Regression to an Ellipse

As for the circle, both algebraic and geometric minimization can be performed.

The quadratic equation of an ellipse is

$$f(\mathbf{x}) = \mathbf{x}^\top \mathbf{A} \mathbf{x} + \mathbf{b}^{\top} \mathbf{x} + c = 0$$ (4.113)

where $\mathbf{A}$ is a symmetric, positive-definite matrix. Here too, solving the homogeneous problem (4.113) makes it possible to determine the system's six unknowns (defined up to a multiplicative factor).

The nonlinear solution that minimizes the geometric quantity can be obtained using the parametric representation of the ellipse:

$$\mathbf{x} = \begin{bmatrix} x_0 \\ y_0 \end{bmatrix} ... ...{bmatrix} a \cos \varphi \\ b \sin \varphi \end{bmatrix}$$ (4.114)

where $(x_0,y_0)$ represents the center of the ellipse, $(a,b)$ the lengths of its two semiaxes, and $\alpha$ the rotation of the ellipse about its center. As for the circle, the $\varphi_i$ are auxiliary variables, and the nonlinear problem has $5+n$ unknowns and $2n$ equations.