Tschirnhausen Cubic

Author

Eric W. Weisstein
June 6, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/TschirnhausenCubic.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TschirnhausenCubic.html.

©2014 Wolfram Research, Inc. except for portions noted otherwise

Plot

Parametric

ParametricPlot[{1-3t^2,t(3-t^2)},{t,-10 Pi,10 Pi},Ticks->None,
PlotRange->{{-12,2},{-3,3}},
PlotStyle->Red,
AspectRatio->Automatic]

Polar

PolarPlot[Sec[q/3]^3,{q,-4,4}]

Coordinates

curve=a{1-3t^2,t(3-t^2)};

Polar

PolarPlot[Sec[θ/3]^3,{θ,-4,4},
PlotRange->{{-12,2},{-3,3}},
Ticks->None,PlotStyle->Red]

Parametric

par=Factor[FullSimplify[a Sec[θ/3]^3{Cos[θ],Sin[θ]}/.θ->3ArcTan[t]]]

shifted=FullSimplify[-{par[[1]]+8a,par[[2]]}]

ParametricPlot[curve/.a->1,{t,-2,2},Ticks->None,PlotStyle->Red]

Cartesian

FullSimplify[Eliminate[{x,y}==par,t]]

FullSimplify[Eliminate[{x,y}==shifted,t]]

Ordinary Double Point

par/.t->Sqrt[3]

Loop

ParametricPlot[{1-3t^2,t(3-t^2)},{t,-Sqrt[3],Sqrt[3]},Ticks->None,
PlotRange->{{-9,2},{-3,3}},
PlotStyle->Red,
AspectRatio->Automatic]

With[
{
x=Function[t,a(1-3t^2)],
y=Function[t,a t(3-t^2)]
},
-FullSimplify[y[t]x'[t]-x[t]y'[t]]
]

Integrate[3a^2 (((1+(t^2)))^2),{t,0,Sqrt[3]}]

Properties

tschirn=a{1-3(t^2),t(3-(t^2))};
eqn=(a-x) (((8 a+x))^2)-27 a (y^2);

Arc Length

v4.2.1

FullSimplify[ArcLength[tschirn,t,Assumptions->a>0&&t∈Reals],a>0&&t∈Reals]

v5

ArcLength[tschirn,t]//Timing

...ran overnight...

v6

ArcLength[tschirn,t]

ArcLength[tschirn,t,Assumptions->a>0&&t∈Reals]//Timing

V6.1

FullSimplify[ArcLength[curve,t],a>0&&t∈Reals]

Curvature

Parametric

FullSimplify[Curvature[curve,t],a>0&&t∈Reals]

Polar

FullSimplify[Curvature[a Sec[q/3]^3,q],a>0&&q∈Reals]

Implicit

SortBy[FullSimplify[Curvature[eqn,{x,y}],a>0&&(x|y)∈Reals],ByteCount]

ByteCount/@%

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[a {1-3 t^2,t (3-t^2)},t]//FullSimplify

Slope[a Sec[q/3]^3,q]//FullSimplify

Slope[(a-x) (8 a+x)^2==27 a y^2,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[tschirn,t],a>0&&t∈Reals]

Tangential Angle

v5

TangentialAngle[tschirn,t]//Timing

v6

TangentialAngle[curve,t]

Vector Properties

parm=a {1-3 t^2,t (3-t^2)};
assum=a>0&&t∈Reals;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,Sqrt[x_^(n_/;Mod[n,2]==0)]:>Abs[x]^(n/2)};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]/.xform

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]/.xform

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]/.xform