lemniscate of Bernoulli

The resulting curve resembles a lemniscate of Bernoulli.

In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli.

For the antiparallelogram formed by the sides and diagonals of a square, it is the lemniscate of Bernoulli.

The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694.

Examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.

Watt's curve is the zero set of the degree-six polynomial equation and has the lemniscate of Bernoulli as a special case.

Cassini ovals (including the lemniscate of Bernoulli), toric sections and limaçons (including the cardioid) are bicircular quartics.

The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate.

Rather, it traces out Watt's curve, a lemniscate or figure eight shaped curve; when the lengths of its bars and its base are chosen to form a crossed square, it traces the lemniscate of Bernoulli.

If the center of inversion is chosen as the hyperbola's own center, the inverse curve is the lemniscate of Bernoulli; the lemniscate is also the envelope of circles centered on a rectangular hyperbola and passing through the origin.

The inverse of a sinusoidal spiral with respect to a circle with center at the origin is another sinusoidal spiral whose value of n is the negative of the original curve's value of n. For example, the inverse of the lemniscate of Bernoulli is a hyperbola.