2D Rotations in Space

Rather than representing an arbitrary vector as a combination of two basis vectors: \(\mathbf{a} = a_x\mathbf{x} + a_y\mathbf{y}\), in this section we will see that we can simplify the algebra by representing an arbitrary vector as a scale and rotation of a single basis vector: \(\mathbf{a} = r_a\mathbf{x}e^{\mathbf{xy}\theta_a}\)

Summary so far

A basis for Geometric Algebra in two dimensions

In the introduction to Geometric Algebra we saw that we can define a 2D space using the basis vectors \(\mathbf{x}\) and \(\mathbf{y}\), where each basis vector's magnitude-squared is 1:

\[ \mathbf{x}^2 = \mathbf{y}^2 = 1 \]

and where each basis vector is orthogonal to the other, which we represented by their anti-commutativity:

\[ \mathbf{x}\mathbf{y} = -\mathbf{y}\mathbf{x} \]

so that any 2D vector constructed as a combination of these basis vectors:

\[ \mathbf{a} = x_a\mathbf{x} + y_a\mathbf{y} \]

still satisfies the requirement that the magnitude-squared of the vector is simply the square of the vector: \(|\mathbf{a}|^2 = \mathbf{a}^2\).

Using Euler's formula to rotate vectors in two dimensions

After some background on Euler's formula itself, we then investigated combining Euler's formula with Geometric Algebra, to see that

\[ e^{\mathbf{xy}\theta} = \cos\theta + \mathbf{xy}\sin\theta \]

defines a rotation in two-dimensional space, which can be applied to an arbitrary vector to rotate it by \(\theta\) in the \(\mathbf{xy}\) plane:

\[ \mathbf{a}e^{\mathbf{xy}\theta} \]

Representing vectors as scaled rotations of a basis

This rotation provided by Euler's formula also allows us another way to represent arbitrary vectors. So far we have been representing an arbitrary vector \(\mathbf{a}\) as a combination of the two basis vectors for the two-dimensional space:

\[ \mathbf{a} = x_a\mathbf{x} + y_a\mathbf{y} \]

But we can now additionally represent an arbitrary vector as a magnitude and orientation applied to a basis vector:

\[ \mathbf{a} = r_a \mathbf{x}e^{\mathbf{xy}\theta_a} = r_a e^{-\mathbf{xy}\theta_a}\mathbf{x} \]

where \(r_a\) is the magnitude and \(\theta_a\) the orientation of \(\mathbf{a}\).

Properties

Using the properties of spatial rotations which we noted earlier, this allows more intuitive results, such as the magnitude-squared of a vector:

\[ \begin{aligned} \mathbf{a}^2 &= (r_a e^{-\mathbf{xy}\theta_a}\mathbf{x})(r_a e^{-\mathbf{x}\mathbf{y}\theta_a}\mathbf{x}) \\ &= r_a^2 e^{-\mathbf{xy}\theta_a} e^{\mathbf{xy}\theta_a}\mathbf{xx} \\ &= r_a^2 \end{aligned} \]

or the product of two vectors:

\[ \begin{aligned} \mathbf{a}\mathbf{b} &= r_a r_b \mathbf{x}e^{\mathbf{x}\mathbf{y}\theta_a} \mathbf{x}e^{\mathbf{x}\mathbf{y}\theta_b} \\ &= r_a r_b \mathbf{x}^2 e^{-\mathbf{x}\mathbf{y}\theta_a} e^{\mathbf{x}\mathbf{y}\theta_b} \\ &= r_a r_b e^{\mathbf{x}\mathbf{y}(\theta_b -\theta_a)} \end{aligned} \]

This product of two vectors is now more intuitively recognised, without expanding, as a scale (\(r_a r_b\)) and rotation in the \(\mathbf{x}\mathbf{y}\) plane by \(\theta_b - \theta_a\) which we will apply in the next section.

This also makes other properties more intuitive, such as

\[ \begin{aligned} \mathbf{ab} + \mathbf{ba} &= r_a r_b e^{\mathbf{x}\mathbf{y}(\theta_b -\theta_a)} + r_a r_b e^{\mathbf{x}\mathbf{y}(\theta_a -\theta_b)} \\ &= r_a r_b (e^{\mathbf{x}\mathbf{y}(\theta_b -\theta_a)} + e^{-\mathbf{x}\mathbf{y}(\theta_b -\theta_a)}) \\ &= 2 r_a r_b \cos (\theta_b - \theta_a) \end{aligned} \]

and

\[ \begin{aligned} \mathbf{ab} - \mathbf{ba} &= r_a r_b e^{\mathbf{x}\mathbf{y}(\theta_b -\theta_a)} - r_a r_b e^{\mathbf{x}\mathbf{y}(\theta_a -\theta_b)} \\ &= r_a r_b (e^{\mathbf{x}\mathbf{y}(\theta_b -\theta_a)} - e^{-\mathbf{x}\mathbf{y}(\theta_b -\theta_a)}) \\ &= 2 r_a r_b \sin (\theta_b - \theta_a) \end{aligned} \]

which in turn allows a very simple derivation of the cosine rule for the magnitude of the third side of an arbitrary triangle given two sides:

\[ \begin{aligned} (\mathbf{a} + \mathbf{b})^2 &= \mathbf{a}^2 + \mathbf{ab} + \mathbf{ba} + \mathbf{b}^2 \\ &= r_a^2 + r_b^2 + 2r_a r_b \cos(\theta_b - \theta_a) \\ &= r_a^2 + r_b^2 - 2r_a r_b \cos(\pi -(\theta_b - \theta_a)) \end{aligned} \]

Rotating a vector in two dimensions

With this definition of arbitrary vectors, it is much easier to see that the product of two vectors defines a scale and rotation, which can be applied to a third vector:

\[ \begin{aligned} \mathbf{abc} &= r_a\mathbf{x}e^{\mathbf{xy}\theta_a}r_b\mathbf{x}e^{\mathbf{xy}\theta_b}\mathbf{c} \\ &= r_a r_b e^{-\mathbf{xy}\theta_a}e^{\mathbf{xy}\theta_b}\mathbf{c} \\ &= r_a r_b e^{\mathbf{xy}(\theta_b - \theta_a)}\mathbf{c} \end{aligned} \]

We can see this is in fact a scale and rotation if we also expand \(\mathbf{c}\)

\[ \begin{aligned} \mathbf{abc} &= r_a r_b e^{\mathbf{xy}(\theta_b - \theta_a)}r_c\mathbf{x}e^{\mathbf{xy}\theta_c} \\ &= r_a r_b r_c \mathbf{x}e^{\mathbf{xy}(\theta_c - (\theta_b - \theta_a))} \end{aligned} \]

showing that \(\mathbf{c}\) is scaled by a factor of \(r_a r_b\) and rotated by an amount \(\theta_b - \theta_a\).

This can also be viewed as the vector \(\mathbf{a}\) being scaled by an amount \(r_b r_c\) and rotated by an amount \(\theta_b - \theta_c\), as we can rearrange the above to the equivalent:

\[ \mathbf{abc} = r_a r_b r_c \mathbf{x}e^{\mathbf{xy}(\theta_a - (\theta_b - \theta_c))} \]

So we can represent a vector in two dimensional space. We can add, scale and rotate such vectors. But is there a way to inherently represent a vector in space-time?