Quaternionic Splines of Paths Robert Shuttleworth Youngstown State University Professor George Francis, Director illiMath2001 NSF VIGRE REU UIUC-NCSA

Order of Events History of the quaternions What is a quaternion? Significance to Computer Graphics Splining of Paths RTICA

History of the Quaternions Sir William Rowan Hamilton ( ) Royal Canal, Dublin – October 16, 1843 First example of a Lie Group Gibbs – vector dot and cross product

What is a quaternion? Generalizations of the complex numbers into 4D i 2 = j 2 = k 2 = ijk = -1 Multiplication of quaternions is not commutative. Complex Numbers (C) Quaternions (H) z = a+bi; a,b in R q = [s,v], s in R, v in R 3 zz’ = (aa’ – bb’) + (ab’ – a’b)i qq’ = [ss’-v.v’, sv’+s’v+v x v’]

Rotation Matrices In 2D:

What is SO(3)? Orthogonal : U T =U -1 SO(n) = special orthogonal group SO(2) = {rotations about the origin in 2D} SO(3) = {set of rotations in 3D}

Rotations with Quaternions S 3 2:1 SO(3) S 3 in R 4 is a Lie Group under Quaternionic Multiplication In R 3, p qpq -1 Rotation:

Advantages of Quaternions in Computer Graphics Coordinate system independent Easy to represent rotations Less values need to be stored when compared to matrices Allows efficient splining of paths

Linear Interpolation (LERP) q0q0 q1q1 qtqt

Spherical Linear Interpolation (SLERP) where: d= acos (A.B)

Geometry of SLERP in the Plane A B

A B L 1 (t) K SLERP with Three Points L 2 (t) L 3 (t) L 4 (t)

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