HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the witch of Agnesi () is a
cubic plane curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...
defined from two diametrically opposite points of a circle. It gets its name from Italian mathematician
Maria Gaetana Agnesi Maria Gaetana Agnesi ( , , ; 16 May 1718 – 9 January 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. She was the first woman to write a mathematics handbook and the first woman appointed as a mathematics profe ...
, and from a mistranslation of an Italian word for a sailing sheet. Before Agnesi, the same curve was studied by
Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is ...
, Grandi, and Newton. The
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of the derivative of the
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
function forms an example of the witch of Agnesi. As the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
, the witch of Agnesi has applications in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
. It also gives rise to
Runge's phenomenon In the mathematical field of numerical analysis, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation ...
in the approximation of functions by polynomials, has been used to approximate the energy distribution of
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...
s, and models the shape of hills. The witch is tangent to its defining circle at one of the two defining points, and
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
to the tangent line to the circle at the other point. It has a unique
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
(a point of extreme curvature) at the point of tangency with its defining circle, which is also its
osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...
at that point. It also has two finite
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s and one infinite inflection point. The area between the witch and its asymptotic line is four times the area of the defining circle, and the volume of revolution of the curve around its defining line is twice the volume of the
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 n ...
of revolution of its defining circle.


Construction

To construct this curve, start with any two points ''O'' and ''M'', and draw a circle with ''OM'' as diameter. For any other point ''A'' on the circle, let ''N'' be the point of intersection of the
secant line Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
''OA'' and the tangent line at ''M''. Let ''P'' be the point of intersection of a line perpendicular to ''OM'' through ''A'', and a line parallel to ''OM'' through ''N''. Then ''P'' lies on the witch of Agnesi. The witch consists of all the points ''P'' that can be constructed in this way from the same choice of ''O'' and ''M''. It includes, as a limiting case, the point ''M'' itself.


Equations

Suppose that point ''O'' is at the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
and point ''M'' lies on the positive y-axis, and that the circle with diameter ''OM'' has Then the witch constructed from ''O'' has the Cartesian equation y = \frac. This equation can be simplified, by choosing to the form y = \frac. or equivalently, by clearing denominators, as the cubic
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
(x^2+1)y=1. In its simplified form, this curve is the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
of the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spe ...
function. The witch of Agnesi can also be described by
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 obj ...
s whose parameter is the angle between ''OM'' and ''OA'', measured clockwise: \begin x &= 2a \tan \theta, \\ y &= 2a \cos ^2 \theta. \end


Properties

The main properties of this curve can be derived from
integral calculus In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
. The area between the witch and its asymptotic line is four times the area of the fixed circle, The volume of revolution of the witch of Agnesi about its asymptote This is two times the volume of the
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 n ...
formed by revolving the defining circle of the witch around the same line. The curve has a unique
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
at the point of tangency with its defining circle. That is, this point is the only point where the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
reaches a local minimum or local maximum. The defining circle of the witch is also its
osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...
at the vertex, the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature. Because this is an osculating circle at the vertex of the curve, it has third-order contact with the curve. The curve has two
inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...
s, at the points \left( \pm\frac, \frac\right) corresponding to the When considered as a curve in the
projective plane In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that d ...
there is also a third infinite inflection point, at the point where the
line at infinity In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The ...
is crossed by the asymptotic line. Because one of its inflection points is infinite, the witch has the minimum possible number of finite real inflection points of any non-singular cubic The largest area of a
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram contain ...
that can be inscribed between the witch and its asymptote for a rectangle whose height is the radius of the defining circle and whose width is twice the diameter of the


History


Early studies

The curve was studied by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
in his 1659 treatise on quadrature. In it, Fermat computes the area under the curve and (without details) claims that the same method extends as well to the
cissoid of Diocles In geometry, the cissoid of Diocles (; named for Diocles) is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be de ...
. Fermat writes that the curve was suggested to him "''ab erudito geometra''" y a learned geometer speculate that the geometer who suggested this curve to Fermat might have been Antoine de Laloubère. The construction given above for this curve was found by ; the same construction was also found earlier by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
, but only published posthumously later, in 1779. also suggested the name ''versiera'' (in Italian) or ''versoria'' (in Latin) for the curve.In his notes to Galileo's "Trattato del moto naturalmente accelerato," Grandi had referred to "quella curva che io descrivo nel mio libro delle quadrature
703 __NOTOC__ Year 703 ( DCCIII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar, the 703rd year of the Common Era (CE) and Anno Domini (AD) designations, the 703rd year of the 1st millennium, the 3 ...
alla prop. IV, nata da' seni versi, che da me suole chiamarsi ''Versiera'', in latino però ''Versoria''." See Galilei, ''Opere'', 3: 393. One finds the new term in Lorenzo Lorenzini, ''Exercitatio geometrica'', xxxi: "sit pro exemplo curva illa, quam Doctissimus magnusque geometra Guido Grandus versoria nominat."
The Latin term is also used for a sheet, the rope which turns the sail, but Grandi may have instead intended merely to refer to the
versine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',Maria Gaetana Agnesi Maria Gaetana Agnesi ( , , ; 16 May 1718 – 9 January 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. She was the first woman to write a mathematics handbook and the first woman appointed as a mathematics profe ...
published ''Instituzioni analitiche ad uso della gioventù italiana'', an early textbook on
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
. In it, after first considering two other curves, she includes a study of this curve. She defines the curve geometrically as the locus of points satisfying a certain proportion, determines its algebraic equation, and finds its vertex, asymptotic line, and inflection points.


Etymology

Maria Gaetana Agnesi Maria Gaetana Agnesi ( , , ; 16 May 1718 – 9 January 1799) was an Italian mathematician, philosopher, theologian, and humanitarian. She was the first woman to write a mathematics handbook and the first woman appointed as a mathematics profe ...
named the curve according to Grandi, ''versiera''. Coincidentally, at that time in Italy it was common to speak of the
Devil A devil is the personification of evil as it is conceived in various cultures and religious traditions. It is seen as the objectification of a hostile and destructive force. Jeffrey Burton Russell states that the different conceptions of ...
through other words like ''aversiero'' or ''versiero'', derived from Latin ''adversarius'', the "adversary" of God. ''Versiera'', in particular, was used to indicate the wife of the devil, or "witch". Because of this, Cambridge professor
John Colson John Colson (1680 – 20 January 1760) was an English clergyman, mathematician, and the Lucasian Professor of Mathematics at Cambridge University. Life John Colson was educated at Lichfield School before becoming an undergraduate at Christ Chu ...
mistranslated the name of the curve as "witch". Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened. Struik mentions that: On the other hand,
Stephen Stigler Stephen Mack Stigler (born August 10, 1941) is Ernest DeWitt Burton Distinguished Service Professor at the Department of Statistics of the University of Chicago. He has authored several books on the history of statistics; he is the son of the e ...
suggests that Grandi himself "may have been indulging in a play on words", a double pun connecting the devil to the versine and the sine function to the shape of the female breast (both of which can be written as "seno" in Italian).


Applications

A scaled version of the curve is the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
. This is the probability distribution on the
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
x determined by the following random experiment: for a fixed point p above the choose uniformly at random a line and let x be the coordinate of the point where this random line crosses the axis. The Cauchy distribution has a peaked distribution visually resembling the
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
, but its heavy tails prevent it from having an
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
by the usual definitions, despite its symmetry. In terms of the witch itself, this means that the of the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any ...
of the region between the curve and its asymptotic line is not well-defined, despite this region's symmetry and finite area. In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
, when approximating functions using
polynomial interpolation In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of data points (x_0,y_0), \ldots, (x_n,y_n), with no ...
with equally spaced interpolation points, it may be the case for some functions that using more points creates worse approximations, so that the interpolation diverges from the function it is trying to approximate rather than converging to it. This paradoxical behavior is called
Runge's phenomenon In the mathematical field of numerical analysis, Runge's phenomenon () is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation ...
. It was first discovered by
Carl David Tolmé Runge Carl David Tolmé Runge (; 30 August 1856 – 3 January 1927) was a German mathematician, physicist, and spectroscopist. He was co-developer and co- eponym of the Runge–Kutta method (German pronunciation: ), in the field of what is today know ...
for Runge's function another scaled version of the witch of Agnesi, when interpolating this function over the The same phenomenon occurs for the witch y=1/(1+x^2) itself over the wider The witch of Agnesi approximates the
spectral energy distribution A spectral energy distribution (SED) is a plot of energy versus frequency or wavelength of light (not to be confused with a 'spectrum' of flux density vs frequency or wavelength). It is used in many branches of astronomy to characterize astron ...
of
spectral lines A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to ident ...
, particularly
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation. Most X-rays have a wavelength ranging from 10 picometers to 10  nanometers, corresponding to frequencies in the range 30&nb ...
lines. The cross-section of a smooth
hill A hill is a landform that extends above the surrounding terrain. It often has a distinct summit. Terminology The distinction between a hill and a mountain is unclear and largely subjective, but a hill is universally considered to be not a ...
has a similar shape to the witch. Curves with this shape have been used as the generic topographic obstacle in a flow in mathematical modeling. Solitary waves in deep water can also take this shape. A version of this curve was used by
Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ...
to derive the Leibniz formula for . This formula, the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
\frac = 1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots, can be derived by equating the area under the curve with the integral of the using the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion of this function as the infinite
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
and integrating term-by-term.


In popular culture

''The Witch of Agnesi'' is the title of a novel by Robert Spiller. It includes a scene in which a teacher gives a version of the history of the term. ''Witch of Agnesi'' is also the title of a music album by jazz quartet Radius. The cover of the album features an image of the construction of the witch.


Notes


External links


"Witch of Agnesi" at MacTutor's Famous Curves Index
*
Witch of Agnesi
by Chris Boucher based on work by
Eric W. Weisstein Eric Wolfgang Weisstein (born March 18, 1969) is an American mathematician and encyclopedist who created and maintains the encyclopedias ''MathWorld'' and ''ScienceWorld''. In addition, he is the author of the '' CRC Concise Encyclopedia of M ...
,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
.
"Witch of Agnesi" at "mathcurve"
* {{good article Algebraic curves