Semicubical Parabola

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot of Evolute

ParametricPlot[{{3/4 (4t^2)^(1/3)+1/2,t},{t^2,t}},{t,-1,1},Ticks->None,AspectRatio->Automatic,PlotStyle->{{RGBColor[1,0,0]},{Dashing[{0.05,0.05}]}}]

Plot of Simplified Curve

Cartesian

Plot[{-x^(3/2),x^(3/2)},{x,0,5},Ticks->None,PlotStyle->Red]

Parametric

ParametricPlot[{t^2,t^3},{t,-1,1},
Ticks->None,PlotStyle->Red,PlotRange->{{-.02,.2},Automatic}]

ParametricPlot[{t^2,t^3},{t,-1,1},Ticks->None,PlotStyle->Red]

Equation

GroebnerBasis[{x==t^2,y==a t^3},{x,y},t,MonomialOrder->EliminationOrder]

eqn=(a^2) (x^3)-(y^2);

Polar

{t^2, t^3}

θ->Tan[t]

FullSimplify[Simplify[Sqrt[t^4+t^6],t∈Reals]/.t->Tan[θ]]

PolarPlot[((Tan[θ])^2)Sec[θ],{θ,0,2π}]

Curve Properties

semicubical=a{t^2,t^3};

Arc Length Function

V5

FullSimplify[ArcLength[semicubical,t,Assumptions->t>0]]

V6.1

FullSimplify[ArcLength[semicubical,t,Assumptions->t>0],t>0&&a>0]

-8+4 (Sqrt[4+9 (t^2)])+9  (t^2) (Sqrt[4+9 (t^2)])//Simplify

4 (Sqrt[4+9 (t^2)])+9  (t^2) (Sqrt[4+9  (t^2)])//Simplify

(1/27) a (-8+(((4+9 (t^2)))^(3/2)))

V10

FullSimplify[MyArcLength[semicubical,t,Assumptions->t>0],t>0&&a>0]

Curvature

v5

Curvature[semicubical,t]

v6.1

Simplify[Curvature[semicubical,t],a>0&&t>0]

Polar

FullSimplify[Curvature[a Tan[q]^2 Sec[q],q],a>0&&0<q<2π]

Implicit

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

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[a {t^2,t^3},t]//FullSimplify

Slope[a Tan[q]^2 Sec[q],q]//FullSimplify

Slope[y^2==a^2 x^3,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

v5

TangentialAngle[semicubical,t,Assumptions->t∈Reals]

TangentialAngle[semicubical,t,Assumptions->t>0]

v6.1

TangentialAngle[semicubical,t,Assumptions->t>0&&a>0]

Vector Properties

parm=a {t^2,t^3};
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