Definition:Lemniscate of Bernoulli/Cartesian Definition/Also defined as

Definition

Some sources define the Cartesian equation for the lemniscate of Bernoulli as:

$\paren {x^2 + y^2}^2 = a^2 \paren {x^2 - y^2}$

which is the same but for a scale factor:

Its associated parametric equation is:

$\begin {cases} x = \dfrac {a \cos t} {\sin^2 t + 1} \\ \\ y = \dfrac {a \cos t \sin t} {\sin^2 t + 1} \end{cases}$

Also see

• Equivalence of Definitions of Lemniscate of Bernoulli

Source of Name

This entry was named for Jacob Bernoulli.

Historical Note

The lemniscate of Bernoulli was investigated in some depth by Jacob Bernoulli, from whom it was given its name.

Linguistic Note

The word lemniscate comes from the Latin word lemniscus, which means pendant ribbon.

The word may ultimately derive from the Latin lēmniscātus, which means decorated with ribbons.

This may in turn come from the ancient Greek island of Lemnos where ribbons were worn as decorations.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Lemniscate: $11.2$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): lemniscate or lemniscate of Bernoulli
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): lemniscate
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): lemniscate
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 9$: Special Plane Curves: Lemniscate: $9.2.$