Editing Pedal curve (section)

===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}}