Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2000 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["ParaPlot`"]
(* MathSource Item#:/Enhancements/Graphics/2D/0207-986 *)

Needs["PlaneCurveFormulas`"]
(* Not available on MathSource as of 1996/02/01 *)

Needs["PlaneCurveGenerator`"]
(* Not available on MathSource as of 1996/02/01 *)

Needs["Graphics`ImplicitPlot`"]
(* MathSource Item#:/Enhancements/Graphics/2D/0200-529 *)

Deltoid

Description

Deltoid (aka Tricuspid) is a special case of Hypocycloid and Hypotrochoid. (see Hypotrochoid)
Parametric formula: {(2 Cos[t])/3 + Cos[2 t]/3, (2 Sin[t])/3 - Sin[2 t]/3},
0 < t <= 2 Pi.
Cartesian equation: (x^2+y^2)^2-8 x (x^2 - 3 y^2) + 18 (x^2 + y^2) - 27 == 0.

(* secants of deltoid *)
ParaPlot[
	Rotate[Deltoid[][t], -Pi/2],
	{t,0, 2 Pi, 2 Pi/120},
	SecantConnection->{60,1},
	PlotDot->False, PlotCurve->False,
	Axes->False,
	AspectRatio->Automatic
];

DeltoidGenerator[ {0, 5.4, .05}, LastFrameOnly->True];

ParaPlot[ Deltoid[][t], {t,0, 2 Pi}, AspectRatio->Automatic];

ImplicitPlot[
	Scale[
		(x^2+y^2)^2-8 x (x^2 - 3 y^2) + 18  (x^2 + y^2) - 27 == 0,
		{x,y},
		{1/3,1/3}
	],
	{x,-1,1}
];

Properties

Deltoid and Astroid

The Caustic of a Deltoid with rays at Infinity is an Astroid.
The following cell generate a animation of the caustic of an deltoid, with parallel rays. The parallel rays rotate 180 degrees. Change the value of step to adjust the number of frames.

Clear[ gp1, gp2, lightSource, d, p, step];

step = 5;
d = N@ Degree;
p = N[2 Pi/12];

Do[
lightSource = 20000 {Cos[#], Sin[#]}& [theta];
gp1 = First@ ParaPlot[ Deltoid[][t],
	{t,0, 2 Pi},
	Which[
		N[0 <= theta < 30 d],
			Range@@{-4 p+.001, 4 p-.001, .1},
		N[30 d <= theta < 90 d],
			Range@@{0 +.001, 4 p-.001, .1},
		N[90 d <= theta < 150 d],
			Range@@{0 +.001, 8 p-.001, .1},
		N[150 d <= theta <= 180 d],
			Range@@{4 p+.001, 8 p-.001, .1}
	],
	CausticLineLength->30000,
	CausticOrigin->lightSource,
	CausticLineStyle->{Hue[0]},
	PlotCurve->False,
	PlotDot->False,
	CurveColorFunction->(GrayLevel[1]&),
	AspectRatio->Automatic,
	PlotRange->{{-1.5,1.6},{-1.6,1.6}},
	DisplayFunction->Identity
];

gp2 = First@ ParaPlot[ 
	Deltoid[][t],
	{t,0 , 2 Pi},
	Which[
		N[0 <= theta < 30 d],
			Range@@{4 p+.001, 8 p-.001, .1},
		N[30 d <= theta < 90 d],
			Range@@{4 p+.001, 12 p-.001, .1},
		N[90 d <= theta < 150 d],
			Range@@{-4 p +.001, 0-.001, .1},
		N[150 d <= theta <= 180 d],
			Range@@{-4 p+.001, 4 p-.001, .1}
	],
	CausticLineLength->30000,
	CausticOrigin-> -lightSource,
	PlotCurve->False,
	CausticLineStyle->{Hue[0]},
	PlotCurve->False,
	PlotDot->False,
	CurveColorFunction->(GrayLevel[1]&),
	AspectRatio->Automatic,
	PlotRange->{{-1.5,1.6},{-1.6,1.6}},
	DisplayFunction->Identity
];

Show[ Graphics[{gp1, gp2}],
	PlotRange->{{-2,2},{-2,2}},
	Axes->False,
	AspectRatio->Automatic,
	PlotRange->{{-1.5,1.6},{-1.6,1.6}},
	Background->Hue[0,0,0],
	DisplayFunction->$DisplayFunction
];
, {theta, 0, 180 d- 180d/step, 180 d/step}
]

Clear[ gp1, gp2, lightSource, d, p, step]

Evolute

The Evolute of a Deltoid is another Deltoid.

ParaPlot[ Deltoid[][t], {t, 0, 2 Pi, 2 Pi/(3 20)},
	PlotEvolute->True, AspectRatio->Automatic];

Clear[myCurves];

myCurves = NestList[ Evolute[#,t]&, Deltoid[][t], 2];

ParaPlot[ myCurves, {t,0, 2 Pi},
	PlotRange->All, Axes->False, AspectRatio->Automatic
];
Clear[myCurves];

Clear[gp, gp2]
gp = First@ ParaPlot[ 
	Rotate[ Deltoid[][t], Pi/2],
	{t, 0, 2 Pi, 2 Pi/(3 30)},
	PlotCurve->False, PlotDot->False,
	PlotOsculatingCircle->True,
	LineStyle->{RGBColor[Random[],Random[], Random[]]},
	CurveColorFunction->{Hue[0]&},
	DisplayFunction->Identity
];

gp2 = First@ ParaPlot[ 
	Rotate[ Deltoid[][t], Pi/2],
	{t, 0, 2 Pi, 2 Pi/(3 30)},
	PlotCurve->False,
	PlotEvolute->True,
	LineStyle->{GrayLevel[0]},
	DisplayFunction->Identity
];
gp2 = DeleteCases[gp2, Line[___], -1];

Show[
	Graphics[ {gp,gp2}
	],
	Background->Hue[0,0,0],
	PlotRange->All,
	AspectRatio->Automatic,
	Axes->False
];

Clear[gp,gp2]

Inversion

The Inversion of Deltoid is Cotes' Spiral.

ParaPlot[ Deltoid[][t], {t, 0, 2 Pi, 2 Pi/(3 100)},
	InversionCircle->{{0,0},1},
	PlotDot->False,
	CurveColorFunction->{Hue[.7,.9,.9]&},
	AspectRatio->Automatic ];

ParaPlot[ 
	{ Deltoid[][t], Inversion[ Deltoid[][t], {{0,0},.5}] },
	{t, 0, 2 Pi},
	PlotStyle->{{Hue[.65]}, {Hue[.0], Thickness[.005]}},
	Epilog->{Hue[.18], Circle[{0,0},.5]},
	AspectRatio->Automatic,
	Ticks->{{-1,-.5,.5,1},{-1,-.5,.5,1}}
];

Pedal

Deltoid's pedal is trifolium.

(* deltoid normal beauty *)
ParaPlot[
	Rotate[Deltoid[][t], -Pi/2],
	{t,0, 2 Pi, 2 Pi/(30 3)},
	PedalPoint->{0,0},
	PlotDot->False, PlotCurve->False,
	Axes->False,
	AspectRatio->Automatic
];

Radial

Deltoid's radial is a trifolium.

ParaPlot[
	Rotate[Deltoid[][t], -Pi/2],
	{t,0, 2 Pi, 2 Pi/(30 3)},
	PlotRadial->True,
	PlotDot->False,
	CurveColorFunction->{Hue[0,0,0]&},
	LineStyle->{RGBColor[Random[],Random[],Random[]]},
	Axes->False,
	AspectRatio->Automatic
];

Simson Lines

Deltoid is the Envelope of Simson Lines of any fixed triangle. (Robert Simson, 1687-1768)

Explaination:
(1) Let there be a triangle inscribed in a circle.
(2) For any point P on the circle, draw a point Q1 such that the line P Q1 is perpendicular to one side of the triangle.  Extend the side if necessary.
(3) Similarly, find point Q2 and Q3 with respect to other sides.
(4) The points Q1, Q2, and Q3 are colinear. The line passing them is called Simson line of the triangle with respect to P on the circumscribing circle.
(5) Repeat the process on other point P on the circle. The Envelope of Simson lines is the Deltoid.

DeltoidGenerator2[{1, 3, 3.5}, {0,  5.2,  5/30}, 
        LastFrameOnly -> True];

DeltoidGenerator2[{1, 3, 4.8}, {0,  5.7,  5/30}, 
        LastFrameOnly -> True];

DeltoidGenerator2[{1, 3, 4}, {2, 6}, LastFrameOnly -> True, 
        NumberOfFrames -> 1];

The derivatives of deltoid seems to be the other hypo/epi trochoids.

ParaPlot[ 
        Evaluate@{D[Deltoid[][t], {t, 2}], 
            D[Deltoid[][t], {t, 1}], Deltoid[][t]}, {t, 0,  
          2 π}, AspectRatio -> Automatic];