Limaçon Trisectrix

Author

Eric W. Weisstein
August 24, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Properties

ArcLength

MyArcLength[1+2Cos[q],{q,0,2Pi}]

With[{r=1+2Cos[q]},FullSimplify[Sqrt[r^2+D[r,q]^2]]]

Integrate[(Sqrt[5+4 Cos[q]]),{q,0,2π}]

N[%]

ArcLengthFunction

Integrate[(Sqrt[5+4 Cos[q]]),q]

%/.q->2π

Area

Solve[(1+2Cos[t])==0,t]

p1=ParametricPlot[(1+2Cos[t]){Cos[t],Sin[t]},{t,0,2Pi/3}]

p2=ParametricPlot[(1+2Cos[t]){Cos[t],Sin[t]},{t,2Pi/3,4Pi/3},PlotStyle->Dashed]

Show[{p1,p2},PlotRange->All]

ParametricPlot[(1+2Cos[t]){Cos[t],Sin[t]},{t,0,2π}]

2(a^2)(1/2)Integrate[(((1+2Cos[t]))^2),{t,0,2Pi/3}]//FullSimplify

Loop

ParametricPlot[(1+2Cos[t]){Cos[t],Sin[t]},{t,2Pi/3,Pi}]

ParametricPlot[(1+2Cos[t]){Cos[t],Sin[t]},{t,2Pi/3,4Pi/3}]

(1/2)(a^2)Integrate[(((1+2Cos[t]))^2),{t,2Pi/3,4Pi/3}]//FullSimplify

2 (1/2)(a^2)Integrate[(((1+2Cos[t]))^2),{t,2Pi/3,π}]//FullSimplify

CartesianEquation

ParametricPlot[(1+2 Cos[t]) {Cos[t],Sin[t]},{t,0,2π}]

x==a(1+2Cos[t])Cos[t]
y==a(1+2Cos[t])Sin[t]

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

expr1=FullSimplify[gb[[1]]]

expr2=((((x^2)+(y^2)-2a x))^2)-(a^2)((x^2)+(y^2));

FullSimplify[expr1-expr2]

Curvature

FullSimplify[Curvature[a(1+2 Cos[t]) {Cos[t],Sin[t]},t],a>0]

ImplicitCurvature

FullSimplify[Curvature[(a^2) (3 (x^2)-(y^2))-4 a x ((x^2)+(y^2))+((((x^2)+(y^2)))^2)==0,{x,y}],a>0]

ImplicitSlope

FullSimplify[Slope[(a^2) (3 (x^2)-(y^2))-4 a x ((x^2)+(y^2))+((((x^2)+(y^2)))^2)==0,{x,y}],a>0]

NormalVector

FullSimplify[NormalVector[a (1+2Cos[t]){Cos[t],Sin[t]},t],a>0&&t>0]

ParametricEquations

Block[{a=1},ParametricPlot[a (1+2Cos[t]){Cos[t],Sin[t]},{t,0,2π}]]

PolarCurvature

TrigReduce[Sin[(1/2) q]]

FullSimplify[Curvature[a Sin[(1/2) q],q],a>0]

PolarEquation

Block[{a=1},PolarPlot[a Sin[(q/3)],{q,0,3π},PlotStyle->Red,PlotLabel->Style[TraditionalForm[r==Sin[(θ/3)]],Directive[FontFamily->"Times",Black,14]]]]

Block[{a=1},PolarPlot[a (2Cos[(q/3)]),{q,0,3π}]]

Block[{a=1},PolarPlot[a(1+2Cos[q]),{q,0,2π},PlotStyle->Red,PlotLabel->Style[TraditionalForm[r==(1+2Cos[θ])],Directive[FontFamily->"Times",Black,14]]]]

PolarSlope

FullSimplify[Slope[a Sin[(1/2) q],q],a>0]

Slope

FullSimplify[Slope[a (1+2Cos[t]){Cos[t],Sin[t]},t],a>0]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[a (1+2Cos[t]){Cos[t],Sin[t]},t],a>0]

TangentVector

FullSimplify[TangentVector[a (1+2Cos[t]){Cos[t],Sin[t]},t],a>0&&t>0]

VectorLength

FullSimplify[VectorLength[a (1+2Cos[t]){Cos[t],Sin[t]},t],a>0&&t>0]