

• Can normalise the rotations to remove the scaling
  effect
• But still left with inaccurate angles

• Normalising quaternions is trivial
• Gram-Schmidt orthogonalisation used to normalise
  matrices (see Van Verth 1.2.7)

• Allows us to use linear interpolation of
  rotations, if the overall rotation is small
  enough
• Otherwise need linear interpolation of the angles

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Slerp

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

• 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
• Returns the interpolated rotation

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Slerp for Matrices

• 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 ? t, then convert back
• Very expensive
• More than 100 operations

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Slerp for Quaternions

• Can show (10.6.2) that slerp for quaternions is
• Expensive, mainly because of the sin functions
• But sin ? can be precalculated
• Constant for any pair of key frames

• Has potential accuracy problems for small ?
• However, more usable than the matrix version

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Interpolation Summary

• We want to interpolate between two key frames
• Can use simple linear interpolation (lerp) for
  the position and scaling
• Consider angle of rotation 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 slow, angles ineffective
• Note newer methods avoid slerp see 10.6.3