Right Strophoid

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Initialization

RightStrophoid[c_][t_]:={-c,t}(t^2-c^2)/(t^2+c^2)

Plot

ParametricPlot[Evaluate[RightStrophoid[t]],{t,-2,2},Ticks->None,AspectRatio->Automatic,PlotRange->All,PlotStyle->Red]

<<Graphics`Graphics`

PolarPlot[Cos[2 t] Sec[t],{t,-Pi/2+.01,Pi/2-.01},Ticks->None,PlotRange->{{-1,1},{-2,2}},PlotStyle->Red,
AspectRatio->Automatic]

PolarPlot[Cos[2 t] Sec[t],{t,-Pi/2,Pi/2}]

Equations

Cartesian

Factor/@GroebnerBasis[Numerator[Factor[{x,y}-RightStrophoid[c][t]]],{x,y},t]

AlgebraicDegree[c (x^2)-(x^3)-c (y^2)-x (y^2),{x,y}]

-Expand[y^2(c+x)-(c-x)x^2]

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

Polar

FullSimplify[y^2-((c-x)/(c+x))x^2/.Thread[{x,y}->c Cos[2θ]Sec[θ]{Cos[θ],Sin[θ]}]]

Properties

curve={-c,t} (t^2-c^2)/(t^2+c^2);

Arc Length Function

FullSimplify[ArcLength[RightStrophoid[c,t],t,Method->blah][[1]],(c|t)∈Reals]

v4.2

ArcLength[RightStrophoid[c,t],t]//Timing

FullSimplify[Integrate[((Sqrt[(c^4)+6 (c^2) (t^2)+(t^4)])/((c^2)+(t^2))),{t,0,t}],(c|t)∈Reals]

FullSimplify[ArcLength[RightStrophoid[1,t],t],t∈Reals]

v5.0

ArcLength[RightStrophoid[c,t],t]//Timing

Integrate[((Sqrt[(c^4)+6 (c^2) (t^2)+(t^4)])/((c^2)+(t^2))),{t,0,t},Assumptions->t∈Reals]//Timing

ArcLength[RightStrophoid[c,t],t,Assumptions->(c|t)∈Reals]//Timing

FullSimplify[Integrate[((Sqrt[(c^4)+6 (c^2) (t^2)+(t^4)])/((c^2)+(t^2))),{t,0,t}],(c|t)∈Reals]//Timing

...crash...

v5.1

ArcLength[RightStrophoid[c,t],t]//Timing

ArcLength[RightStrophoid[c,t],t,Assumptions->c∈Reals]//Timing

ArcLength[RightStrophoid[c,t],t,Assumptions->(c|t)∈Reals]//Timing

FullSimplify[(1/(Abs[c]))(I (Sqrt[3+2 (Sqrt[2])]) (c^2) ((-3+2 (Sqrt[2])) EllipticE[I ArcSinh[(t/((Sqrt[3+2 (Sqrt[2])]) Abs[c]))],((3+2 (Sqrt[2]))/(3-2 (Sqrt[2])))]-2 ((1+(Sqrt[2])) EllipticF[I ArcSinh[(t/((Sqrt[3+2 (Sqrt[2])]) Abs[c]))],((3+2 (Sqrt[2]))/(3-2 (Sqrt[2])))]-2 EllipticPi[3+2 (Sqrt[2]),I ArcSinh[(t/((Sqrt[3+2 (Sqrt[2])]) Abs[c]))],((3+2 (Sqrt[2]))/(3-2 (Sqrt[2])))]))),(c|t)∈Integers]

Log[1+Sqrt[2],17+12Sqrt[2]]//N

FullSimplify[Integrate[((Sqrt[(c^4)+6 (c^2) (t^2)+(t^4)])/((c^2)+(t^2))),{t,0,t}],(c|t)∈Reals]//Timing

...crash...

V6

Integrate[((Sqrt[(c^4)+6 (c^2) (t^2)+(t^4)])/((c^2)+(t^2))),{t,0,t},Assumptions->c>0&&t>0]//Timing

Curvature

v5

FullSimplify[Curvature[RightStrophoid[c,t],t],(c|t)∈Reals]//Factor

v6

FullSimplify[Curvature[RightStrophoid[c][t],t],(c|t)∈Reals]//Factor

Polar

FullSimplify[Curvature[c Cos[2 q] Sec[q],q],c>0&&0<q<2π]

Implicit

FullSimplify[ImplicitCurvature[c (x^2)-(x^3)-c (y^2)-x (y^2),{x,y}]]

Inflection Points

Union[{x,y}/.Solve[eqn==2 (c+x) ((((2 c-3 x))^2) (x^2)+2 (-2 (c^2)+6 c x+3 (x^2)) (y^2)-3 (y^4))==0,{x,y}]]

InflectionPoints[eqn,{x,y}]

Singularities

SingularPoints[eqn,{x,y}]

Slope

Parametric

Slope[{-c,t} (t^2-c^2)/(t^2+c^2),t]//FullSimplify

Polar

Slope[Cos[2 q] Sec[q],q]//FullSimplify

Implicit

Slope[y^2==(c-x)/(c+x) x^2,{x,y}]//FullSimplify

SpecialAffineCurvature

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

ToRadicals[%]

Tangential Angle

v5

TangentialAngle[RightStrophoid[c,t],t]//Timing

...ran overnight...

c==1

TangentialAngle[RightStrophoid[1,t],t]

Plot[-2 ArcTan[t]-ArcTan[((2 t)/(1+(t^2)))],{t,-5,5}]

v5.1

TangentialAngle[RightStrophoid[c,t],t]//Timing

TangentialAngle[RightStrophoid[c,t],t,Assumptions->(c|t)∈Reals]//Timing

FullSimplify[Last[%],(c|t)∈Reals]

FullSimplify[Subtract@@{-2 ArcTan[t]+ArcTan[((2 t)/(1+(t^2)))],-2 ArcTan[(t/c)]+ArcTan[(((-1+(Sqrt[2])) t)/c)]-ArcTan[(((1+(Sqrt[2])) t)/c)]}/.c->1,t∈Integers]

Plot[-2 ArcTan[(t/c)]+ArcTan[(((-1+(Sqrt[2])) t)/c)]-ArcTan[(((1+(Sqrt[2])) t)/c)]/.c->1,{t,-5,5}]

v6

TangentialAngle[RightStrophoid[c][t],t]

TangentialAngle[RightStrophoid[c][t],t,Assumptions->c>0&&-Pi/2<t<Pi/2]

FullSimplify[%,c>0&&-Pi/2<t<Pi/2]

ϕ=(1/8) (6 (Sqrt[2]) ArcCot[(((1+(Sqrt[2])) c)/t)]-6 (Sqrt[2]) ArcCot[((c-(Sqrt[2]) c)/t)]-16 ArcTan[(t/c)]+I ((-4+3 (Sqrt[2])) Log[I c+t-(Sqrt[2]) t]-(4+3 (Sqrt[2])) Log[I c+t+(Sqrt[2]) t]+(4-3 (Sqrt[2])) Log[I c+(-1+(Sqrt[2])) t]+(4+3 (Sqrt[2])) Log[I c-(1+(Sqrt[2])) t]));

ExpToTrig[ϕ]//FullSimplify

%/.c->1

Plot[(1/8) (-6 (Sqrt[2]) ArcCot[((1-(Sqrt[2]))/t)]+6 (Sqrt[2]) ArcCot[((1+(Sqrt[2]))/t)]-16 ArcTan[t]+I ((-4+3 (Sqrt[2])) Log[I+t-(Sqrt[2]) t]-(4+3 (Sqrt[2])) Log[I+t+(Sqrt[2]) t]+(4-3 (Sqrt[2])) Log[I+(-1+(Sqrt[2])) t]+(4+3 (Sqrt[2])) Log[I-(1+(Sqrt[2])) t])),{t,-1,1}]

Plot[(1/8) (-6 (Sqrt[2]) ArcCot[((1-(Sqrt[2]))/t)]+6 (Sqrt[2]) ArcCot[((1+(Sqrt[2]))/t)]-16 ArcTan[t]+I ((4-3 (Sqrt[2])) Log[((I+(-1+(Sqrt[2])) t)/(I+t-(Sqrt[2]) t))]+(4+3 (Sqrt[2])) Log[((I-(1+(Sqrt[2])) t)/(I+t+(Sqrt[2]) t))])),{t,-1,1}]

FullSimplify[(1/8) (-6 (Sqrt[2]) ArcCot[((1-(Sqrt[2]))/t)]+6 (Sqrt[2]) ArcCot[((1+(Sqrt[2]))/t)]-16 ArcTan[t]+I ((4-3 (Sqrt[2])) Log[((I+(-1+(Sqrt[2])) t)/(I+t-(Sqrt[2]) t))]+(4+3 (Sqrt[2])) Log[((I-(1+(Sqrt[2])) t)/(I+t+(Sqrt[2]) t))])),t∈Reals]

ExpToTrig[%]

FullSimplify[%]

-2 ArcTan[t/c]+ArcTan[((-1+Sqrt[2]) t)/c]-ArcTan[((1+Sqrt[2]) t)/c]/.c->1

Plot[-2 ArcTan[t]+ArcTan[(-1+(Sqrt[2])) t]-ArcTan[(1+(Sqrt[2])) t],{t,-1,1}]

Loop

Function[t,{c((c^2-(t^2))/((t^2)+c^2)),((  t((t^2)-c^2))/((t^2)+c^2))}]/@{-c,c}

x[t_]:=c((c^2-(t^2))/((t^2)+c^2))
y[t_]:=((  t((t^2)-c^2))/((t^2)+c^2))

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

Arc Length

v5

ArcLength[RightStrophoid[c,t],{t,-c,c},Assumptions->c∈Reals]//Timing

ArcLength[RightStrophoid[c,t],{t,-c,c},Assumptions->c>0]//Timing

FullSimplify[Last[%],c>0]

Log[1+Sqrt[2],17+12Sqrt[2]]//N

v5.1

ArcLength[RightStrophoid[c,t],{t,-c,c},Assumptions->c∈Reals]//Timing

ArcLength[RightStrophoid[c,t],{t,-c,c},Assumptions->c>0]//Timing

FullSimplify[Last[%],c>0]

Log[1+Sqrt[2],#]&/@{17+12Sqrt[2],3+2Sqrt[2]}//N

Area

Simplify[y[t]x'[t]-x[t]y'[t]]

Simplify[Integrate[((c ((((c^2)-(t^2)))^2))/((((c^2)+(t^2)))^2)),{t,-c,c}]]/2

Integrate[((1-u^2)/(1+u^2))^2,{u,0,1}]

Vector Properties

parm={-c,t} (t^2-c^2)/(t^2+c^2);
assum=c>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