Negative Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Derivation

The line segment through (f, g) perpendicular to the (0, 0)-(f, g) line.

{x1,y1}={f[t],g[t]};
{x2,y2}={f[t],g[t]}+{-(g[t]-y0),f[t]-x0};

{x2,y2}

eqn=y-y1-(y2-y1)/(x2-x1)(x-x1)

The equation of the envelope

FullSimplify[Solve[{eqn==0,D[eqn,t]==0},{x,y}]]

{y->((((((x0-f(t)))^2)-((g(t))^2)+y0 g(t)) f'(t)+(x0-f(t)) (y0-2 g(t)) g'(t))/((y0-g(t)) f'(t)+(f(t)-x0) g'(t))),x->-(((x0-2 f(t)) (y0-g(t)) f'(t)+((((y0-g(t)))^2)+(x0-f(t)) f(t)) g'(t))/((y0-g(t)) f'(t)+(f(t)-x0) g'(t)))}

NegativePedalCurve[{f[t],g[t]},{(Subscript[x, 0]),(Subscript[y, 0])},t]/.{g[t]->g,f[t]->f,g'[t]->g',f'[t]->f'}//TraditionalForm

With[{f=Function[t,Sin[t]/2],g=Function[t,Cos[t]],
x0=0.,y0=1/2.
},Show[
{
ParametricPlot[
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])},{t,0,2Pi}],
Graphics[Line/@Table[{{f[t],g[t]},{f[t],g[t]}+{-(g[t]-y0),f[t]-x0}},{t,0,2Pi,.1}]]
},AspectRatio->Automatic]
]

With[{f=Function[t,Sin[t]/2],g=Function[t,Cos[t]],
x0=0.,y0=1/2.
},
ParametricPlot[
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])},{t,0,2Pi},
Ticks->False,AspectRatio->Automatic]
]

<<Graphics`Colors`

With[{f=Function[t,Cos[t]],g=Function[t,Sin[t].5],
x0=0,y0=0
},
ParametricPlot[
{{f[t],g[t]},
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])}
},{t,0,2Pi},
PlotStyle->{Black,Red},
Ticks->False,AspectRatio->Automatic]
]

With[{f=Function[t,Cos[t]],g=Function[t,Sin[t].7],
x0=Sqrt[1-.7^2],y0=0
},
ParametricPlot[
{{f[t],g[t]},
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])}
},{t,0,2Pi},
PlotStyle->{Black,Red},
Ticks->False,AspectRatio->Automatic]
]

With[{f=Function[t,Cos[t]],g=Function[t,Sin[t]/2],
x0=1/2,y0=0
},
ParametricPlot[
{{f[t],g[t]},
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])}
},{t,0,2Pi},
Ticks->False,AspectRatio->Automatic]
]

With[{f=Function[t,Cos[t]],g=Function[t,Sin[t]/2],
x0=1,y0=0
},
ParametricPlot[
{{f[t],g[t]},
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])}
},{t,0,2Pi},
Ticks->False,AspectRatio->Automatic]
]

With[{f=Function[t,Cos[t]],g=Function[t,Sin[t]/2],
x0=1.05,y0=0
},
ParametricPlot[
{{f[t],g[t]},
{-((x0-2 f[t]) (y0-g[t]) f'[t]+((x0-f[t]) f[t]+(((y0-g[t]))^2)) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t]),
(((((x0-f[t]))^2)+y0 g[t]-((g[t])^2)) f'[t]+(x0-f[t]) (y0-2 g[t]) g'[t])/((y0-g[t]) f'[t]+(-x0+f[t]) g'[t])}
},{t,0,2Pi},
Ticks->False,AspectRatio->Automatic]
]