Maltese Cross Cruve

Author

Eric W. Weisstein
December 28, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Plot

ContourPlot

ContourPlot[x y(x^2-y^2)-(x^2+y^2)==0,{x,-5,5},{y,-5,5},ContourStyle->Red,AspectRatio->Automatic,Axes->True,ImageSize->Small]

ParametricPlot

ParametricPlot[2{Cos[t],Sin[t]}/Sqrt[Sin[4t]],{t,0,2Pi},Ticks->None,PlotStyle->Red,ImageSize->Small]

Reduce[Sin[4t]==0&&0<t<2π,t]

ParametricPlot[2{Cos[t],Sin[t]}/Sqrt[Sin[4t]],{t,0,Pi/2},Ticks->None,PlotStyle->Red,ImageSize->Small,Exclusions->{0}]

ParametricPlot[2{Cos[t],Sin[t]}/Sqrt[Sin[4t]],{t,0,2Pi},Ticks->None,PlotStyle->Red,ImageSize->Small,Exclusions->Sin[4t]==0]

PolarPlot

PolarPlot[1/Sqrt[Sin[4t]],{t,0,2Pi},Ticks->None,PlotStyle->Red,ImageSize->Small]

Properties

r=a/Sqrt[Sin[4q]];

curve=2a/Sqrt[Sin[4 t]]{Cos[t],Sin[t]};

AlgebraicDegree

Entity["PlaneCurve","MalteseCrossCurve"]["AlgebraicDegree"]

ArcLengthFunction

ArcLength[r,t]

CartesianEquation

Entity["PlaneCurve","MalteseCrossCurve"]["CartesianEquation"][a][x,y]

eqn=x y(x^2-y^2)==(a^2)(x^2+y^2);

Expand[Subtract@@eqn]

AlgebraicDegree[eqn,{x,y}]

Classes

Entity["PlaneCurve","MalteseCrossCurve"]["Classes"]

Curvature

FullSimplify[Curvature[curve,t],a>0&&t∈Reals]

ImplicitCurvature

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

Show[{ContourPlot[(2 ((x^2) (-4+5 (x^4))-2 (x^3) (4+3 (x^4)) y+(-4+15 (x^4)) (y^2)+2 x (4-3 (x^4)) (y^3)+15 (x^2) (y^4)+6 (x^3) (y^5)+5 (y^6)+6 x (y^7)))/((((x^6)-16 (x^3) y+4 (y^2)+3 (x^4) (y^2)+16 x (y^3)+(y^6)+(x^2) (4+3 (y^4))))^(3/2)),{x,-2,2},{y,-2,2},ColorFunction->Hue],
PolarPlot[1/Sqrt[Sin[4t]],{t,0,2Pi},Ticks->None,PlotStyle->Red,PlotRange->2]}]

ImplicitSlope

FullSimplify[Slope[eqn,{x,y}],a>0]//Union

PolarCurvature

FullSimplify[Curvature[r,q],a>0&&q∈Reals]

PolarEquation

Entity["PlaneCurve","MalteseCrossCurve"]["PolarEquation"][a][r,θ]

Polar

Solve[Entity["PlaneCurve","MalteseCrossCurve"]["CartesianEquation"][a][x,y]/.{x->r Cos[θ],y->r Sin[θ]},r]//FullSimplify

PolarSlope

FullSimplify[Slope[r,q],a>0&&q∈Reals]

Slope

FullSimplify[Slope[curve,t],a>0]

SpecialAffineCurvature

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

FullSimplify[((-567-588 ((Cot[4 t])^2)-80 ((Cot[4 t])^4)-64 ((Cot[4 t])^6))/((a^(4/3)) ((Csc[4 t])^2) (((22-6 Cos[8 t]))^(2/3)) (((7+4 ((Cot[4 t])^2)))^2)))]

Plot[Csc[4 t],{t,0,2π}]

Limit[Csc[4 t],t->0,Direction->1]

TangentialAngle

TangentialAngle[curve,t]

Vector Properties

parm=2a/Sqrt[Sin[4 t]]{Cos[t],Sin[t]};
assum=0<t<2π&&a>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]]