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In polar coordinates

If p > 0, the parabola with equation y 2 = 2 p x {\displaystyle y^{2}=2px} (opening to the right) has the polar representation

where r 2 = x 2 + y 2, x = r cos φ {\displaystyle r^{2}=x^{2}+y^{2},\ x=r\cos \varphi }.

Its vertex is V = (0, 0) {\displaystyle V=(0,0)}, and its focus is F = (p 2, 0) {\displaystyle F=\left({\tfrac {p}{2}},0\right)}.

If one shifts the origin into the focus, that is, F = (0, 0) {\displaystyle F=(0,0)}, one obtains the equation

Remark 1: Inverting this polar form shows that a parabola is the inverse of a cardioid.

Remark 2: The second polar form is a special case of a pencil of conics with focus F = (0, 0) {\displaystyle F=(0,0)} (see picture):

(e {\displaystyle e} is the eccentricity).