1. UW Extension Certificate Program in Game Development 2nd quarter: Advanced Graphics
Animation

2. Goals Learn what is at the core of animation math
See some implementation strategies

3. Animatable values We say we animate one value
In practice, we could be animating several values at once
A color has three components (R, G, B)
A 3D position has three coordinates
A transformation matrix has up to 12 degrees of freedom (DoF)
Addition and multiplication by scalar gives us lerp
Note that a rotation matrix or quaternion has three DoF
Conceptually, rotations have “addition” (composition) and slerp
We will assume that we animate a single float

4. Interpolation vs. extrapolation
Interpolation is figuring out values for instants in between already-known values
lerp, with a weight between 0 and 1
Extrapolation is figuring it out for instants outside of the range of the known values
Lerp with a weight that’s negative or >1
When animating, we normally want to do interpolation
Interpolation
Extrapolation X1 X0 t0 t1 t

5. Piece-wise linear interpolation
Start with a few known samples
Figure out the two closest ones
Lerp between them
Simple and effective
Continuity of movement
No continuity of velocities! X2 X1 X4 X0 X3 t0 t1 t2 t3 t4 t

6. Piece-wise cubic spline interpolation
Start with a few known samples
Figure out the four closest ones
They describe a cubic
A*t3 + B*t2 + C*t + D
Fairly simple, very effective
Continuity of movement
Also continuity of velocities! X2 X1 X4 X0 X3 t0 t1 t2 t3 t4 t

7. Cubic spline interpolation
Interpolate between X1 and X2 with a cubic polynomial
D3DX provides functions
“Hermite” if tangents are provided
“Catmull-Rom” if adjacent points are provided X1 X3 X2 X0 t0 t1 t2 t3 t

8. Cubic spline interpolation
Can be done as a series of lerps:
A = lerp(X0,X1,a+1)
B = lerp(X1,X2,a)
C = lerp(X2,X3,a–1)
D = lerp(A,B,(a+1)/2)
E = lerp(B,C,a/2)
lerp(D,E,a)
Can use slerp for rotations!
Not very common (use linear) X1 X3 X2 X0 t0 t1 t2 t3 t

9. Keyframed animation Same concept as piecewise
But eliminate unnecessary samples
They will be interpolated!
Animation code will need to figure out which two keyframes are the closest
Need to perform a search
Cannot use Catmull-Rom
Samples are not equidistant anymore t t0 X0 t1 X1 t2 X2 t3 X3 t t0 t1 X0 X1 t2 X2 t3 X3 t4 X4