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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the symbol. Its name is from , which is
Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
for "decorated with hanging ribbons". It is a special case of the
Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the d ...
and is a rational
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of degree 4. This
lemniscate 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 alternativel ...
was first described in 1694 by
Jakob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Lei ...
as a modification of an
ellipse In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse i ...
, which is the
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...
of points for which the sum of the
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
s to each of two fixed ''focal points'' is a constant. A
Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the d ...
, by contrast, is the locus of points for which the ''product'' of these distances is constant. In the case where the curve passes through the point midway between the foci, the oval is a lemniscate of Bernoulli. This curve can be obtained as the inverse transform of a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
, with the inversion
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
centered at the center of the hyperbola (bisector of its two foci). It may also be drawn by a mechanical linkage in the form of
Watt's linkage In kinematics, Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel on a nearly straight line. It was described in ...
, with the lengths of the three bars of the linkage and the distance between its endpoints chosen to form a crossed parallelogram.


Equations

The equations can be stated in terms of the focal distance or the half-width of a lemniscate. These parameters are related as . * Its Cartesian
equation In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
is (up to translation and rotation): *:\begin \left(x^2 + y^2\right)^2 &= a^2 \left(x^2 - y^2\right) \\ &= 2 c^2 \left(x^2 - y^2\right) \end * As a
parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ob ...
: *:x = \frac; \qquad y = \frac * A rational parametrization: *:x = a \frac; \qquad y = a\frac * In
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
: *:r^2 = a^2 \cos 2\theta * Its equation in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
is: *:, z-c, , z+c, =c^2 *In
two-center bipolar coordinates In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers c_1 and c_2. This system is very useful in some scientific applications (e.g. calculating the electric field ...
: *:rr' = c^2 *In rational polar coordinates: *:Q = 2s-1


Arc length and elliptic functions

The determination of the
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
of arcs of the lemniscate leads to
elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
s, as was discovered in the eighteenth century. Around 1800, the
elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
s inverting those integrals were studied by
C. F. Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
(largely unpublished at the time, but allusions in the notes to his ''
Disquisitiones Arithmeticae The (Latin for "Arithmetical Investigations") is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24. It is notable for having had a revolutionary impact on th ...
''). The period lattices are of a very special form, being proportional to the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s. For this reason the case of elliptic functions with complex multiplication by is called the '' lemniscatic case'' in some sources. Using the elliptic integral :\operatornamex \stackrel \int_0^x\frac the formula of the arc length can be given as :\begin L &= 4\sqrt\,c\int_^1\frac = 4\sqrt\,c\,\operatorname1 \\ pt&= \frac\,c =\fracc\approx 7416 \cdot c \end where \Gamma is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except t ...
and \operatorname is the arithmetic–geometric mean.


Angles

Given two distinct points \rm A and \rm B, let \rm M be the midpoint of \rm AB. Then the lemniscate of
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
\rm AB can also be defined as the set of points \rm A, \rm B, \rm M, together with the locus of the points \rm P such that , \widehat-\widehat, is a right angle (cf.
Thales' theorem In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved ...
and its converse). The following theorem about angles occurring in the lemniscate is due to German mathematician Gerhard Christoph Hermann Vechtmann, who described it 1843 in his dissertation on lemniscates.Alexander Ostermann, Gerhard Wanner: ''Geometry by Its History.'' Springer, 2012, pp
207-208
/ref> : and are the foci of the lemniscate, is the midpoint of the line segment and is any point on the lemniscate outside the line connecting and . The normal of the lemniscate in intersects the line connecting and in . Now the interior angle of the triangle at is one third of the triangle's exterior angle at (see also
angle trisection Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
). In addition the interior angle at is twice the interior angle at .


Further properties

*The lemniscate is symmetric to the line connecting its foci and and as well to the perpendicular bisector of the line segment . *The lemniscate is symmetric to the midpoint of the line segment . *The area enclosed by the lemniscate is . *The lemniscate is the
circle inversion A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
of a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
and vice versa. *The two tangents at the midpoint are perpendicular, and each of them forms an angle of with the line connecting and . *The planar cross-section of a standard
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
tangent to its inner equator is a lemniscate.


Applications

Dynamics on this curve and its more generalized versions are studied in quasi-one-dimensional models.


See also

*
Lemniscate of Booth In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. ...
*
Lemniscate of Gerono In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an \infty symbol, or figure eight. It has equation :x ...
* Gauss's constant *
Lemniscatic elliptic function In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among ot ...
*
Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points (foci) is constant. This may be contrasted with an ellipse, for which the ''sum'' of the d ...


Notes


References

*


External links

* {{MathWorld, title=Lemniscate, urlname=Lemniscate
"Lemniscate of Bernoulli" at The MacTutor History of Mathematics archive

"Lemniscate of Bernoulli"
at MathCurve.

(in French) Plane curves Algebraic curves Spiric sections