Swastika Curve

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Equation

FullSimplify[(Sqrt[((Sin[θ]Cos[θ])/(((Sin[θ])^4)-((Cos[θ])^4)))]),0<θ<2π]

{eqn}=Factor/@GroebnerBasis[{r^2(((Sin[θ])^4)-((Cos[θ])^4))==Sin[θ]Cos[θ],x==r Cos[θ],y==r Sin[θ],Sin[θ]^2+Cos[θ]^2==1},{x,y},{r,Cos[θ],Sin[θ]}]

eqn=(x^4)+x y-(y^4);

Plot

r[t_]:=Abs[Sin[2 t]] (Sqrt[-Csc[4 t]])

V5

Implicit Cartesian

<<Graphics`ImplicitPlot`

ImplicitPlot[y^4-x^4==x y,{x,-1,1},{y,-1,1},PlotStyle->Red,Ticks->None,AxesOrigin->{0,0}]

Polar

<<Graphics`Graphics`

PolarPlot[(Sqrt[((Sin[θ]Cos[θ])/(((Sin[θ])^4)-((Cos[θ])^4)))]),{θ,0,2π},PlotStyle->Red,ImageSize->200,Ticks->None]

V6

PolarPlot[Abs[Sin[2 θ]] (Sqrt[-Csc[4 θ]]),{θ,0,2π},PlotRange->1,ImageSize->Small]

ContourPlot[(x^4)+x y-(y^4)==0,{x,-1,1},{y,-1,1},ImageSize->Small]

ParametricPlot[{Cos[t],Sin[t]}Abs[Sin[2 t]] Sqrt[-Csc[4 t]],{t,0,2π}]

Properties

curve={Cos[t],Sin[t]}Abs[Sin[2 t]] Sqrt[-Csc[4 t]];

Arc Length Function

ArcLength[r[t]/.Abs[x_]:>Sqrt[x^2],t]//Timing

Curvature

Paramateirc

FullSimplify[Curvature[r[t]/.Abs[x_]:>Sqrt[x^2],t],Assumptions->t>0]

Polar

abs[x_]:=Sqrt[x^2]

FullSimplify[Curvature[abs[Sin[2 q]] Sqrt[-Csc[4 q]],q],0<q<2π]

Implicit

FullSimplify[ImplicitCurvature[(x^4)+x y-(y^4),{x,y}],(x|y)∈Reas]

Inflection Points

pts=InflectionPoints[eqn,{x,y}]

ContourPlot[eqn==0,{x,-2,2},{y,-2,2},Epilog->{Point[pts]}]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[{Cos[t],Sin[t]}Sqrt[Sin[2 t]^2] Sqrt[-Csc[4 t]],t]//FullSimplify

Slope[Sqrt[Sin[2 q]^2] Sqrt[-Csc[4 q]],q]//FullSimplify

Slope[x^4+x y-y^4==0,{x,y}]//FullSimplify

SpecialAffineCurvature

ByteCount[k=SpecialAffineCurvature[{Cos[t],Sin[t]}Sqrt[Sin[2 t]^2] Sqrt[-Csc[4 t]],t]]//Timing

ByteCount[k2=Simplify[k,t∈Reals]]//Timing

k2

FullSimplify[k2,0<t<2π]

Tangential Angle

FullSimplify[TangentialAngle[r[t],t],Assumptions->t>0]

Vector Properties

parm={Cos[t],Sin[t]}Sqrt[Sin[2 t]^2] Sqrt[-Csc[4 t]];
assum=0<t<2π
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