Cornoid

Author

Eric W. Weisstein
June 27, 2013

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

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

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

Equations

par[t_]:=a{Cos[t](1-2Sin[t]^2),Sin[t](1+2Cos[t]^2)}

eqns=Append[Thread[{x,y}==par[t]],Sin[t]^2+Cos[t]^2==1]

{xy}=Factor/@GroebnerBasis[eqns,{x,y},{Cos[t],Sin[t]}];

eqn=-4 (a^6)+3 (a^2) (x^4)+(x^6)+8 (a^4) (y^2)-6 (a^2) (x^2) (y^2)+3 (x^4) (y^2)-5 (a^2) (y^4)+3 (x^2) (y^4)+(y^6);

AlgebraicDegree[xy,{x,y}]

Polar

FullSimplify[Sqrt[#.#]&@{Cos[t](1-2Sin[t]^2),Sin[t](1+2Cos[t]^2)}]

FullSimplify[Solve[q==1/Divide@@{Cos[t](1-2Sin[t]^2),Sin[t](1+2Cos[t]^2)},t]]

Plot

V5

ParametricPlot[par[t],{t,0,2Pi},AspectRatio->Automatic,Ticks->None,PlotStyle->Red]

V6

ParametricPlot[par[t],{t,Pi/4,3Pi/4},AspectRatio->Automatic,Ticks->None,PlotStyle->Red]

ParametricPlot[par[t],{t,3Pi/4,5π/4},AspectRatio->Automatic,Ticks->None,PlotStyle->Red]

Properties

curve=a{Cos[t](1-2Sin[t]^2),Sin[t](1+2Cos[t]^2)};

Arc Length

ArcLength[par[t],{t,0,2π}]

ArcLength[par[t],{t,0,2π}]//N

FullSimplify[((((Cos[t] (1+2 ((Cos[t])^2))-4 Cos[t] ((Sin[t])^2)))^2)+(((-4 ((Cos[t])^2) Sin[t]-Sin[t] (1-2 ((Sin[t])^2))))^2)),0<t<2π]

NIntegrate[(Sqrt[(1/2) (7+8 Cos[2 t]+3 Cos[4 t])]),{t,0,2π}]

Integrate[(Sqrt[(1/2) (7+8 Cos[2 t]+3 Cos[4 t])]),{t,0,2π}]//Timing

ad[u_]:=2 Sqrt[(1-u^2) (1+2 u^2)]/(-2+u)+2 EllipticE[ArcSin[u],-2]-6 EllipticF[ArcSin[u],-2]+6 EllipticPi[1/4,ArcSin[u],-2]+2/3 Sqrt[3] ArcTan[2-5 u^2,2 Sqrt[3] Sqrt[(1-u^2) (1+2 u^2)]];

D[ad[u],u]//FullSimplify

ad[1]-ad[-1]//Simplify

N[%,20]

Arc Length Function

ArcLength[par[t],t]

Integrate[Sqrt[(1/2)(7+8Cos[2t]+3Cos[4t])],t]//Timing

FullSimplify[%,0<t<2π]

ad1[t_]:=((Sqrt[7+8 Cos[2 t]+3 Cos[4 t]]) ((1+I) (Sqrt[I+2 (Sqrt[2])]) ((-I+2 (Sqrt[2])) EllipticE[ArcSin[((Tan[t])/(Sqrt[1-2 I (Sqrt[2])]))],(1/9) (-7-4 I (Sqrt[2]))]-2 ((Sqrt[2]) EllipticF[ArcSin[((Tan[t])/(Sqrt[1-2 I (Sqrt[2])]))],(1/9) (-7-4 I (Sqrt[2]))]+4 I EllipticPi[-1+2 I (Sqrt[2]),ArcSin[((Tan[t])/(Sqrt[1-2 I (Sqrt[2])]))],(1/9) (-7-4 I (Sqrt[2]))])) ((Sec[t])^2) (Sqrt[1+(((Tan[t])^2)/(-1+2 I (Sqrt[2])))]) (Sqrt[9+(-1+2 I (Sqrt[2])) ((Tan[t])^2)])+3 (Sqrt[2]) Tan[t] (9-2 ((Tan[t])^2)+((Tan[t])^4))))/(12 (9-2 ((Tan[t])^2)+((Tan[t])^4)))

Plot[ad1[t],{t,0,2π}]

Plot[Through[{Re,Im}[ad1[t]]],{t,0,π/2}]

substitution of t -> 1/2 ArcCos[(1 - 2*u)/(u - 2)] brings the integrand into an algebraic form

4 NIntegrate[3/2 Sqrt[(1+2*u^2)/(1-u^2)]/(u-2)^2,{u,-1,1}]

FullSimplify[Integrate[3/2 Sqrt[(1+2*u^2)/(1-u^2)]/(u-2)^2,u],-1<u<1]

The following is an anti - derivative of the above integrand:

ad2[u_]:=2 Sqrt[(1-u^2) (1+2 u^2)]/(-2+u)+2 EllipticE[ArcSin[u],-2]-6 EllipticF[ArcSin[u],-2]+6 EllipticPi[1/4,ArcSin[u],-2]+2/3 Sqrt[3] ArcTan[2-5 u^2,2 Sqrt[3] Sqrt[(1-u^2) (1+2 u^2)]];

Plot[ad2[u],{u,-1,1}]

D[ad[u],u]//FullSimplify

Solve[t==1/2 ArcCos[(1-2*u)/(u-2)],u]

Function[t,((1+2 Cos[2 t])/(2+Cos[2 t]))]/@{0,π/2}

FullSimplify[ad2[u]/.u->((1+2 Cos[2 t])/(2+Cos[2 t])),0<t<2π]

ad3[t_]:=(1/(Sqrt[3]))2 ArcTan[-2-4 Cos[2 t]-3 Cos[4 t],(Sqrt[6]) Abs[Sin[2 t]] (Sqrt[7+8 Cos[2 t]+3 Cos[4 t]])]-(((Sqrt[2]) Abs[Sin[2 t]] (Sqrt[7+8 Cos[2 t]+3 Cos[4 t]]))/(2+Cos[2 t]))+2 EllipticE[ArcSin[2-(3/(2+Cos[2 t]))],-2]-6 EllipticF[ArcSin[2-(3/(2+Cos[2 t]))],-2]+6 EllipticPi[(1/4),ArcSin[2-(3/(2+Cos[2 t]))],-2]

Plot[ad3[t]/4,{t,0,2π}]

Area

Area[curve,{t,0,2π}]

Pieces

a1=Area[par[t],{t,Pi/4,3Pi/4}]//FullSimplify

a2=2Area[par[t],{t,3Pi/4,5π/4}]//FullSimplify

a2-2a1//Simplify

Inequality

NIntegrate[Boole[Evaluate[xy/.a->1]<0],{y,-2,2},{x,-2,2}]

RegionPlot[xy<0,{x,-2,2},{y,-2,2},MaxRecursion->10]

AreaInertiaTensor

AreaInertiaTensor[curve,{t,0,2π}]//Timing

Centroid

Centroid[curve,{t,0,2π}]

Curvature

FullSimplify[Curvature[par[t],t],a>0]

Plot[((3 (Sqrt[2]) (5+4 Cos[2 t]+Cos[4 t]))/(((7+8 Cos[2 t]+3 Cos[4 t]))^(3/2))),{t,0,2π}]

Implicit Curvature

FullSimplify[ImplicitCurvature[xy,{x,y}],(x|y)∈Reals&&a>0]

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

ByteCount/@%

Inflection Points

InflectionPoints[eqn,{x,y}]//Union

Singular Points

SingularPoints[eqn,{x,y}]//Union

Slope

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

Slope[-4 a^6+3 a^2 x^4+x^6+8 a^4 y^2-6 a^2 x^2 y^2+3 x^4 y^2-5 a^2 y^4+3 x^2 y^4+y^6==0,{x,y}]//FullSimplify

Slope[-4 a^6+3 a^2 x^4+x^6+8 a^4 y^2-6 a^2 x^2 y^2+3 x^4 y^2-5 a^2 y^4+3 x^2 y^4+y^6==0,{x,y}]//FullSimplify

SpecialAffineCurvature

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

TangentialAngle

FullSimplify[TangentialAngle[par[t],t,Assumptions->0<t<2π]]

Plot[(1/3) (t+ArcCot[2 Cot[2 t]+Csc[2 t]]),{t,0,2π}]

Vector Properties

parm=a {Cos[t] (1-2 Sin[t]^2),Sin[t] (1+2 Cos[t]^2)};
assum=a>0&&0<t<2π;
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]]