Lemniscata

lemniscata

What is the meaning of lemniscate?

Definition of lemniscate : a figure-eight shaped curve whose equation in polar coordinates is ρ2=a2 cos 2θ or ρ2=a2 sin 2θ First Known Use of lemniscate circa 1781, in the meaning defined above

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.

How do you find the lemniscate of Bernoulli?

2 ), circle ( n = 1) and lemniscate of Bernoulli ( n = 2 ), where rn = −1n cos nθ in polar coordinates and their equivalents in rectangular coordinates. In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2 c from each other as the locus of points P so that PF1 · PF2 = c2.

What is the difference between a Cassini oval and a lemniscate?

This lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A Cassini oval, by contrast, is the locus of points for which the product of these distances is constant.

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 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 the lemniscate used for?

The lemniscate, reduced in size to that of typographical characters, is commonly used as the symbol for infinity, or for a value that increases without limit. Eric Weissteins World of Mathematics explains the lemniscate more fully.

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