Is it possible to express the parametric equation of an ellipse with geometric algebra?

Question

The following $\mathbb{R}^2$ curve $$ \mathbf{r}(t) = \begin{bmatrix} \alpha & -\mu \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \cos t \\ \sin t \end{bmatrix}, \,t \in[0,2 \pi] $$ describes an ellipse (the parameters $\alpha > 0$ and $\mu \in \mathbb{R}$ control its shape). I am wondering if such an equation can be expressed with geometric algebra... Is it possible ? I tried starting from the definition of the exponential $$e^{I \, t} = \cos t + I \sin t, \; I = e_1 e_2$$ and then use the remapping \begin{align} e_1 \mapsto e_1' &= \alpha e_1 - \mu e_2 \\ e_2 \mapsto e_2' &= e_2 \end{align} so that \begin{align} e^{I t} &= \cos t + (\alpha e_1 - \mu e_2)e_2 \sin t \\ &= \cos t - \mu \sin t + e_1 e_2 \alpha \sin t \end{align} but then I would expect the scalar to match the x-coordinate of the curve and the bivector to match the y-coordinate of the curve. This is not what happens: \begin{align} \langle e^{I t} \rangle \equiv \cos t - \mu \sin t &\neq \alpha \cos t - \mu \sin t \equiv x_r \\ \langle e^{I t} \rangle_2 \equiv \alpha \sin t &\neq \sin t \equiv y_r \end{align} I've just started learning geometric algebra, so maybe what I'm trying to do doesn't make sense... Any help would be appreciated :)

Answer 1

Part of [1] ch. 3, problem (8.6), is to show that an ellipse can be parameterized by

$$\mathbf{r}(t) = \mathbf{c} \cosh( \mu + i t ).$$

Here $ i $ is a unit bivector, and $ i \wedge \mathbf{c} $ is zero (i.e. $ \mathbf{c} $ must be in the plane of the bivector $ i $). Note that $ \mu, t, i $ all commute since $ \mu, t $ are both scalars. That allows a complex-like expansion of the hyperbolic cosine to be used

$$\begin{aligned}\cosh( \mu + i t )&=\frac{1}{2} \left( { e^{\mu + i t} + e^{-\mu -i t} } \right) \\ &=\frac{1}{2} \left( { e^{\mu} (\cos t + i \sin t) + e^{-\mu} (\cos t -i \sin t) } \right) \\ &= \cosh \mu \cos t + i \sinh \mu \sin t.\end{aligned}$$

Since an ellipse can be parameterized as

$$\mathbf{r}(t) = \mathbf{a} \cos t + \mathbf{b} \sin t,$$

where the vector directions $ \mathbf{a} $ and $ \mathbf{b} $ are perpendicular, the multivector hyperbolic cosine representation parameterizes the ellipse provided

$$\begin{aligned}\mathbf{a} &= \mathbf{c} \cosh \mu \\ \mathbf{b} &= \mathbf{c} i \sinh \mu.\end{aligned}$$

It is desirable to relate the parameters $ \mu, i $ to the vectors $ \mathbf{a}, \mathbf{b} $. Because $ \mathbf{c} \wedge i = 0 $, the vector $ \mathbf{c} $ anticommutes with $ i $, and therefore $ (\mathbf{c} i)^2 = -\mathbf{c} i i \mathbf{c} = \mathbf{c}^2 $, which means

$$\begin{aligned}\mathbf{a}^2 &= \mathbf{c}^2 \cosh^2 \mu \\ \mathbf{b}^2 &= \mathbf{c}^2 \sinh^2 \mu,\end{aligned}$$

or $$\mu = \tanh^{-1} \frac{\left\lvert {\mathbf{b}} \right\rvert}{\left\lvert {\mathbf{a}} \right\rvert}.$$

The bivector $ i $ is just the unit bivector for the plane containing $ \mathbf{a} $ and $ \mathbf{b} $

$$\begin{aligned}\mathbf{a} \wedge \mathbf{b} &= \cosh \mu \sinh \mu \mathbf{c} \wedge (\mathbf{c} i) \\ &= \cosh \mu \sinh \mu {\left\langle{{ \mathbf{c} \mathbf{c} i }}\right\rangle}_{2} \\ &= \cosh \mu \sinh \mu i \mathbf{c}^2 \\ &= \cosh \mu \sinh \mu i \frac{ \mathbf{a}^2 }{ \cosh^2 \mu } \\ &= \mathbf{a}^2 \tanh \mu i \\ &= \mathbf{a}^2 i \frac{\left\lvert {\mathbf{b}} \right\rvert}{\left\lvert {\mathbf{a}} \right\rvert},\end{aligned}$$

so $$i = \frac{ \mathbf{a} \wedge \mathbf{b} }{\left\lvert {\mathbf{a}} \right\rvert\left\lvert {\mathbf{b}} \right\rvert}.$$

Observe that $ i $ is a unit bivector provided the vectors $ \mathbf{a}, \mathbf{b} $ are perpendicular, as required

$$\begin{aligned}(\mathbf{a} \wedge \mathbf{b})^2&= (\mathbf{a} \wedge \mathbf{b}) \cdot (\mathbf{a} \wedge \mathbf{b}) \\ &= ( (\mathbf{a} \wedge \mathbf{b}) \cdot \mathbf{a} ) \cdot \mathbf{b} \\ &= ( \mathbf{a} (\mathbf{b} \cdot \mathbf{a}) - \mathbf{b} \mathbf{a}^2 ) \cdot \mathbf{b} \\ &= (\mathbf{a} \cdot \mathbf{b})^2 - \mathbf{b}^2 \mathbf{a}^2 \\ &= - \mathbf{b}^2 \mathbf{a}^2.\end{aligned}$$

References

[1] D. Hestenes. New Foundations for Classical Mechanics. Kluwer Academic Publishers, 1999.