Deltoid

Author

Eric W. Weisstein
October 27, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equation

Cartesian

eqn=FullSimplify[Eliminate[{
x==TrigExpand[a(2/3 Cos[t]+1/3 Cos[2t])],
y==TrigExpand[a(2/3 Sin[t]-1/3 Sin[2t])],
Sin[t]^2+Cos[t]^2==1},
{Sin[t],Cos[t]}]]

FullSimplify[Subtract@@eqn]

eqn=Expand[%]

AlgebraicDegree[eqn,{x,y}]

Polar

deltoid={2/3 Cos[t]+1/3 Cos[2t],2/3 Sin[t]-1/3 Sin[2t]};

FullSimplify[deltoid.deltoid]

FullSimplify[1/Divide@@deltoid]

Together[TrigExpand[%]]

Solve[Tan[θ]==%,t]

ImplicitRegion (V10)

R=ImplicitRegion[(a-x)^3 (a+3 x)-6 (a^2+4 a x+x^2) y^2-3 y^4>=0,{x,y}]/.a->1

{RegionDimension[R],RegionEmbeddingDimension[R]}

ParametricRegion (V10)

P=ParametricRegion[{{2/3 Cos[t]+1/3 Cos[2t],2/3 Sin[t]-1/3 Sin[2t]},0<=t<2π},{t}]

{RegionDimension[P],RegionEmbeddingDimension[P]}

Plot

Parametric

ParametricPlot[
{2/3 Cos[t]+1/3 Cos[2t],2/3 Sin[t]-1/3 Sin[2t]},{t,0,2Pi},AspectRatio->Automatic,Ticks->None,PlotStyle->Red,ImageSize->200]

ImplicitRegion

Region`SemialgebraicRegionPlot[R,{{-1,1.5},{-1.,1}}]

ParamericRegion

Region`SemialgebraicRegionPlot[P,{{-1,1.5},{-1.,1}}]

Animation

Show[HypocycloidFrames[{3,1},Table[x,{x,0,2Pi-.01,2Pi/5}]],
ImageSize->200]

Show/@(movie=Hypocycloid[{3,1},{0,2Pi-.001,2Pi/40.}])

<<Utilities`GIF`

Export[gifs<>"deltoid.gif",movie]//Timing

Properties

deltoid={2/3 Cos[t]+1/3 Cos[2t],2/3 Sin[t]-1/3 Sin[2t]};

ArcLength

ArcLength[deltoid,{t,0,2π}]

MyArcLength[deltoid,{t,0,2π}]

Integrate ParametricRegion

RegionMeasure[P]

Integrate[1,t∈P]

ArcLengthFunction

V4.2

FullSimplify[ArcLength[deltoid,t],0<t<2π/3]

V5.0

FullSimplify[ArcLength[deltoid,t]]

FullSimplify[ArcLength[deltoid,t],0<t<2π/3]

V8

ArcLength[deltoid,t]

FullSimplify[%,0<t<2π/3]

FunctionExpand[ConditionalExpression[(16/9) ((Sin[((3 t)/4)])^2),Cos[3 t]<1],0<t<2π/3]

Assuming[0<t<2π/3,ConditionalExpression[(16/9) ((Sin[((3 t)/4)])^2),Cos[3 t]<1]]

Plot[Cos[3t],{t,0,2π/3}]

Area

Green's Theorem

Area[deltoid,{t,0,2π}]

N[%]

Numerical

With[{a=1},RegionPlot[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,-.6,1},{y,-.9,.9},MaxRecursion->5,ImageSize->150,AspectRatio->Automatic]]

With[{a=1},NIntegrate[Boole[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4)],{x,-.6,1},{y,-.9,.9}]]//Chop

Inequality

Assuming[a>0,Area[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,y}]]//Timing

Assuming[a>0,Area[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{y,x}]]//Timing

With[{a=1},(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)-3 (y^4)]

NIntegrate[Boole[(((1-x))^3) (1+3 x)>6 (1+4 x+(x^2)) (y^2)+3 (y^4)],{x,-.6,1},{y,-.9,.9}]//Chop

RegionMeasure

RegionMeasure[R]//Simplify//Timing

Integrate ImplicitRegion

Integrate[1,{x,y}∈R]//Simplify//Timing

AreaInertiaTensor

Parametric

AreaInertiaTensor[deltoid,{t,0,2π}]

Numerical

With[{a=1},AreaInertiaTensor[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,y}]]//N

Inequality

Assuming[a>0,AreaInertiaTensor[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,y}]]//Timing

With[{a=1},AreaInertiaTensor[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,y}]]//Timing

Assuming[a>0,AreaInertiaTensor[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{y,x}]]//Timing

With[{a=1},AreaInertiaTensor[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{y,x}]]//Timing

Centroid

Parametric

Centroid[deltoid,{t,0,2π}]

Inequality

Assuming[a>0,Centroid[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,y}]]//Timing

With[{a=1},Centroid[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{x,y}]]//Timing

N[%138]//Chop

Assuming[a>0,Centroid[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{y,x}]]//Timing

With[{a=1},Centroid[(((a-x))^3) (a+3 x)>6 ((a^2)+4 a x+(x^2)) (y^2)+3 (y^4),{y,x}]]//Timing

N[%140]//Chop

RegionCentroid (V10)

RegionCentroid[R]//Simplify//Timing

Convex

ConvexRegionQ[R]

Curvature

FullSimplify[Curvature[deltoid,t],0<t<2π]

FullSimplify[Curvature[deltoid,t],0<t<2π/3]

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

ByteCount/@%

Singular Points

SingularPoints[eqn,{x,y}]//Union

Slope

Slope[{2/3 Cos[t]+1/3 Cos[2 t],2/3 Sin[t]-1/3 Sin[2 t]},t]//FullSimplify

Slope[(a-x)^3 (a+3 x)-6 (a^2+4 a x+x^2) y^2-3 y^4==0,{x,y}]//FullSimplify

Slope[(a-x)^3 (a+3 x)-6 (a^2+4 a x+x^2) y^2-3 y^4==0,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

V4.2

TangentialAngle[deltoid,t]

V5.0.1

TangentialAngle[deltoid,t]

V8

TangentialAngle[deltoid,t]

Vector Properties

parm=a {2/3 Cos[t]+1/3 Cos[2 t],2/3 Sin[t]-1/3 Sin[2 t]};
assum=a>0&&0<t<2π;
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]]