Dürer Folium

Author

Eric W. Weisstein
August 24, 2021

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

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

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

Properties

ArcLength

MyArcLength[Sin[q/2],{q,0,4Pi}]

FullSimplify[Sqrt[Sin[q/2]^2+D[Sin[q/2],q]^2]]

Integrate[((Sqrt[5-3 Cos[q]])/(2 (Sqrt[2]))),{q,0,4π}]

N[%]

ArcLengthFunction

Assuming[q>0,MyArcLength[Sin[q/2],q]]

???

Integrate[((Sqrt[5-3 Cos[q]])/(2 (Sqrt[2]))),q]

%/.q->4π

Area

p1=ParametricPlot[Sin[(1/2) t] {Cos[t],Sin[t]},{t,0,π/2},PlotStyle->Dashed];

p2=ParametricPlot[Sin[(1/2) t] {Cos[t],Sin[t]},{t,π/2,π}];

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

4(a^2)(1/2)Integrate[((Sin[(1/2) t])^2),{t,π/2,π}]

CartesianFunction

ParametricPlot[Sin[(1/2) t] {Cos[t],Sin[t]},{t,0,4π}]

x==a(Sqrt[((1-Cos[t])/2)])Cos[t]
y==a(Sqrt[1-((Cos[t])/2)])Sin[t]

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

FullSimplify[-4 (a^2) (x^4)+4 (x^6)+(a^4) (y^2)-8 (a^2) (x^2) (y^2)+12 (x^4) (y^2)-4 (a^2) (y^4)+12 (x^2) (y^4)+4 (y^6)]

FullSimplify[(x^2+y^2)(2(x^2+y^2)-a^2)^2-a^4x^2]

Curvature

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

ImplicitCurvature

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

ImplicitSlope

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

NormalVector

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

ParametricEquations

Block[{a=1,n=1/2},ParametricPlot[a Sin[n t] {Cos[t],Sin[t]},{t,0,4π}]]

PolarCurvature

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

PolarSlope

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

PolarEquations

Block[{a=1,n=(1/2)},PolarPlot[a Sin[n t],{t,0,4π},PlotStyle->Red,Ticks->None]]

Slope

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

SpecialAffineCurvature

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

TangentVector

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

VectorLength

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