
06429556
j
2015e.00652
Akopyan, Arseniy V.
The lemniscate of Bernoulli, without formulas.
Math. Intell. 36, No. 4, 4750 (2014).
2014
Springer US, New York, NY
EN
G70
polynomial lemniscate
radius of a lemniscate
Cassini oval
lemniscate of Bernoulli
equilateral hyperbola
doi:10.1007/s0028301494455
``A polynomial lemniscate with foci $F_1,F_2,\dots,F_n$ is a locus of points $X$ such that the product of distances from $X$ to the foci is constant ($\prod_{i=1,\dots,n}F_iX=\mathrm{const}$). The $n$th root of this value is called the {\it radius} of the lemniscate. It is clear that a lemniscate is an algebraic curve of degree (at most) $2n$" (from the text). Using purely synthetic arguments, the author presents three constructions of the Bernoulli lemniscate ($n=2$, $\mathrm{const}=(1/4)F_1F_2^2$), one is based on a threebar linkage invented by James Watt. In the same way it is proved that the Bernoulli lemniscate is an inversion image of an equilateral hyperbola. Finally, a very simple construction of the normal of the Bernoulli lemniscate is described.
Rolf Riesinger (Wien)