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:0207-986*)

Needs["PlaneCurveFormulas`"](*0208-482*)

Needs["PlaneCurveGenerator`"](*0208-314*)

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

$HistoryLength = 10;

Nephroid

Description

Nephroid is a special case of Epitrochoid.

Nephroid[][t]

NephroidGenerator[{0,5.8,.05}, LastFrameOnly->True];

ParaPlot[ Nephroid[][x], {x,0, 2 Pi},AspectRatio->Automatic];

ImplicitPlot[(((-4 + (x^2) + (y^2)))^3) == 108 (x^2), {x, -5, 
          5}];

Properties

Catacaustic of a circle

It is a Catacaustic of a circle with light rays parallel to a line that connects its two cusps.

ParaPlot[ {Cos[t],Sin[t]},
	{t, Pi+.00001, 2 Pi+.0001, Pi/40},
	CausticLineLength->5100, CausticOrigin->{0,5000},
	PlotDot->False,
	PlotCurve->False
	CurveColorFunction->{Hue[.7]&},
	CurveStyle->Thickness[.007],
	CausticLineStyle->{
		{RGBColor[Random[],Random[], Random[]]},
		{RGBColor[Random[],Random[], Random[]]}
	},
	AspectRatio->Automatic,
	PlotRange->{{-1.5,1.5},{-2,.5}},
	Axes->False,
	Background->Hue[0,0,0]
];

Catacaustic of a cardioid

Nephroid is the catacaustic of a cardioid with a luminous cusp.

ParaPlot[ Cardioid[][t],
	{t, .00001, 2 Pi, Pi/30},
	CausticLineLength->51,
	CausticOrigin->{-1,0},
	PlotDot->False,
	PlotCurve->True,
	CurveColorFunction->{Hue[.7,.5,.9]&},
	CurveStyle->Thickness[.007],
	CausticLineStyle->{{RGBColor[Random[],Random[], Random[]]}},
	AspectRatio->Automatic,
	PlotRange->{{-2,3.5},{-1,1}3},
	Axes->False
];

Envelope of circles

It is the Envelope of a set of circles with centers on a given circle C and each member of the set is tangent to a chord that bisect C.

Clear[circles];

circles =
	Table[ Circle[{Cos@i, Sin@i}, Abs@ Cos@ i ], {i, -Pi, Pi, 2 Pi/30} ];

Show[ Graphics[ {Hue[Random[],1,.7], circles}],AspectRatio->Automatic];
Clear[ circles];

Evolute is itself

Its Evolute is itself rotated Pi/2 radians and scaled by 1/2. (see Epitrochoid)

ParaPlot[ Nephroid[][t], {t, 0, 2 Pi, 2 Pi/50},
	PlotEvolute->True,
	PlotOsculatingCircle->True,
	PlotDot->True,
	PlotCurve->False,
	CurveColorFunction->{Hue[0,0,0]&},
	LineStyle->{RGBColor[Random[],Random[], Random[]]},
	AspectRatio->Automatic,
	Axes->False
];

ParaPlot[ Nephroid[][t], {t, 0, 2 Pi, 2 Pi/60},
	NormalLineLength->20,
	PlotDot->True,
	PlotCurve->False,
	CurveColorFunction->{Hue[0,0,0]&},
	LineStyle->{RGBColor[Random[],Random[], Random[]]},
	AspectRatio->Automatic,
	PlotRange->{{-2,2}1.1,{-2,2}},
	Axes->False
];

pedal and rose

The pedal of a nephroid with respect to its center is the rose r==Cos[1/2*theta].

ParametricPlot[
        Evaluate@(2*
              Cos[1/2*θ]*{Cos@θ, Sin@θ}), {θ, 
          0, 4*Pi}, PlotStyle -> Hue[0], 
        Epilog -> (DeleteCases[#, _Point, Infinity] &)@First@
              ParaPlot[Nephroid[][t], {t, 0, 2*Pi}, 
                Table[i, {i, 0, Pi/2, Pi/2/20}], PedalPoint -> {0, 0}, 
                PlotDot -> False, PlotCurve -> True, 
                LineStyle -> {GrayLevel[0]}, 
                CurveColorFunction -> (Hue[ .7] &), 
                DisplayFunction -> Identity], AspectRatio -> Automatic, 
        Axes -> False];

Graphics Gallery

Pedal of nephroid

A pedal of nephroid. Try different pedal points

ParaPlot[ Rotate[Nephroid[][t],0],
	{t, 0.0001, 2 Pi, 2 Pi/160},
	PlotEvolute->False,
	PlotOsculatingCircle->False,
	PedalPoint->{0,2},
	PlotDot->True,
	PlotCurve->True,
	CurveColorFunction->{Hue[.7]&},
	LineStyle->{RGBColor[Random[],Random[], Random[]]},
	AspectRatio->Automatic,
	Axes->False
];