Arbitrary rotations in 3D space

Introduction

In this article we review the known formulaes for rotations along the , and axes and generalize this knowledge to apply it in the context of rotations about am arbitrary axis in 3D space. For this purpose we make use of infinitesimal rotations to derive our final result, the Rodrigues rotation formula.

Rotations about the coordinate axes

From elementary linear algebra we know that the rotations about the coordinate axes are expressed as matrices , and given by:

(1)

It is clear that our generalization should yield this results as a special case. Let's retrict ourselves to the --plane and only consider for now, which we shall derive by a different method.

A circular motion, and thus a rotation, is always described by a tangential velocity vector perpendicular to the position vector at every point along the curve. This can be easily proven as follows. A point in a circumference can be written as

(2)

Thus, the derivative of is given by

(3)

From which our claim is evident. . Here we considered, without loss of generality, only two dimensions for simplicity.

Using this result, we can express the result of rotating a vector about the axis by a infinitesimal angle as follows:

(4)

We use the cross product to construct a vector perpendicular to .

To construct a rotation from infinitesimal rotations we need a method to apply the latter multiple times. For this purpose, we rewrite the cross product with as a matrix . Equation (4) becomes

(5)

Thus, the composition of multiple applications of the infinitesimal rotation is expressed as a product of the transformation matrix

(6)

where . Equation (6) is analogous to the definition of the exponential function. Therefore, we rewrite it as

(7)

The argument is sometimes called the generator of the rotation. Finally, we can use the Cayley-Hamilton theorem to compute the result of the exponential function. From the characteristic polynomial of , we know that

(8)

Thus,

(9)

To generalize this result to a arbitrary axis it suffices to prove that equation (8) holds for any skew symmetric matrix

(10)

where are the components of the axis. A Laplace expansion over the first column of the determinant yields

(11)

Hence, any skew-symmetric matrix fulfilles equation (8) if and only if is a unit vector. However, because only the direction of is relevant, this restriction has little effect.

Finally, we can convert back to the cross product. This yields

(12)

This result is known as the rodriges rotation formula.