Editing Pedal curve (section)

===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>