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

(* Minimum packages you need available on MathSource *)

Needs["ParaPlot`"]
(*MathSource Item:0207-986*)

Needs["PlaneCurveFormulas`"]


Needs["PlaneCurveGenerator`"]

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

Pedal, Negative Pedal

Pedal of a given curve C and a point O can be generated by the following steps:

1.Give a curve and a fixed point O. 
2.Draw a tangent at any point P on the curve. 
3.Mark a point Q on this tangent so that line PQ and line OQ are penpendicular. 
4.Repeat step (2) and (3) for every point P on the curve. The locus of Q is the pedal of the given curve with respect to point O.

Pedal of a Cosine.

ParaPlot[{t, -Cos[t]}, {t, -2 π, 2 π, Pi/34}, 
        PedalPoint -> {0, -3}, PlotDot -> False, 
        CurveColorFunction -> {Hue[ .68,  .8, 1] &}, 
        DotStyle -> {{Hue[0], PointSize[ .01]}}, 
        LineStyle -> {{Thickness[ .003], 
              RGBColor[Random[], Random[], Random[]]}}, Axes -> False, 
        AspectRatio -> Automatic];

Pedal and negative pedal are inverse concepts. Negative pedal of a curve C can be defined as a curve C' such that the pedal of C is C'. 

Step by step description: 

    1.Give a curve and a fixed point O. 
    2.Draw a line from O to any point P on the curve. 
    3.Draw a second line perpendicular to line OP and passing P. 
    4.Repeat step (2) and (3) for every point P on the curve. The envelope of lines is the negative pedal of the given curve with respect to point O. 

For example, the Pedal of a parabola with respect to its focus is a line.
The negative pedal of a line is then a parabola.

Clear[unitV];

ParaPlot[
	{t, 1/4 t^2}, {t, -6,6,.5},
	PedalPoint->{0,1},
	CurveColorFunction->{Hue[.67,.6,1]&},
	CurveStyle->Thickness[.005],
	LineStyle->{GrayLevel[1]},
	PlotRange->{{-1,1} 3.1,{-.5,2.5}},
	Axes->False,
	Background->Hue[0,0,0]
];

unitV[v_]:= v/Sqrt[v.v]

Show[
	Graphics[{
		{Hue[.75], Thickness[.007], Line[{{-10,0},{10,0}}]},
		{	Line[{{0,1},#}],
			Line[{
				50 Reverse[unitV[N[#-{0,1}]]{1,-1}] +#,
				-50 Reverse[unitV[N[#-{0,1}]]{1,-1}] +#
			}],
			{Hue[0],PointSize[.02], Point[#]}
		}& /@ Table[{i,0},{i,-4,4,.25}],
		{Hue[.7],PointSize[.02], Point[{0,1}]}
	}],
	PlotRange-> {{-1,1} 3.1,{-.5,2.5}}
];
Clear[unitV];

Family of pedals of Cosine.

\(\(Table[
        Pedal[{t, \(-Cos[t]\)}, t, {0, \(-i\)}], {i, 0, 
          10}];\)
\)
ParaPlot[Evaluate@Join[%, {{t, Cos[t]}}], {t, -3 Pi, 3 Pi}, 
        AspectRatio -> Automatic, 
        PlotRange -> {{-2 Pi, 2 Pi} 1.2, Automatic}, 
        PlotStyle -> 
          Join[Table[{Hue[i], Thickness[ .005]}, {i, 0, 1, 1/10}], {GrayLevel[
                1]}], Background -> Hue[0, 0, 0]];