# egg curve estimation

Let $p_1,...,p_3$ be three points on an ellipse, and $t_1,...,t_3$ be their tangent lines. For $i={1..2}$, let $M_i$ be the point of intersection of $t_i$ and $t_{(i+1)\%2}$, and $K_i$ be the midpoint of $p_i$ and $p_{(i+1)\%2}$. In the case of non-parallel tangent lines, the ellipse center is given by the point of intersection of the two lines $\overline{M_1 K_1}$ and $\overline{M_2 K_2}$. If the ellipse equation is modified slightly, such that it takes the form of an egg curve defined by $\frac{x^2}{a} + \frac{y^2}{b}t(x) = 1$, where $t(x)= 1 + (2dx+d²)/a²$, What strategies would let me efficiently estimate the "egg" center (origin) from points on the curve $p_1,...,p_n$ and their tangents $t_1,...,t_n$?

Brief context: I'm performing image based ellipse detection inspired by the RANSAC based approach outlined in this paper, but I need to adapt the approach to more almond like shapes. As described in the paper, the above approach is used to center the ellipse before estimating the remaining parameters $a$, $b$, and $\theta$ -- since the shapes are not axis aligned. I'm assuming the third order curve I refer to will be easier to work with than some of the more egg like 4th order curves shown in the previous link, but any would probably get the job done.