Confocal conics

co-focal curves

Curves of the second order with common foci. If $F$ and $F_1$ are two given points in the plane, then through every point of the plane there are one ellipse and one hyperbola that have $F$ and $F_1$ as their foci (Fig. a).

Every ellipse is orthogonal to every hyperbola confocal with it, that is, they intersect (in four points) at right angles. In a suitable coordinate system, all the confocal ellipses and hyperbolas can be given by the equation

$$\frac{x^2}{\lambda}+\frac{y^2}{\lambda-c^2}=1,\tag{*}$$

where $c$ is the distance of the foci from the coordinate origin and $\lambda$ is a variable parameter. If $\lambda>c^2$, this equation defines an ellipse, and if $0<\lambda<c^2$, it defines a hyperbola (if $\lambda<0$, it is an imaginary curve of the second order). If one of the foci tends to infinity, then in the limit one obtains two families of confocal parabolas (Fig. b).

Figure: c024710a

Figure: c024710b

Any two parabolas from different families are also orthogonal to one another. Confocal ellipses and hyperbolas can be used to introduce a so-called elliptic coordinate system in the plane as follows. If $M(x,y)$ is any point of the plane, then by substituting its coordinates $x$ and $y$ into \ref{*}, one obtains a quadratic equation for $\lambda$; its roots $\lambda_1,\lambda_2$ are the elliptic coordinates of $M$. The confocal ellipses and hyperbolas themselves form the coordinate net of this coordinate system, that is, they are defined by the equations $\lambda_1=\text{const}$ and $\lambda_2=\text{const}$.

References

[a1]	D.J. Struik, "Lectures on analytic and projective geometry" , Addison-Wesley (1953) pp. 157–160