Definition:Ovals of Cassini/Shape

Definition

Let $P_1$ and $P_2$ be points in the plane such that $P_1 P_2 = 2 a$ for some constant $a$.

Let the ovals of Cassini be defined as the loci of points $M$ in the plane such that:

$P_1 M \times P_2 M = b^2$

for real constant $b$.

When $b > a$, $M$ is in one continuous piece, either oval or bone-shaped.

When $b < a$, $M$ is in two separate pieces, each surrounding one of the foci of $M$.

When $b = a$, $M$ degenerates into the lemniscate of Bernoulli.

Also see

• Lemniscate of Bernoulli is Special Case of Ovals of Cassini

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 11$: Special Plane Curves: Ovals of Cassini: $11.31$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cassini's ovals
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Cassini's ovals

• Weisstein, Eric W. "Cassini Ovals." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CassiniOvals.html