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| In [[geometry]], the '''Tschirnhausen cubic''' is a [[plane curve]] defined by the polar equation
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| :<math>r = a\sec^3(\theta/3).</math> | |
| [[Image:Tschirnhausen cubic.svg|right|thumb|500px|The Tschirnhausen cubic, <math>y^2=x^3+3x^2.</math>]]
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| ==History==
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| The curve was studied by [[Ehrenfried Walther von Tschirnhaus|von Tschirnhaus]], [[Guillaume de l'Hôpital|de L'Hôpital]] and [[Eugène Charles Catalan|Catalan]]. It was given the name Tschirnhausen cubic in a 1900 paper by R C Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.
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| ==Other equations==
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| Put <math>t=\tan(\theta/3)</math>. Then applying [[De Moivre's formula|triple-angle formulas]] gives
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| :<math>x=a\cos \theta \sec^3 \frac{\theta}{3} = a(\cos^3 \frac{\theta}{3} - 3 \cos \frac{\theta}{3} \sin^2 \frac{\theta}{3}) \sec^3 \frac{\theta}{3}</math>
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| ::<math>= a\left(1 - 3 \tan^2 \frac{\theta}{3}\right)= a(1 - 3t^2) </math>
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| :<math>y=a\sin \theta \sec^3 \frac{\theta}{3} = a \left(3 \cos^2 \frac{\theta}{3}\sin \frac{\theta}{3} - \sin^3 \frac{\theta}{3} \right) \sec^3 \frac{\theta}{3}</math>
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| ::<math>= a \left(3 \tan \frac{\theta}{3} - \tan^3 \frac{\theta}{3} \right) = at(3-t^2)</math>
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| giving a [[parametric equation|parametric]] form for the curve. The parameter t can be eliminated easily giving the [[Cartesian coordinate system|Cartesian equation]]
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| :<math>27ay^2 = (a-x)(8a+x)^2</math>. | |
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| If the curve is translated horizontally by 8''a'' then the equations become
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| :<math>x = 3a(3-t^2)</math>
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| :<math>y = at(3-t^2)</math>
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| or
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| :<math>x^3=9a \left(x^2-3y^2 \right)</math>.
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| This gives an alternate polar form of
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| :<math>r=9a \left(\sec \theta - 3\sec \theta \tan^2 \theta \right)</math>.
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| ==References==
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| * J. D. Lawrence, ''A Catalog of Special Plane Curves''. New York: Dover, 1972, pp. 87-90.
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| ==External links==
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| * {{MathWorld|title=Tschirnhausen Cubic|urlname=TschirnhausenCubic}}
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| * [http://www-groups.dcs.st-and.ac.uk/~history/Curves/Tschirnhaus.html "Tschirnhaus' Cubic" at MacTutor History of Mathematics Archive]
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| * [http://mathcurve.com/courbes2d/tschirnhausen/tschirnhausen.shtml "Cubique de Tschirnhausen" at Encyclopédie des Formes Mathématiques Remarquables] (in French)
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| {{geometry-stub}}
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| [[Category:Curves]]
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