Introduction

The purpose of this paper is to demonstrate how quaternion operations are used to perform rotation in Euclidean space.

Rotation is demonstrated using nothing but simple vector operations (dot products, cross products and scalar multiplication).

At no point will I rely on imaginary numbers, calculus or any other so-called “non-Euclidean” concepts.

Definitions

Quaternions are specified thusly:

q = (v,s), where v is a vector in ordinary 3-dimensional space and s is a real-valued scalar.

For a vector v, the equivalent quaternion is specified as:

v = (v,0)

Quaternion multiplication is performed thusly:

(v,s)(v’,s’) = (vÄv’ + sv’ + vs’, ss’ – v·v’) where Ä is the vector cross-product and · is the vector dot-product.

Multiplication is associative but not commutative.

Scalar multiplication is performed thusly:

x(v,s) = (xv,xs)

Scalar multiplication is both associative and commutative.

The conjugate of a quaternion is defined thusly:

(v,s)^* = (-v,s)

These are the only definitions we need to demonstrate the rotation formula.

Quaternion Rotation

Let v₀, v₁ and v₂ be unit vectors in 3-dimensional space. v₀, v₁ and v₂ are coplanar such that v₀Äv₁ = v₁Äv₂ and v₀·v₁ = v₁·v₂.

Let q = cos^-1(v₀·v₁) be the angle between v₀ and v₁. It is clear from the diagram (fig 1) that v₂ represents a rotation of v₀ about v₀Äv₁ by angle 2q.

Theorem

Given the quaternion q = (v₀Äv₁, v₀·v₁), show that the operation qv₀q^* = v₂.

Proof

First, we find some equivalent forms for q:

q = (v₀Äv₁, v₀·v₁)

v₁v₀* = (-v₁Äv₀,v₁·v₀) = (v₀Äv₁,v₀·v₁)

\ q = v₁v₀*

v₁Äv_{2 =}v₀Äv₁

v₁·v₂ = v₀·v₁

\ q = (v₁Äv₂,v₁·v₂) = v₂v₁^*

Now, substituting q = v₁v₀^* into qv₀q^* gives:

qv₀q^* = (v₁v₀^*)v₀q^* = v₁(v₀^*v₀)q^*

For unit a unit vector v, v^*v = 1, so:

qv₀q^* = v₁q^*

Now we express q^* in terms of v₁ and v₂:

q^* = (v₂v₁^*)^* = v₁v₂^*

Plugging this form of q^* into v₁q^* gives:

qv₀q^* = v₁q^* = v₁²v₂^*

For a unit vector v, v² = -1, so

qv₀q^* = -v₂^*

For a vector, v^* = -v, so

qv₀q^* = v₂

QED

Okay, so we’ve established that qv₀q^* = v₂ for v₀, v₁ and v₂ as unit vectors. Generalizing this for all vectors is trivial. Just multiply both sides of the equation by a scalar:

qv₀q^* = v2

\ qAv₀q^* = Av₂ for some scalar A.

Matrices and Slerps

The basis for quaternion-matrix conversions is finding the matrix M such that Mv = qvq*. Since qvq* is a linear operation, it stands to reason that the matrix M exists.

The basis for the much-touted spherical linear interpolation (slerp) is the fact that quaternions as rotations all have the form:

q = (vsin(q/2),cos(q/2))

where v is a unit vector in the direction of the axis of rotation and q is the angle of rotation. These quaternions all have unit length. Therefore we can apply spherical linear interpolation the way we would for any other collection of n-dimensional unit vectors and be sure that we a) always get a valid rotation b) always follow the shortest path from one rotation to another and c) the interpolation will be smooth.