id: 06429556
dt: j
an: 2015e.00652
au: Akopyan, Arseniy V.
ti: The lemniscate of Bernoulli, without formulas.
so: Math. Intell. 36, No. 4, 47-50 (2014).
py: 2014
pu: Springer US, New York, NY
la: EN
cc: G70
ut: polynomial lemniscate; radius of a lemniscate; Cassini oval; lemniscate of
Bernoulli; equilateral hyperbola
ci:
li: doi:10.1007/s00283-014-9445-5
ab: “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 three-bar 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.
rv: Rolf Riesinger (Wien)