Limaçon Evolute

Author

Eric W. Weisstein
June 28, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equation

limacon=(b+a Cos[t]){Cos[t],Sin[t]};

FullSimplify[Evolute[limacon,t]]

curve={((a (4 ((a^2)+(b^2))+9 a b Cos[t]-a b Cos[3 t]))/(4 (2 (a^2)+(b^2)+3 a b Cos[t]))),(((a^2) b ((Sin[t])^3))/(2 (a^2)+(b^2)+3 a b Cos[t]))};

Plot

EvaluatePlot[Function[{a,b},{Cos[t] (b+a Cos[t]),(b+a Cos[t]) Sin[t]}],{t,0,2 π}][1,2]

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},EvolutePlot[limacon/.{a->1,b->#},{t,0,2π},PlotRange->{{-1.1,3.1},{-2.5,2.5}}]&/@{2,1.3,1,.5}]
]]

Function[{a,b},ParametricPlot[{((a (4 ((a^2)+(b^2))+9 a b Cos[t]-a b Cos[3 t]))/(4 (2 (a^2)+(b^2)+3 a b Cos[t]))),(((a^2) b ((Sin[t])^3))/(2 (a^2)+(b^2)+3 a b Cos[t]))},{t,0,2π},PlotStyle->Red,ImageSize->Tiny]]@@@{{1,2},{1,1.3},{1,1},{1,.5}}

Equations

eqns=TrigExpand[{4 (2 (a^2)+(b^2)+3 a b Cos[t])x==a (4 ((a^2)+(b^2))+9 a b Cos[t]-a b Cos[3 t]),y(2 (a^2)+(b^2)+3 a b Cos[t])==(a^2) b ((Sin[t])^3)}]

Factor/@GroebnerBasis[Append[eqns,Sin[t]^2+Cos[t]^2==1],{x,y},{Cos[t],Sin[t]},MonomialOrder->EliminationOrder]

eqn=((a^12)-3 (a^10) (b^2)+3 (a^8) (b^4)-(a^6) (b^6)-12 (a^11) x+30 (a^9) (b^2) x-24 (a^7) (b^4) x+6 (a^5) (b^6) x+60 (a^10) (x^2)-123 (a^8) (b^2) (x^2)+78 (a^6) (b^4) (x^2)-15 (a^4) (b^6) (x^2)-160 (a^9) (x^3)+264 (a^7) (b^2) (x^3)-132 (a^5) (b^4) (x^3)+20 (a^3) (b^6) (x^3)+240 (a^8) (x^4)-312 (a^6) (b^2) (x^4)+123 (a^4) (b^4) (x^4)-15 (a^2) (b^6) (x^4)-192 (a^7) (x^5)+192 (a^5) (b^2) (x^5)-60 (a^3) (b^4) (x^5)+6 a (b^6) (x^5)+64 (a^6) (x^6)-48 (a^4) (b^2) (x^6)+12 (a^2) (b^4) (x^6)-(b^6) (x^6)+12 (a^10) (y^2)-27 (a^8) (b^2) (y^2)+18 (a^6) (b^4) (y^2)-3 (a^4) (b^6) (y^2)-96 (a^9) x (y^2)+168 (a^7) (b^2) x (y^2)-84 (a^5) (b^4) x (y^2)+12 (a^3) (b^6) x (y^2)+288 (a^8) (x^2) (y^2)-384 (a^6) (b^2) (x^2) (y^2)+123 (a^4) (b^4) (x^2) (y^2)-18 (a^2) (b^6) (x^2) (y^2)-384 (a^7) (x^3) (y^2)+384 (a^5) (b^2) (x^3) (y^2)-66 (a^3) (b^4) (x^3) (y^2)+12 a (b^6) (x^3) (y^2)+192 (a^6) (x^4) (y^2)-144 (a^4) (b^2) (x^4) (y^2)+9 (a^2) (b^4) (x^4) (y^2)-3 (b^6) (x^4) (y^2)+48 (a^8) (y^4)-72 (a^6) (b^2) (y^4)+27 (a^4) (b^4) (y^4)-3 (a^2) (b^6) (y^4)-192 (a^7) x (y^4)+192 (a^5) (b^2) x (y^4)-6 (a^3) (b^4) x (y^4)+6 a (b^6) x (y^4)+192 (a^6) (x^2) (y^4)-144 (a^4) (b^2) (x^2) (y^4)-18 (a^2) (b^4) (x^2) (y^4)-3 (b^6) (x^2) (y^4)+64 (a^6) (y^6)-48 (a^4) (b^2) (y^6)-15 (a^2) (b^4) (y^6)-(b^6) (y^6));

AlgebraicDegree[eqn,{x,y}]

Properties

curve={((a (4 ((a^2)+(b^2))+9 a b Cos[t]-a b Cos[3 t]))/(4 (2 (a^2)+(b^2)+3 a b Cos[t]))),(((a^2) b ((Sin[t])^3))/(2 (a^2)+(b^2)+3 a b Cos[t]))};

Arc Length Function

ArcLength[curve,t,Assumptions->a>0&&b>0&&0<t<2π]

Area

Assuming[a>0&&b>0,Area[curve,{t,0,2π}]]//Timing

AreaInertiaTensor

Assuming[a>0&&b>0,FullSimplify@AreaInertiaTensor[curve,{t,0,2π}]]//Timing

Centroid

Assuming[a>0&&b>0,FullSimplify@Centroid[curve,{t,0,2π}]]//Timing

Curvature

FullSimplify[Curvature[curve,t],Assumptions->a>0&&b>0&&0<t<2π]

Plot[((Abs[((Csc[t])/(a+b Cos[t]))] (((2 (a^2)+(b^2)+3 a b Cos[t]))^3))/(3 (a^2) b ((((a^2)+(b^2)+2 a b Cos[t]))^(3/2))))/.{a->1,b->1},{t,0,2π}]

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

Slope

Slope[{(a (4 (a^2+b^2)+9 a b Cos[t]-a b Cos[3 t]))/(4 (2 a^2+b^2+3 a b Cos[t])),(a^2 b Sin[t]^3)/(2 a^2+b^2+3 a b Cos[t])},t]//FullSimplify

Slope[(a^12-3 a^10 b^2+3 a^8 b^4-a^6 b^6-12 a^11 x+30 a^9 b^2 x-24 a^7 b^4 x+6 a^5 b^6 x+60 a^10 x^2-123 a^8 b^2 x^2+78 a^6 b^4 x^2-15 a^4 b^6 x^2-160 a^9 x^3+264 a^7 b^2 x^3-132 a^5 b^4 x^3+20 a^3 b^6 x^3+240 a^8 x^4-312 a^6 b^2 x^4+123 a^4 b^4 x^4-15 a^2 b^6 x^4-192 a^7 x^5+192 a^5 b^2 x^5-60 a^3 b^4 x^5+6 a b^6 x^5+64 a^6 x^6-48 a^4 b^2 x^6+12 a^2 b^4 x^6-b^6 x^6+12 a^10 y^2-27 a^8 b^2 y^2+18 a^6 b^4 y^2-3 a^4 b^6 y^2-96 a^9 x y^2+168 a^7 b^2 x y^2-84 a^5 b^4 x y^2+12 a^3 b^6 x y^2+288 a^8 x^2 y^2-384 a^6 b^2 x^2 y^2+123 a^4 b^4 x^2 y^2-18 a^2 b^6 x^2 y^2-384 a^7 x^3 y^2+384 a^5 b^2 x^3 y^2-66 a^3 b^4 x^3 y^2+12 a b^6 x^3 y^2+192 a^6 x^4 y^2-144 a^4 b^2 x^4 y^2+9 a^2 b^4 x^4 y^2-3 b^6 x^4 y^2+48 a^8 y^4-72 a^6 b^2 y^4+27 a^4 b^4 y^4-3 a^2 b^6 y^4-192 a^7 x y^4+192 a^5 b^2 x y^4-6 a^3 b^4 x y^4+6 a b^6 x y^4+192 a^6 x^2 y^4-144 a^4 b^2 x^2 y^4-18 a^2 b^4 x^2 y^4-3 b^6 x^2 y^4+64 a^6 y^6-48 a^4 b^2 y^6-15 a^2 b^4 y^6-b^6 y^6)==0,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&b>0&&0<t<2π]

TangentialAngle

TangentialAngle[curve,t,Assumptions->a>0&&b>0&&0<t<2π]

FullSimplify[Curvature[curve,t] Sqrt[#.#&[D[curve,t]]],a>0&&b>0&&0<t<2π]

Integrate[((2 (a^2)+(b^2)+3 a b Cos[t])/((a^2)+(b^2)+2 a b Cos[t])),{t,0,t},Assumptions->a>0&&b>0&&0<t<2π]

FullSimplify[((((a^2)+(b^2)+2 a b Cos[t])/(2 a b-2 a b Cos[t]))∉Reals||((a^2)+(b^2)+2 a b Cos[t]) Re[(1/(-2 a b+2 a b Cos[t]))]≤0)&&2 a b+(((a+b))^2) Re[(1/(-1+Cos[t]))]≤0&&(a^2)+(b^2)+2 a b Cos[t]≠0,a>0&&b>0&&0<t<2π]

Integrate[((2 (a^2)+(b^2)+3 a b Cos[t])/((a^2)+(b^2)+2 a b Cos[t])),{t,0,x},Assumptions->a>0&&b>0&&0<x<2π]

Integrate[((2 (a^2)+(b^2)+3 a b Cos[t])/((a^2)+(b^2)+2 a b Cos[t])),{t,0,x},Assumptions->a>0&&b>0&&0<x<π]

Integrate[((2 (a^2)+(b^2)+3 a b Cos[t])/((a^2)+(b^2)+2 a b Cos[t])),{t,0,x},Assumptions->a>0&&b>0&&π<x<2π]

(1/2) (3 x-2 ArcTan[(((a+b) Cot[(x/2)])/(a-b))]+π Sign[a-b])/.x->t

Vector Properties

parm={(a (4 (a^2+b^2)+9 a b Cos[t]-a b Cos[3 t]))/(4 (2 a^2+b^2+3 a b Cos[t])),(a^2 b Sin[t]^3)/(2 a^2+b^2+3 a b Cos[t])};
assum=a>0&&b>0&&t>0;
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]]