The "witch of Agnesi" is a curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uso della gioventù italiana (the first surviving mathematical work written by a woman). x

New York: Dover, pp. x -coordinate of the centroid of the region between the curve and its asymptotic line is not well-defined, despite this region's symmetry and finite area. https://mathworld.wolfram.com/WitchofAgnesi.html.

Witch of Agnesi. It has a unique vertex (a point of extreme curvature) at the point of tangency with its defining circle, which is also its osculating circle at that point. 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. . p The volume of revolution of the witch of Agnesi about its asymptote is Witch of Agnesi. 1 [20][21] Struik mentions that:[17]. [13], The construction given above for this curve was found by Grandi (1718); the same construction was also found earlier by Isaac Newton, but only published posthumously later, in 1779. Graph. ± Witch of Agnesi. The "witch of Agnesi" is a curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uso della gioventù italiana (the first This paradoxical behavior is called Runge's phenomenon.

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 function that appeared in his construction. 2

The line is an asymptote

The cover of the album features an image of the construction of the witch.[32]. 1 pp. intersection of the extension of line with the line has been used to approximate the energy distribution of spectral lines, and models the shape of hills. 1

[29][30], A version of this curve was used by Gottfried Wilhelm Leibniz to derive the Leibniz formula for π. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.

{\displaystyle x} {\displaystyle x} The point M is diametrically opposite to O. It was first discovered by Carl David Tolmé Runge for Runge's function π

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6 equations, giving. 1 − x Explore anything with the first computational knowledge engine. 4 The curve is also known as cubique d'Agnesi or agnésienne, and had been studied earlier by Fermat and Guido Grandi in 1703. 2: Special Topics of Elementary Mathematics. [17], Maria Gaetana Agnesi named the curve according to Grandi, versiera. 2 The Cartesian equation can be obtained by eliminating in the parametric 5 . surviving mathematical work written by a woman). 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. x Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. The curve has inflection points at . x 4 + 2 {\displaystyle 1/(1+x^{2})} Gray, S. "History of the Name 'Witch.' The Cauchy distribution has a peaked distribution visually resembling the normal distribution, but its heavy tails prevent it from having an expected value by the usual definitions, despite its symmetry. This is the probability distribution on the random variable York: Dover, p. 331, 1958. [6] The defining circle of the witch is also its osculating circle at the vertex,[7] the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature. Starting with a fixed circle, a point O on the circle is chosen. -axis, choose uniformly at random a line through

Lawrence, J. D. A

+ [4], The witch of Agnesi can also be described by parametric equations whose parameter θ is the angle between OM and OA, measured clockwise:[2][3], The main properties of this curve can be derived from integral calculus. [19] Different modern works about Agnesi and about the curve suggest slightly different guesses how exactly this mistranslation happened. 25 4 using the Taylor series expansion of this function as the infinite geometric series The same phenomenon occurs for the witch ,

Ann Arbor, MI: J. W. Edwards, = As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory. Solitary waves in deep water can also take this shape. Then the witch constructed from O and M has the Cartesian equation[2][3], This equation can be simplified, by choosing a = 1/2, to the form, In its simplified form, this curve is the graph of the derivative of the arctangent function. , another scaled version of the witch of Agnesi, when interpolating this function over the interval The Instituzioni has plates of carefully numbered illustrations attached at the back of each volume. above the x 1 [14][23], In numerical analysis, when approximating functions using polynomial interpolation 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. 3 θ [5], The curve was studied by Pierre de Fermat in his 1659 treatise on quadrature.

{\displaystyle x} . [31], Witch of Agnesi is also the title of a music album by jazz quartet Radius. Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Yates, R. C. "Witch of Agnesi." π , and integrating term-by-term.

New angle in the second parametrization are given by. π The first to use the term 'witch' in this sense may have been B. Williamson, Integral calculus, 7 (1875), 173;[22] see Oxford English Dictionary.

[8] Because this is an osculating circle at the vertex of the curve, it has third-order contact with the curve. x [15][17] Coincidentally, at that time in Italy it was common to speak of the Devil, the adversary of God, through other words like aversiero or versiera, derived from Latin adversarius. [11] The curve had already appeared in the writings of Fermat (Oeuvres, I, 279–280; III, 233–234) and of others; the name versiera is from Guido Grandi (Quadratura circuli et hyperbolae, Pisa, 1703). Discover Resources. 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 of revolution of its defining circle. 1 [ [2][3] When considered as a curve in the projective plane there is also a third infinite inflection point, at the point where the line at infinity is crossed by the asymptotic line.

Basically just a nice curve I found in a book. Weisstein, Eric W. "Witch of Agnesi." Witch of Agnesi. [5], The curve has a unique vertex at the point of tangency with its defining circle. [ d'Agnesi or agnésienne, and had been studied earlier by Fermat and Guido Grandi through the circle of radius and center , then picking the point with the coordinate of the " https://instructional1.calstatela.edu/sgray/Agnesi/WitchHistory/Historynamewitch.html. Knowledge-based programming for everyone. Fermat writes that the curve was suggested to him "ab erudito geometra" [by a learned geometer]. From MathWorld--A Wolfram Web Resource. The name "witch" derives from a mistranslation of the term averisera ("versed sine curve," from the Latin vertere, "to turn") ( ]

[2] This is two times the volume of the torus formed by revolving the defining circle of the witch around the same line. a A program for graphing the "Witch of Agnesi. 1. a = 2. {\displaystyle 4\pi ^{2}a^{3}} to the curve.

"Witch of Agnesi." − Join the initiative for modernizing math education. [18] Because of this, Cambridge professor John Colson mistranslated the name of the curve as "witch". Practice online or make a printable study sheet. The curve is also known as cubique {\displaystyle 4a^{2}} To construct this curve, start with any two points O and M, and draw a circle with OM as diameter. 1 . a History of Mathematics, Vol. 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 curve. [26] Curves with this shape have been used as the generic topographic obstacle in a flow in mathematical modeling. [2][3][5] It gets its name from Italian mathematician Maria Gaetana Agnesi, and from a mistranslation of an Italian word for a sailing sheet. Rigid Motions on the Plane; OUR Math 8.1.3.3 Cool-down: Some are Translations and Some Aren't

be the coordinate of the point where this random line crosses the axis. https://mathworld.wolfram.com/WitchofAgnesi.html. The area between the witch and its asymptotic line is four times the area of the fixed circle, . = Grandi (1718) also suggested the name versiera (in Italian) or versoria (in Latin) for the curve.

, {\displaystyle 4\pi a^{2}} ) . 4 A program for graphing the "Witch of Agnesi." , This is typical of 18th century mathematics books. Witch of Agnesi. . Suppose that point O is at the origin and point M lies on the positive y-axis, and that the circle with diameter OM has radius a.

Smith, D. E. History of Mathematics, Vol. [10], The largest area of a rectangle that can be inscribed between the witch and its asymptote is devil") in an 1801 translation of the work by Cambridge Lucasian Professor of The area between the curve and the -axis is. , and let In it, after first considering two other curves, she includes a study of this curve. I don't know why it's called the Witch but I know it's named after an Italian mathematician ( https: ... A subreddit dedicated to sharing graphs creating using the Desmos graphing calculator. x [5][14][15][16], In 1748, Maria Gaetana Agnesi published Instituzioni analitiche ad uso della gioventù italiana, an early textbook on calculus.

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