Lemniscates

Plane algebraic curves of order $2n$ such that the product of the distances of each point of the curve from $n$ given points (foci) $F_1,\dotsc,F_n$ is equal to the $n$-th power of a given number $r$ (the radius of the lemniscate). The equation of a lemniscate in rectangular Cartesian coordinates is

$$|(z-z_1)\dotsm(z-z_n)|=r^n,\quad r>0,\quad z=x+iy.$$

A circle is a lemniscate with one focus, and a Cassini oval is a lemniscate with two foci. See also Bernoulli lemniscate and Booth lemniscate.

References

[a1]	F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)

A lemniscate is a level curve of a polynomial. If all the foci $F_k$: $z_k=x_k+iy_k$, $k=1,\dotsc,n$, are distinct and the radius of the lemniscate is sufficiently small, then the lemniscate consists of $n$ continua that have pairwise no common points. For a sufficiently large radius a lemniscate consists of one connected component. As D. Hilbert showed in 1897, the boundary $\Gamma$ of an arbitrary simply-connected finite domain can be arbitrarily closely approximated by a lemniscate, that is, for any $\epsilon>0$ one can find a lemniscate $\Lambda$ such that in the $\epsilon$-neighbourhood of each point of $\Gamma$ there are points of $\Lambda$ and every point of $\Lambda$ is in the $\epsilon$-neighbourhood of an appropriate point of $\Gamma$.

References

[1]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[2]	J.L. Walsh, "Interpolation and approximation by rational functions in the complex domain" , Amer. Math. Soc. (1965)

E.D. Solomentsev