Versiera of the Agnesi

The Versiera der Agnesi , also Versiera der Maria Agnesi , is a special flat curve , an algebraic curve of the 3rd order, which is generated with the help of constructive methods on the basis of a circle. The curve itself corresponds to the curve of the Cauchy distribution .

The curve was examined by Pierre de Fermat in 1653 and by Guido Grandi in 1703 . It is named after the mathematician Maria Agnesi , who published it in 1748. The Italian name la versiera di Agnesi is based on the Latin versoria ( sheet for sailing ships) and the Sinus versus . This was read by Cambridge professor John Colson as l'avversiera di Agnesi , where avversiera means "woman who is directed against God" and was interpreted as a "witch" ( witch ), which is why the curve in English witch of Agnesi ("witch from Agnesi “) is called.

construction

The Versiera of the Agnesi with named points

Starting with a solid circle, a point O is chosen on the circle. For every other point A on the circle, the secant OA is drawn. The point M is diametrically opposite to O . The line OA intersects the tangent at M to the point N . The line parallel to OM by N , and the line perpendicular to OM by A intersect in P . If point A is changed, the way from P is the versiera of Agnesi.

The curve is asymptotic to the tangent to the circle at the point O .

Equations of the Versiera of Agnesi

An animation showing the construction of the Versiera of the Agnesi

Assume that the Cartesian coordinate system has the origin in O and M lies on the positive y axis; further the diameter of the circle is equal to a . Then the following equations of the Versiera of Agnesi result:

Cartesian coordinates: or ${\ displaystyle (x ^ {2} + a ^ {2}) ya ^ {3} = \, 0}$ ${\ displaystyle y = {\ frac {a ^ {3}} {x ^ {2} + a ^ {2}}}}$
Parametric equation: ${\ displaystyle x = at \ ;, y = {a \ over t ^ {2} +1}}$
Parametric equation with the angle when the angle is between OM and OA (measured clockwise): ${\ displaystyle \ theta}$ ${\ displaystyle \ theta}$ ${\ displaystyle x = a \ tan \ theta, \ y = a \ cos ^ {2} \ theta. \,}$

Parametric equation with the angle , if the angle is between OA and the x -axis, increasing counterclockwise: ${\ displaystyle \ theta}$ ${\ displaystyle \ theta}$ ${\ displaystyle x = a \ cot \ theta, \ y = a \ sin ^ {2} \ theta. \,}$

Here the parameter is . ${\ displaystyle a \ in \ mathbb {R}}$

properties

The Versiera of Agnesi with parameters a = 2, a = 4, a = 8, and a = 16

Asymptote :

{\ displaystyle y = 0}

Area between curve and asymptote:

{\ displaystyle \ pi a ^ {2}}

Volume of rotation of the curve around its asymptote:

{\ displaystyle {\ frac {1} {2}} \ pi ^ {2} a ^ {3}}

Radius of curvature at the apex : .

{\ displaystyle (x, \, y) = (0, \, a)}

{\ displaystyle R = {\ frac {a} {2}}}

Two turning points :

{\ displaystyle (x, \, y) = \ left (\ pm {\ frac {a} {\ sqrt {3}}}, {\ frac {3a} {4}} \ right)}

If one the representation in Cartesian coordinates to y in order, we obtain Thus, a primitive function of y (x) , ie . ${\ displaystyle y (x) = {\ frac {a ^ {3}} {x ^ {2} + a ^ {2}}}.}$ ${\ displaystyle Y (x) = a ^ {2} \ arctan {\ frac {x} {a}}}$ ${\ displaystyle Y '(x) = y (x)}$

variant

Occasionally the horizontal straight line (above MN ) is not laid through the north pole of the circle, but through its center. The versiera then runs for points above this straight line inside the generating circle, its equation is in Cartesian coordinates , where r is the radius of the circle. The astonishing fact emerges that the volume of the solid of revolution that arises when the curve rotates around the x axis is exactly the same as that of the torus that the circle produces when it rotates around the x axis, namely the same . ${\ displaystyle y = {\ frac {2r ^ {3}} {x ^ {2} + r ^ {2}}}}$ ${\ displaystyle 2r ^ {3} \ pi ^ {2}}$

history

Pierre de Fermat studied the curve in his 1659 treatise on quadrature . In it, Fermat calculates the area below the curve and claims (without details) that the method used is also suitable for the cissoids of Diocles . Fermat writes that the curve was suggested to him by a trained geometer (" ab erudito geometra "). Paradís / Pla / Viader (2008) speculate that the proposing geometer could have been Antoine de Laloubère .

The construction of the curve described above was developed by Grandi in 1718; Isaac Newton had previously found the same construction, but it was not published until 1779, after Newton's death. In 1748 Maria Gaetana Agnesi published Instituzioni analitiche ad uso della gioventù italiana , an early textbook on infinitesimal calculus . This book, after considering two other curves, contained a study of the Versiera. It defines the curve geometrically as the geometrical location of points that meet a certain condition, determines the algebraic equation, the vertex, the asymptote and the inflection points.

Applications

A scaled version of the curve corresponds to the probability density function of the Cauchy distribution . This is the probability distribution of the random variable , which is defined by the following random experiment : For a fixed point above the axis, a straight line through is chosen at random (uniform distribution) ; be the coordinate of the intersection of this straight line with the -axis. The Cauchy distribution determined by this is similar to the normal distribution , but due to the heavy-tailed distribution there is no expected value according to the usual definitions, despite the symmetry. This means that the coordinate of the center of gravity of the surface between the curve and its asymptote is not well defined, although the surface is symmetrical and its content is finite. ${\ displaystyle x}$ ${\ displaystyle P}$ ${\ displaystyle x}$ ${\ displaystyle P}$ ${\ displaystyle x}$ ${\ displaystyle x}$ ${\ displaystyle x}$

In numerical mathematics , when approximating functions using polynomials ( polynomial interpolation ) with evenly distributed interpolation points, it can happen with some functions that the use of more points provides poorer approximations. This paradoxical behavior is called Runge's phenomenon . It was first discovered by Carl Runge for the Runge function , another scaled version of the Versiera of Agnesi, with the interpolation of the function over the interval . The same phenomenon occurs for using the larger interval . ${\ displaystyle y = 1 / (1 + 25x ^ {2})}$ ${\ displaystyle [-1.1]}$ ${\ displaystyle y = 1 / (1 + x ^ {2})}$ ${\ displaystyle [-5.5]}$

The Versiera of Agnesi describes the energy distribution of spectral lines , especially with X-rays .

The cross-section of a gentle hill resembles the Versiera. Curves of this type have been used in the mathematical modeling of landscapes.

Solitons in deep water can also have this shape.

A version of the curve was of Gottfried Wilhelm Leibniz used the Leibniz formula for the circle constant ${\ displaystyle \ pi}$ derive. This formula, the infinite series

{\ displaystyle {\ frac {\ pi} {4}} = 1 \, - \, {\ frac {1} {3}} \, + \, {\ frac {1} {5}} \, - \ , {\ frac {1} {7}} \, + \, {\ frac {1} {9}} \, - \, \ cdots,}

results from the area between the curve and its asymptote, i.e. from the integral of the function , if the geometric series is set up as a Taylor series development of this function and the series members are individually integrated. ${\ displaystyle 1 / (1 + x ^ {2})}$ ${\ displaystyle 1-x ^ {2} + x ^ {4} -x ^ {6} + \ cdots}$

literature

Ulrike Klens: Mathematicians in the 18th Century: Maria Gaetana Agnesi, Gabrielle-Emilie du Châtelet, Sophie Germain: Case studies on the interaction between science and philosophy in the Age of Enlightenment. Centaurus, Pfaffenweiler 1998, ISBN 3-89085-826-0 (also dissertation at the University of Augsburg 1992).

Web links

John J. O'Connor, Edmund F. Robertson : Witch of Agnesi. In: MacTutor History of Mathematics archive .
Eric W. Weisstein : Witch of Agnesi . In: MathWorld (English).
DynaGeoX applet of the curve
The Witch of Agnesi - Mathforum.org Java-Applet (English)

Individual evidence

^ Lynn M. Osen: Women in Mathematics. MIT Press, Cambridge MA 1975, ISBN 0-262-15014-X , p. 45.
↑ Simon Singh : Fermat's Enigma. The quest to solve the world's greatest mathematical problem. Walker Books, New York 1997, ISBN 0-471-27047-4 , p. 100.
↑ David J. Darling: The universal book of mathematics. From Abracadabra to Zeno's paradoxes. Wiley International, Hoboken NJ 2004, ISBN 0-8027-1331-9 , p. 8.
↑ Hermann Schmidt: Selected higher curves . Kesselringsche Verlagsbuchhandlung, Wiesbaden 1949, p. 64 ff.

[1] Lynn M. Osen: Women in Mathematics. MIT Press, Cambridge MA 1975, ISBN 0-262-15014-X , p. 45.

[2] Simon Singh : Fermat's Enigma. The quest to solve the world's greatest mathematical problem. Walker Books, New York 1997, ISBN 0-471-27047-4 , p. 100.

[3] David J. Darling: The universal book of mathematics. From Abracadabra to Zeno's paradoxes. Wiley International, Hoboken NJ 2004, ISBN 0-8027-1331-9 , p. 8.

[4] Hermann Schmidt: Selected higher curves . Kesselringsche Verlagsbuchhandlung, Wiesbaden 1949, p. 64 ff.