Lemniscate of Bernoulli

The lemniscate (from the Latin lemniscus, loop ' ) is generally a loop-shaped geometric curve. In particular, it is the symbol of infinity of the form of a figure eight. Unicode: U 221 E ( ∞).

Lemniscate of Bernoulli

The lemniscate of Bernoulli, Swiss mathematician named after Jakob Bernoulli, is a plane curve with the shape of a figure eight. It is an algebraic curve of the fourth order and special case of a Cassini curve.

Definition

The lemniscate of Bernoulli is defined by the following geometric property:

Equations of the lemniscate of Bernoulli

It is assumed for simplicity that the points lie on the x-axis and the center between them just is the origin.

• Equation in Cartesian coordinates:

• Equation in polar coordinates:

• Parameter equation:

The parameter is the distance between the coordinate origin and the two defining points F1 and F2. The route from F1 to F2 so has the length.

Properties

The lemniscate of Bernoulli has the following properties:

• She is axisymmetric for connecting straight line from and.
• It is axially symmetric to the center between the vertical and
• It is point-symmetrical to the middle point between and
• On the connecting straight line from and are from all points of the lemniscate only the middle point between and and this distant two curve points in the distance.
• The mid- point between and is a double point of the curve, so he'll run twice. The two tangents to the straight connecting line of cut it and at an angle of 45 °.
• The lemniscate is the geometric curve of the circle inverted equilateral hyperbola.

Surface

• The two enclosed by the lemniscate sub-areas have the surface area of ​​each.

Arc length

The total arc length of the lemniscate is linear in a and, by using the 1750 studied by Giulio Carlo Fagnano dei Toschi elliptic integral

Be explicitly specified as

Or, with the use of imported in 1798 by Carl Friedrich Gauss lemniscate constant

As

Which is about 5,244 a.

Curvature

The curvature of the lemniscate can be in polar coordinates as a state, so it is always proportional to their distance. In the above parametric representation of this curve but through different! Here is to all generations. If it is given even in implicit Cartesian form, can not say anything about the sign of the curvature - because no free flow sense is given - and thus only their absolute amount is reasonably estimable. If one requires the most natural passing - differential geometry as smooth as possible, analytically ie existence of the highest possible derivatives with respect to arc length along the curve path - the two loops of the curve will pass through each other way around and the sign of the curvature of the lemniscate therefore changes during the passage of the curve through the zero.

Occurrence

The lemniscate curve occurs as a movement in Watt's parallelogram or Watt's linkage to and in Lemniskatenanlenkung of a railway wheel set.

Other lemniscates

• The lemniscate of Booth (Andrew D. Booth )
• The lemniscate of Gerono ( Camille -Christophe Gerono )

Symbolism in Freemasonry

Freemasonry knows the lemniscate as a symbol for the worldwide chain of brotherhood. The loop is formed with the twelve- knot cord or the bond of union ( love rope). They can be found for example in the so-called working carpets of the continental European locust lodges. ( See also: Eight, Endacht ).