Tschirnhausen Cubic -- from Wolfram Mathworld

12/3/13 Tschirnhausen Cubic -- from Wolfram MathWorld

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Discrete Mathematics THINGS TO TRY: Tschirnhausen Cubic tschirnhausen cubic Foundations of Mathematics 6x6 latin squares Geometry circle, diameter=10 History and Terminology

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Topology The Tschirnhausen cubic is a plane curve given by the polar equation Alphabetical Index (1) Interactive Entries

Random Entry Letting gives the parametric equations New in MathWorld (2) MathWorld Classroom (3)

About MathWorld or Contribute to MathWorld (4) Send a Message to the Team (5) MathWorld Book (Lawrence 1972, p. 88).

Wolfram Web Resources » Eliminating from the above equations gives the Cartesian equations

13,191 entries (6) Last updated: Wed Nov 6 2013 (7) Created, developed, and nurtured by Eric Weisstein (Lawrence 1972, p. 88). at Wolfram Research The curve is also known as Catalan's trisectrix and l'Hospital's cubic. The name Tschirnhaus's cubic is given in R. C. Archibald's 1900 paper attempting to classify curves (MacTutor Archive).

The curve has a loop, illustrated above, corresponding to in the above parametrization. The area of the loop is given by

(8)

(9)

(10)

(11)

(Lawrence 1972, p. 89).

In the first parametrization, the arc length, curvature, and tangential angle as a function of are

(12) (13)

(14)

The curve has a single ordinary double point located at in the parametrization of equations (◇) and (◇).

The Tschirnhausen cubic is the negative pedal curve of a parabola with respect to the focus and the catacaustic of a parabola with respect to a point at infinity perpendicular to the symmetry axis.

SEE ALSO: Conchoid of de Sluze, Conchoid of Nicomedes, Fish Curve, Maclaurin Trisectrix, Right Strophoid, Strophoid, Tschirnhausen Cubic Catacaustic, Tschirnhausen Cubic Pedal Curve

REFERENCES: Law rence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 87-90, 1972. Loy, J. "Trisection of an Angle." http://w w w .jimloy.com/geometry/trisect.htm#curves. mathworld.wolfram.com/TschirnhausenCubic.html 1/2 12/3/13 Tschirnhausen Cubic -- from Wolfram MathWorld Loy, J. "Trisection of an Angle." http://w w w .jimloy.com/geometry/trisect.htm#curves. MacTutor History of Mathematics Archive. "Tschirnhaus's Cubic." http://w w w -groups.dcs.st- and.ac.uk/~history/Curves/Tschirnhaus.html.

CITE THIS AS: Weisstein, Eric W. "Tschirnhausen Cubic." From MathWorld--A Wolfram Web Resource. http://mathw orld.w olfram.com/TschirnhausenCubic.html

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