Rectellipse

Author

Eric W. Weisstein
February 20, 2012

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

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

©2012 Wolfram Research, Inc. except for portions noted otherwise

Generalization of Wikipedia Definition

Block[{a=1,b=.8},ContourPlot[((x^4)/(a^4))+((y^4)/(b^4))==1^4,{x,-1,1},{y,-.8,.8},Frame->False,Axes->True,Ticks->None,ContourStyle->Red,AspectRatio->Automatic]]

(((r(Sqrt[Cos[t]])))^4)+(((r(Sqrt[Sin[t]])))^4)//Simplify

{
ParametricPlot[{Sqrt[Cos[t]],Sqrt[Sin[t]]},{t,0,π/2}],
ParametricPlot[{-Sqrt[Cos[t]],Sqrt[Sin[t]]},{t,0,π/2}],
ParametricPlot[{-Sqrt[Cos[t]],-Sqrt[Sin[t]]},{t,0,π/2}],
ParametricPlot[{Sqrt[Cos[t]],-Sqrt[Sin[t]]},{t,0,π/2}]
}

Properties

Inequality

ineq[x_,y_]:=((x^4)/(a^4))+((y^4)/(b^4))≤1

Arc Length Function

Arc Length

Solve[((x^4)/(a^4))+((y^4)/(b^4))==1,y]

FullSimplify[(Sqrt[1+D[((b ((((a^4)-(x^4)))^(1/4)))/a),x]^2]),0<x<a&&b>0]

4Integrate[(Sqrt[1+(((b^2) (x^6))/((a^2) ((((a^4)-(x^4)))^(3/2))))]),{x,0,1}]

Integrate[(Sqrt[1+(((b^2) (x^6))/((a^2) ((((a^4)-(x^4)))^(3/2))))]),x]

Area

Block[{a=1,b=.8},RegionPlot[((x^4)/(a^4))+((y^4)/(b^4))≤1,{x,-a,a},{y,-b,b},AspectRatio->Automatic]]

Assuming[a>0&&b>0,FullSimplify[Area[ineq[x,y],{x,y}]]]//Timing

4Assuming[a>0&&b>0,Integrate[((b ((((a^4)-(x^4)))^(1/4)))/a),{x,0,a}]]

Area Moment of Inertia

Assuming[a>0&&b>0,AreaInertiaTensor[ineq[x,y],{x,y}]]//Timing

Centroid

Assuming[a>0&&b>0,Centroid[ineq[x,y],{x,y}]]//Timing

Curvature

SortBy[FullSimplify[Curvature[((x^4)/(a^4))+((y^4)/(b^4))-1,{x,y}],Assumptions->a>0&&b>0],ByteCount]

RadiiOfGyration

Assuming[a>0&&b>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing

Slope

Slope[x^4/a^4+y^4/b^4==1,{x,y}]//FullSimplify

SpecialAffineCurvature

Tangential Angle

Gusati Ellipse

With[{s=.5,a=1,b=.8},RegionPlot[s x^2/a^2 y^2/b^2-(x^2/a^2+y^2/b^2)>-1,{x,-2,2},{y,-2,2}]]

g1=Block[{a=1,b=.8},ContourPlot[Evaluate[(# ((x^2)/(a^2))((y^2)/(b^2))-(((x^2)/(a^2))+((y^2)/(b^2)))+1==0)&/@Range[0,1,.2]],{x,-1.01,1.01},{y,-1.01,1.01},ContourStyle->{Red,Orange,Yellow,Green,Blue,Purple},Frame->False,Axes->True,Ticks->None]]

g2=Block[{a=1,b=.8},ContourPlot[Evaluate[(# ((x^2)/(a^2))((y^2)/(b^2))-(((x^2)/(a^2))+((y^2)/(b^2)))+1==0)&/@Range[0,1,.2]],{x,-5,5},{y,-5,5},ContourStyle->{Red,Orange,Yellow,Green,Blue,Purple},Frame->False,Axes->True,Ticks->None]]

GraphicsRow[{g1,g2}]

Properties

Inequality

ineq[x_,y_]:=(s^2)((x^2)/(a^2))((y^2)/(b^2))-(((x^2)/(a^2))+((y^2)/(b^2)))+1>=0

Arc Length Function

Arc Length

Area

Block[{s=.8,a=1,b=.8},RegionPlot[(s^2)((x^2)/(a^2))((y^2)/(b^2))-(((x^2)/(a^2))+((y^2)/(b^2)))+1≥0,{x,-a,a},{y,-b,b},AspectRatio->Automatic]]

Block[{s=.8,a=1,b=.8},RegionPlot[(s^2)((x^2)/(a^2))((y^2)/(b^2))-(((x^2)/(a^2))+((y^2)/(b^2)))+1≥0,{x,-5a,5a},{y,-5b,5b},AspectRatio->Automatic]]

Assuming[a>0&&b>0&&0<s<1,FullSimplify[Integrate[Boole[ineq[x,y]],{x,-a,a},{y,-b,b}]]]//Timing

Limit[((4 a b EllipticE[ArcSin[s],(1/(s^2))])/s),s->#]&/@{0,1}

Area Moment of Inertia

AreaInertiaTensor[{x,y}]

Assuming[a>0&&b>0&&0<s<1,FullSimplify[Integrate[{{(y^2),-x y},{-x y,(x^2)}} Boole[ineq[x,y]],{x,-a,a},{y,-b,b}]]]//Timing

Centroid

Assuming[a>0&&b>0&&0<s<1,Integrate[{x,y}Boole[ineq[x,y]],{x,-a,a},{y,-b,b}]]//Timing

Curvature

Assuming[{a>0&&b>0&&0<s<1},SortBy[FullSimplify[Curvature[(s^2)((x^2)/(a^2))((y^2)/(b^2))-(((x^2)/(a^2))+((y^2)/(b^2)))+1,{x,y}]],ByteCount]]

RadiiOfGyration

Assuming[a>0&&b>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing

Slope

Slope[s x^2/a^2 y^2/b^2-(x^2/a^2+y^2/b^2)+1==0,{x,y}]//FullSimplify

SpecialAffineCurvature

Tangential Angle