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Quaternions

Ken Shoemake

Department of Computer and Information Science

University of Pennsylvania

Philadelphia, PA 19104

Abstract

Of the many mathematical tools used in computer graphics, most are well covered

in standard texts; quaternions are not. This article is intended to provide tutorial

material on quaternions, including what they are, why they are useful, how to use

them, where to use them, and when to be careful.

Introduction

Computer graphics uses quaternions as coordinates for rotations and orientations.

At SIGGRAPH 1985, quaternion curve methods were introduced to computer

graphics to facilitate rotation animation. Although this is a rather specialized

environmental niche, quaternions work so well they are able to compete

successfully both with more general coordinates such as matrices, and with other

special coordinates such as Euler angles.

Quaternion use has since expanded to include new curve methods and new

applications, including physically based modeling, constraint systems, and user

interfaces. This is because when a quaternion implementation is compared to other

alternatives, it is usually simpler, cheaper, and better behaved.

That’s the good news. The bad news is that researchers and implementors must

learn some new mathematics, and quaternions are not taught in core mathematics

and science courses. However, neither are homogeneous coordinates; and a broad

view reveals that quaternions are simply a more sophisticated use of the four-

component homogeneous coordinates to which we are already accustomed.

Rotations

What can be so unusual about three-dimensional rotations that they warrant their

own coordinate system? Put simply, because rotations do not commute, they cannot

be treated as vectors. Suppose T

(d) is translation along the x axis by d, and R

(θ) is

rotation around the x axis by θ ; similarly for T

, T

, R

, and R

. Then

(45cm)

(90cm) commutes, meaning it equals T

(90cm)

(45cm); but

(45°)

(90°) ≠ R

(90°)

(45°). (As usual, B

A means the composition of action A

followed by action B.) Note also that if we take the composition R

(180°)

(180°) we

get the same result as R

(180°)!

It takes some study to unravel all the implications of these elementary observations,

but they are far-reaching and often surprising. The set of all 3-dimensional rotations

is not organized as a simple 3-dimensional vector space, but as a closed curved 3-

dimensional manifold, which is also a (continuous) group. This group is known as

SO(3). (For special and orthogonal.) Manifolds are generalizations of surfaces; this

one has the topology of real 3-dimensional projective space, RP

, and also a natural

distance measure related to rotation angle. Although a vector space such as the

translations trivially splits into a product of lines, SO(3) does not split. Instead, it has

a more sophisticated description as a fiber bundle over the sphere of directions, S

with fiber space the planar rotations, SO(2).

Unit quaternions have the remarkable property of capturing all of the geometry,

topology, and group structure of 3-dimensional rotations in the simplest possible

way. (Technically, they form what is called a universal covering.)

Quaternion definitions

Quaternions can be defined in several different, equivalent ways. It is helpful to

know them all, since each form is useful. Historically, quaternions were conceived

by Hamilton as like extended complex numbers, w+ix+jy+kz, with i

= j

= k

= –1, ij

= k = –ji, with real w, x, y, z. (In honor of Hamilton, mathematicians denote the

quaternions by H.) Notice the non-commutative multiplication, their novel feature;

otherwise, quaternion arithmetic is pretty much like real arithmetic. Hamilton was

also quite aware of the more abstract possibility of treating quaternions as simply

quadruples of real numbers [x, y, z, w ], with operations of addition and

multiplication suitably defined. But it happens that the components naturally group

into the imaginary part, (x, y, z), for which Hamilton coined the term vector, and the

purely real part, w , which he called a scalar. Later authors (notably Gibbs)

appropriated Hamilton’s terminology and extracted from the clean operations of

quaternion arithmetic the somewhat messier—but more general—operations of

vector arithmetic. Courses today teach Gibbs’ dot and cross products, so it is

convenient to reverse history and describe the quaternion product using them.

Thus we usually will want to write a quaternion as [v, w], with v = (x, y, z). We will

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