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 (1805-1865)

• 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

i2 = j2 = k2 = 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 R3

zz’ = (aa’ – bb’) + (ab’ – a’b)i

qq’ = [ss’-v.v’, sv’+s’v+vxv’]

Rotation Matrices

)cos(sin(

)sin()cos(

In 2D:

0

What is SO(3)?

• Orthogonal : UT=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

S3

2:1

SO(3)

S3 in R4 is a Lie Group under Quaternionic Multiplication

In R3, p qpq-1

2

sincos

vq

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)

)()1(),,( 1010 tqtqqqtlerp

q0q1

qt

Spherical Linear Interpolation (SLERP)

)sin(

)sin())1(sin(),,(

d

dtBtdABAtslerp

where:

d= acos (A.B)

Geometry of SLERP in the Plane

A B),,( BAtslerp

d

A

B

L1(t)

),,()(1 BAtslerptL

K

SLERP with Three Points

),,()(2 KAtslerptL

L2(t)),,()(3 BKtslerptL

L3(t)

))(),(,()( 324 tLtLtslerptL

L4(t)

RTICA