Lemniscata

lemniscata

What are the properties of a lemniscate?

Further properties. The lemniscate is symmetric to the line connecting its foci F 1 and F 2 and as well to the perpendicular bisector of the line segment F 1F 2. The lemniscate is symmetric to the midpoint of the line segment F 1F 2. The area enclosed by the lemniscate is 2a 2.

What is the lemniscate of a circle?

The lemniscate may be defined as an algebraic curve, the zero set of the quartic polynomial when the parameter d is negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values of d one instead obtains the oval of Booth.

What is the origin of the word lemniscate?

Lemniscate. The study of lemniscates (and in particular the hippopede) dates to ancient Greek mathematics, but the term lemniscate for curves of this type comes from the work of Jacob Bernoulli in the late 17th century.

What is the lemniscate of a torus?

The lemniscate is the circle inversion of a hyperbola and vice versa. 4 with the line connecting F1 and F2. The planar cross-section of a standard torus tangent to its inner equator is a lemniscate. Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.

What is the area enclosed by the lemniscate?

The lemniscate is symmetric to the line connecting its foci F1 and F2 and as well to the perpendicular bisector of the line segment F1F2. The lemniscate is symmetric to the midpoint of the line segment F1F2. The area enclosed by the lemniscate is a2 = 2c2. The lemniscate is the circle inversion of a hyperbola and vice versa.

What are the different types of lemniscates?

Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono.

Is the lemniscate symmetric to the line connecting its foci?

The lemniscate is symmetric to the line connecting its foci F1 and F2 and as well to the perpendicular bisector of the line segment F1F2.

What is lemniscus in physics?

A polar curve also called Lemniscate of Bernoulli which is the Locus of points the product of whose distances from two points (called the Foci) is a constant. Letting the Foci be located at , the Cartesian equation is Jakob Bernoulli published an article in Acta Eruditorum in 1694 in which he called this curve the lemniscus (``a pendant ribbon).

What does lemniscate stand for?

In algebraic geometry, a lemniscate is any of several figure-eight or ∞ -shaped curves. The word comes from the Latin lēmniscātus meaning decorated with ribbons, from the Greek λημνίσκος meaning ribbons, or which alternatively may refer to the wool from which the ribbons were made.

What are the different types of lemniscates?

Curves that have been called a lemniscate include three quartic plane curves: the hippopede or lemniscate of Booth, the lemniscate of Bernoulli, and the lemniscate of Gerono.

What is the equation for a lemniscate?

Definition of lemniscate. lemniscate in British English. a closed plane curve consisting of two symmetrical loops meeting at a node. Equation: (x2 + y2)2 = a2(x2 – y2), where a is the greatest distance from the curve to the origin.

What is Bernoullis lemniscate?

Also called Bernoullis lemniscate. lemniscate. a closed plane curve consisting of two symmetrical loops meeting at a node. Equation: (x ² + y ²)² = a ²(x ² – y ²), where a is the greatest distance from the curve to the origin.

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