CGA Representation of an Ellipse

Question

In conformal geometric algebra (CGA), I know the representation of a circle $s$ with center $c$ is:

$$ S = H(c) - \frac{1}{2}r^2n_{\infty} $$

That $H(c)$ is the representation of $c$ (i.e., $n_0 + c +‌ \frac{1}{2}c^2n_\infty$). Notice that $n_\infty$ is also denoted by $e_\infty$. Now my question is what is the representation of an ellipse with centers of $c_1$ and $c_2$ under these notations in CGA?

Answer 1

CGA has no way to represent a non-circular ellipse.

CGA represents objects by their intersection with the paraboloid centered on the $n_\infty$ axis. Below is a diagram of 2D CGA I've drawn before to illustrate this.

Here, the thick blue point-pair and red circle are two 2D objects represented by CGA, while their 3D (ignoring $n_0$, which is just used to offset from the origin) representations are respectively the red plane and blue line.

The 3D subspaces are intersected with the white paraboloid and projected to the XY plane to obtain their 2D interpretation.

It should be straightforward to see that this system doesn't have enough degrees of freedom to represent arbitrary ellipses.

Answer 2

You might be interested by Double Conformal Geometric Algebra, Cl(8,2) https://link.springer.com/article/10.1007/s00006-017-0784-0 "DCGA is an extension of CGA and has entities representing points and general (quartic) Darboux cyclide surfaces in Euclidean 3D space, including circular tori and all quadrics"