Definition:Lemniscate of Bernoulli/Geometric Definition/Mistake

Source Work

1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables:

Thème et variations

$1,31102 87771 46059 90523 \ldots$

Mistake

La lemniscate est l'ensemble des points $M$ du plan tels que le produit des distances à deux points fixes $P_1$, $P_2$ est $P_1 M \cdot P_2 M = C^{te} = \sqrt {P_1 P_2}$.

In English:

The lemniscate is the set of points $M$ of the plane such that the product of the distances from two fixed points $P_1$, $P_2$ is $P_1 M \cdot P_2 M = C = \sqrt {P_1 P_2}$.

Correction

The constant is incorrect.

The expression ought to read:

$P_1 M \cdot P_2 M = C = \paren {\dfrac {P_1 P_2} 2}^2$

That is,(when the half-distance between $P_1$ and $P_2$ is equal to the square root of the constant.

Sources

• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,31102 87771 46059 90523 \ldots$