Focus (geometry) - Confocal Curves

Confocal Curves

Let P₁, P₂, …, P_m be given as foci of a curve C of class m. Let P be the product of the tangential equations of these points and Q the product of the tangential equations of the circular points at infinity. Then all the lines which are common tangents to both P=0 and Q=0 are tangent to C. So, by the AF+BG theorem, the tangential equation of C has the form HP+KQ=0. Since C has class m, H must be a constant and K but have degree less than or equal to m−2. The case H=0 can be eliminated as degenerate, so the tangential equation of C can be written as P+fQ=0 where f is an arbitrary polynomial of degree m−2.

For example, let P₁=(1,0), P₂=(−1,0). The tangential equations are X+1=0 and X−1=0 so P= X2-1=0. The tangential equations for the circular points at infinity are X+iY=0 and X−iY=0 so Q=X2+Y2. Therefore the tangential equation for a conic with the given foci is X2-1+c(X2+Y2)=0, or (1+c)X2+cY2=1 where c is an arbitrary constant. In point coordinates this becomes