Question

tracking of a central conic formula

Answer 1

If we are given a curve in the form

ax2+2bxy+cy2+2dx+2ey+f=0ax2+2bxy+cy2+2dx+2ey+f=0

and the following determinant

δ=∣∣∣abbc∣∣∣=ac−b2δ=|abbc|=ac−b2

is non-zero, then this is either a curve of elliptic type (ellipse, point or empty set) or a curve of hyperbolic type (a hyperbola or a pair of lines). So in these cases in makes sense to talk about center of the curve, i.e., the point with respect to which this curve is symmetric.

The center can be found as the solution of the following system of equations

ax+by+dbx+cy+e=0,=0.ax+by+d=0,bx+cy+e=0.

(This system has a unique solution, since the determinant of the matrix of this system is δ≠0δ≠0.)

That means, the coordinates of the center can be computed as

xy=be−cdδ,=bd−aeδ.x=be−cdδ,y=bd−aeδ.

Several resources can be found where this way of finding center is described. (Often in the form of solving ∂F∂x=∂F∂y=0∂F∂x=∂F∂y=0, where F(x,y)=ax2+2bxy+cy2+2dx+2ey