Editing Pedal curve (section)

==Equations==

===From the Cartesian equation===
Take ''P'' to be the origin. For a curve given by the equation ''F''(''x'', ''y'')=0, if the equation of the [[tangent line]] at ''R''=(''x''<sub>0</sub>, ''y''<sub>0</sub>) is written in the form
:<math>\cos \alpha x + \sin \alpha y = p</math>
then the vector (cos α, sin α) is parallel to the segment ''PX'', and the length of ''PX'', which is the distance from the tangent line to the origin, is ''p''. So ''X'' is represented by the [[polar coordinates]] (''p'', α) and replacing (''p'', α) by (''r'', θ) produces a polar equation for the pedal curve.<ref>Edwards p. 164</ref>

[[Image:PedalCurve1.gif|500px|right|thumb|Pedal curve (red) of an [[ellipse]] (black). Here ''a''=2 and ''b''=1 so the equation of the pedal curve is 4''x''<sup>2</sup>+y<sup>2</sup>=(''x''<sup>2</sup>+y<sup>2</sup>)<sup>2</sup>]]
For example,<ref>Follows Edwards p. 164 with ''m''=1</ref> for the ellipse
:<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>
the tangent line at ''R''=(''x''<sub>0</sub>, ''y''<sub>0</sub>) is
:<math>\frac{x_0x}{a^2}+\frac{y_0y}{b^2}=1</math>
and writing this in the form given above requires that
:<math>\frac{x_0}{a^2}=\frac{\cos \alpha}{p},\,\frac{y_0}{b^2}=\frac{\sin \alpha}{p}.</math>
The equation for the ellipse can be used to eliminate ''x''<sub>0</sub> and ''y''<sub>0</sub> giving
:<math>a^2 \cos^2 \alpha + b^2 \sin^2 \alpha = p^2,\,</math>
and converting to (''r'', θ) gives
:<math>a^2 \cos^2 \theta + b^2 \sin^2 \theta = r^2,\,</math>
as the polar equation for the pedal. This is easily converted to a Cartesian equation as
:<math>a^2 x^2 + b^2 y^2 = (x^2+y^2)^2.\,</math>
{{Clear}}

===From the polar equation===
For ''P'' the origin and ''C'' given in [[Polar coordinate system|polar coordinates]] by ''r''&nbsp;=&nbsp;''f''(θ). Let ''R''=(''r'', θ) be a point on the curve and let ''X''=(''p'', α) be the corresponding point on the pedal curve. Let ψ denote the angle between the tangent line and the radius vector, sometimes known as the [[Tangential angle#Polar|polar tangential angle]]. It is given by
:<math>r=\frac{dr}{d\theta}\tan \psi.</math>
Then
:<math>p=r\sin \psi</math>
and
:<math>\alpha = \theta + \psi - \frac{\pi}{2}.</math>
These equations may be used to produce an equation in ''p'' and α which, when translated to ''r'' and θ gives a polar equation for the pedal curve.<ref>Edwards p. 164-5</ref>

For example,<ref>Follows Edwards p. 165 with ''m''=1</ref> let the curve be the circle given by ''r'' = ''a'' cos θ. Then
:<math>a \cos \theta = -a \sin \theta \tan \psi</math>
so
:<math>\tan \psi = -\cot \theta,\, \psi = \frac{\pi}{2} + \theta, \alpha = 2 \theta.</math>
Also
:<math>p=r\sin \psi\ = r \cos \theta = a \cos^2 \theta = a \cos^2 {\alpha \over 2}.</math>

So the polar equation of the pedal is
:<math>r = a \cos^2 {\theta \over 2}.</math>

===From the pedal equation===
The [[pedal equation]]s of a curve and its pedal are closely related. If ''P'' is taken as the pedal point and the origin then it can be shown that the angle ψ between the curve and the radius vector at a point ''R'' is equal to the corresponding angle for the pedal curve at the point ''X''. If ''p'' is the length of the perpendicular drawn from ''P'' to the tangent of the curve (i.e. ''PX'') and ''q'' is the length of the corresponding perpendicular drawn from ''P'' to the tangent to the pedal, then by similar triangles
:<math>\frac{p}{r}=\frac{q}{p}.</math>
It follows immediately that the if the pedal equation of the curve is ''f''(''p'',''r'')=0 then the pedal equation for the pedal curve is<ref>Williamson p. 228</ref>
:<math>f(r,\frac{r^2}{p})=0</math>

From this all the positive and negative pedals can be computed easily if the pedal equation of the curve is known.

===From parametric equations===
[[Image:Contrapedal.gif|500px|right|thumb|Contrapedal of the same ellipse]]
[[Image:PedalCurve3.gif|500px|right|thumb|Pedal of the evolute of the ellipse : same as the contrapedal of the original ellipse]]
Let
<math>\vec{v} = P - R</math>
be the vector for ''R'' to ''P'' and write
:<math>\vec{v} = \vec{v}_{\parallel}+\vec{v}_\perp</math>,
the [[tangential and normal components]] of <math>\vec{v}</math> with respect to the curve.
Then <math>\vec{v}_{\parallel}</math> is the vector from ''R'' to ''X'' from which the position of ''X'' can be computed.

Specifically, if ''c'' is a [[parametric curve|parametrization]] of the curve then
:<math>t\mapsto c(t)+{ c'(t) \cdot (P-c(t))\over|c'(t)|^2} c'(t)</math>
parametrises the pedal curve (disregarding points where ''c' ''is zero or undefined).

For a parametrically defined curve, its pedal curve with pedal point (0;0) is defined as
:<math>X[x,y]=\frac{(xy'-yx')y'}{x'^2 + y'^2}</math>

:<math>Y[x,y]=\frac{(yx'-xy')x'}{x'^2 + y'^2}.</math>

The contrapedal curve is given by:
:<math>t\mapsto P-{ c'(t) \cdot (P-c(t))\over|c'(t)|^2} c'(t)</math>
With the same pedal point, the contrapedal curve is the pedal curve of the [[evolute]] of the given curve.