I would like a formula for the foci of the smallest ellipse passing through a set of 3 given points. For example, say we have points $(x_1,y_1), (x_2,y_2), (x_3,y_3)$ , there will be a unique smallest ellipse (where by smallest I mean smallest area). I would like to know the locations of the foci for this ellipse. Thanks,
Mathew
Best Answer
There is a straightforward way to find this smallest ellipse (that we will call "circumscribed" to the triangle) by considering the issue with complex numbers. For a full explanation, see this nice article (Mathematics Mag. 86 (2013) 136-143.)
Here is how it works.
Let $z_k=x_k+iy_k, k=1,2,3$. Let us denote by $T$ the triangle with vertices $z_k$ in the complex plane. Consider the third degree polynomial
$$f(z)=(z-z_1)(z-z_2)(z-z_3)\tag{1}$$
The 2 roots of quadratic equation
$$f'(z)=0 \ \iff \ 3z^2-2\underbrace{(z_1+z_2+z_3)}_{S}z+\underbrace{(z_1z_2+z_2z_3+z_3z_1)}_{R}=0\tag{2}$$
are shown to be the (complex numbers associated with) the foci of an inscribed ellipse which is known as the Steiner ellipse (ellipse with smallest area inscribed into triangle $T$, with the property that it passes through the 3 midpoints of triangle $T$). From there, it suffices to take an homothety (= enlargment) with ratio $2$ centered into the ellipse's center (which is as well the center of gravity of triangle $T$) to find the ellipse with smallest area that is circumscribed to triangle $T$.
Edit :
The explicit solutions of (2) are :
$$f_1=(S-\sqrt{D})/3, \ \ \ f_2=(S+\sqrt{D})/3 \ \ \text{where} \ D:=S^2-3R$$
(take care : $\pm\sqrt{D}$ is to be understood as any root of complex equation $z^2=D$). $f_1$ and $f_2$ are the foci of the inscribed ellipse in triangle $T$.
It remains to do the enlargment, finally giving the foci of the circumscribed ellipse :
Fig. 1 : Right image : The two Steiner ellipses (inscribed and circumscribed). They have been obtained plainly as the images of the left figure through transformation $z \ \to z':=z+a \bar{z}$ for a certain $a$. Stars indicate the foci of the inscribed ellipse which can be shown to be in this case $F=\pm \tfrac12\sqrt{1+a^2}$.