image of Tschirnhausen cubic

Tschirnhausen cubic

Plane Curves

Description

Tschirnhaus's cubic is the negative pedal of a parabola with respect to the focus of the parabola. The caustic of Tschirnhaus's cubic where the radiant point is the pole is Neile's parabola.

Object definitions

Mapping

Mapping of Tschirnhausen cubic
\left\{x\to a \left(1-3 t^2\right),y\to a t \left(3-t^2\right)\right\}
<math> <mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>-</mo> <mrow> <mn>3</mn> <mo>&#8290;</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mi>t</mi> <mo>&#8290;</mo> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>-</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </math>
{x -> a*(1 - 3*t^2), y -> a*t*(3 - t^2)}
[x = a*(1-3*t^2), y = a*t*(3-t^2)]

Constants

Constants of Tschirnhausen cubic
\{a\}
<math> <mrow> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow> </math>
{a}
[a]

Cite this as:

Plane Curves: Tschirnhausen cubic from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/PlaneCurves/Tschirnhaus-Cubic.php

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