Lemniscate of Bernoulli

Conchoids of Lemniscate of Bernoulli, colored with varing levels of gray. Conchoids of Lemniscate

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History

Studied by Jacob Bernoulli.

Description

Lemniscate of Bernoulli is a special case of Cassinian oval. That is, the locus of points P, such that distance[P,F1] * distance[P,F2] == (distance[F1,F2]/2)^2, where F1, F2 are fixed points called foci. It is analogous to the definition of ellipse, where sum of two distances is replace by product.

Lemniscate. Tracing Lemniscate

Formulas

• Parametric: Cos[t]/(1+Sin[t]^2) {1, Sin[t]}, 0 < t ≤ 2 π.
• Polar: r^2 == Cos[2 * θ].
• Cartesian: (x^2 + y^2)^2 == (x^2-y^2).

Foci are at {-1/Sqrt[2],0}, {1/Sqrt[2],0}

Properties

Lemniscate as a Cissoid

Lemniscate of Bernoulli can be generated as a cissoid of two circles.

Step by step description:

1. Let there be a circle with radius r.
2. Let there be a point O, r*Sqrt[2] distant from the center of the circle.
3. Draw a line passing through O and the circle. Let the intersections be Q1, Q2.
4. Let there be a vector with origin at O whose length is distance[Q1,Q2].
5. The locus of the vector is one loop of the lemniscate of Bernoulli. The other loop is symmetric with respect to O.

Lemniscate as a Cissoid

Relation to Rectangular Hyperbola

Its inversion and negative pedal with respect to its center is the rectangular hyperbola. It is also the envelope of circles with centers on a rectangular hyperbola and each circle passing the hyperbola's center.

Slicing a Torus

Lemniscate of Bernoulli is the intersection of a plane tangent to the inner ring of a torus whose inner radius equals to its radius of generating circle. (See Cassinian oval).

Construction of Tangent and Normal

The normal of any point P on the curve makes a angle 2 theta with the radius vector and 3 theta with the polar axis. The tangent of inclination is 2 theta + π/2.

Construction of Normal

Generation by Linkage

Lemniscate of Bernoulli can be generated by these linkages. On the left: AB == ND == OD == c, AO == AN == BD == c/Sqrt[2]. P and Q are midpoints of line OD and ND respectively. The point P traces half a lemniscate and half a cicle. Same with Q. On the right: AB == AC == a, CE == BE == EF == a/Sqrt[2]

Lemniscate Linkage 1	Lemniscate Linkage 2

Trivia

The math symbol for infinity is shaped like a lemniscate. It was first used by John Wallis in 1655 in his De Sectionibus conicis. (See Florian Cajori's “A history of mathematical notations” 1929 amazon)

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive.

Wikipedia: Lemniscate of Bernoulli.