Maclaurin Trisectrix

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Equation

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

Cartesian

curve

Factor/@GroebnerBasis[{x(1+t^2)==a (-3+(t^2)),y(1+t^2)==a t (-3+(t^2))},{x,y},t,MonomialOrder->EliminationOrder]

(y/.Solve[Eliminate[{x==a (((t^2)-3)/((t^2)+1)),y==a (((t^2)-3)/((t^2)+1))t},t],y])^2//Simplify

Polar

FullSimplify[#.#]&[{a(((t^2)-3)/((t^2)+1)),a (((t^2)-3)/((t^2)+1))t}/.t->Tan[θ]]

Simplify[2Sin[3θ]/Sin[2θ]-(1+2 Cos[2 θ]) Sec[θ]]

Plots

Implicit Cartesian

<<Graphics`ImplicitPlot`

ImplicitPlot[y^2==((x^2(x+3))/(1-x)),{x,-3,2},
PlotRange->{{-3,1},{-3,3}},PlotStyle->Red]

Parametric

ParametricPlot[{1,t}(t^2-3)/(t^2+1),{t,-4,4},AspectRatio->Automatic,PlotRange->{{-3,1},{-3,3}},PlotStyle->Red]

ParametricPlot[{1,t}(t^2-3)/(t^2+1),{t,-(Sqrt[3]),(Sqrt[3])},AspectRatio->Automatic,PlotRange->{{-3,1},{-3,3}},PlotStyle->Red,ImageSize->150]

Polar 1

PolarPlot[-(1+2 Cos[2 θ]) Sec[θ],{θ,-π/2,π/2},PlotRange->{-3,3},Ticks->None,PlotStyle->Red,ImageSize->150]

PolarPlot[-(1+2 Cos[2 θ]) Sec[θ],{θ,-π/3,π/3},PlotRange->{-3,3},Ticks->None,PlotStyle->Red,ImageSize->150]

Polar 2

PolarPlot[-Sec[θ/3],{θ,-4,4},PlotRange->{-3,3},PlotStyle->Red]

Loop Properties

mac=a{1,t}(t^2-3)/(t^2+1);
macpolar1=(1+2 Cos[2 θ]) Sec[θ];
macpolar2=Sec[θ/3];

Arc Length

FullSimplify[2ArcLength[curve,{t,0,(Sqrt[3])}],a>0]

N[%]

Area

Paramteric

2Area[curve,{t,0,(Sqrt[3])}]

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

Area[mac,{t,-Sqrt[3],Sqrt[3]}]

Polar1

Solve[(1+2 Cos[2 θ])==0,θ]

PolarPlot[(1+2 Cos[2 θ]) Sec[θ],{θ,-π/3,π/3},PlotRange->{-3,3},Ticks->None,PlotStyle->Red]

Area[a(1+2 Cos[2 θ]) Sec[θ],{θ,-π/3,π/3}]

Polar2

Sec[θ/3]/.θ->π

PolarPlot[Sec[θ/3],{θ,-π,π},PlotRange->{-3,3},PlotStyle->Red]

Area[a Sec[θ/3],{θ,-π,π}]

Properties

eqn=3 a x^2+x^3-a y^2+x y^2;

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

Arc Length

v5

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

v6

FullSimplify[ArcLength[mac,t]]

V6.1

ArcLength[curve,t,Assumptions->a>0&&t>0]

Plot[-3 I EllipticE[I ArcSinh[t],(1/9)],{t,0,2}]

Subtract@@Function[t,-3 I a EllipticE[I ArcSinh[t],(1/9)]]/@{-(Sqrt[3]),(Sqrt[3])}//FullSimplify

N[%]

Curvature

Parametric

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

Polar

FullSimplify[Curvature[-(1+2 Cos[2q]) a Sec[q],q],a>0&&-π/2<q<π/2]

Implicit

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

Singular Point

SingularPoints[eqn,{x,y}]

Slope

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

Slope[-(1+2 Cos[2q]) Sec[q],q]//FullSimplify

Slope[3 a x^2+x^3-a y^2+x y^2==0,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

v5

FullSimplify[TangentialAngle[mac,t]]

v6

FullSimplify[TangentialAngle[curve,t],t∈Reals]

Plot[-(π/2) Sign[t]+ArcTan[(3/t)]+3 ArcTan[t],{t,-10,10},PlotRange->All]

Vector Properties

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

VectorLength

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

NormalVector

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

TangentVector

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