Class 7: Interpolation

Topics:

- Interpolation - floats, shapes, colors, pixels
- Lerp vs smoother interpolation - keys
http://en.wikipedia.org/wiki/Image:Interpolation_example_linear.png

http://en.wikipedia.org/wiki/Image:Interpolation_example_spline.png

http://www.garycmartin.com/phoneme_examples.html

- Blends - multiway interpolation, WEIGHTED interpolation; "mix"
http://www.jgiesen.de/ColorTheory/RGBColorApplet/rgbcolorapplet.html

- smoothstep()
http://www.renderman.ru/2translate/cade_maths_files/smoothstep.jpg

- scattered interpolation - distance-based weighting
http://www.quantdec.com/SYSEN597/GTKAV/section9/exb/interpolation.gif

- natural neighbor interpolation:
http://www.mlahanas.de/CompGeom/vor1-Dateien/image016.jpg

- angle interpolation
http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/slerp/index.htm

http://www.theory.org/software/qfa/writeup/node12.html

http://number-none.com/product/Hacking%20Quaternions/index.html

http://www.gamedev.net/reference/articles/article1095.asp

http://number-none.com/product/Understanding%20Slerp,%20Then%20Not%20Using%20It/

Doug Moore, Cornell, '89 ------------------------------------------------------------------------------- Reply-To: moore@svax.cs.cornell.edu (Doug Moore) Distribution: usa Organization: Cornell Univ. CS Dept, Ithaca NY Lines: 49 Since no one else has mentioned this, I guess I will. Why not use quaternions, rather than rotation matrices, to represent your rotations? Quaternions on the unit sphere and 3-d rotations are isomorphic, and quaternions don't require the redundant storage and calculation that 3x3 matrices do. A quaternion may be thought of as an entity of the form , where s is a scalar and x is a 3-vector. Multiplication of quaternions is given by * = . A unit quaternion is one that satisfies s*s + x dot x = 1. A unit quaternion may also be thought of as a rotation of angle 2 arccos s about the axis v. To rotate a vector v by a rotation quaternion q to get a vector w, use the formula <0,w> = inv(q) * <0,v> * q, where inv(q) * q = <1,0>, and inv() = . Or, if you prefer, form the equivalent rotation matrix 1 - 2 (x2*x2 + x3*x3) 2 (x1*x2 + s * x3) 2 (x1*x3 - s*x2) 2 (x1*x2 - s*x3) 1 - 2 (x1*x1 + x3*x3) 2 (x2*x3 + s*x1) 2 (x1*x3 + s*x2) 2 (x2*x3 - s*x1) 1 - 2 (x1*x1 + x2*x2) and use that. The basic algorithm, then, to display vectors V = rotating by q every frame is rot = <1,0> do-forever R = rotation matrix associated with rot DISP = R * V display all vectors in DISP rot = rot * q norm = rot.s * rot.s + rot.x1 * rot.x1 + rot.x2 * rot.x2 + rot.x3 * rot.x3 if (abs(norm - 1) > tolerance) norm = sqrt(norm) rot.s = rot.s/norm rot.x1 = rot.x1/norm rot.x2 = rot.x2/norm rot.x3 = rot.x3/norm endif enddo That's the general idea, anyway. For a less terse exposition, see Ken Shoemake, "Animating Rotation with Quaternion Curves", COMPUTER GRAPHICS Vol 19 No 3, pp. 245-254. Incidentally, similar quaternion techniques can be used for 4-d rotations. I haven't been able to get a handle on higher dimensions, though.

- extrapolation