α) by (r, θ) produces a polar equation for the pedal curve. For example, for the ellipse the tangent line at R=(x0, y0) is and writing this in the form given above requires that The equation for the ellipse can be used to eliminate x0 and y0 giving and converting to (r, θ) gives as the polar equation for the pedal. This is easily converted to a Cartesian equation as From the polar equation For P the origin and C given in polar coordinates by r = 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 polar tangential angle. It is given by Then and 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. For example, let the curve be the circle given by r = a cos θ. Then so Also So the polar equation of the pedal is From the pedal equation The pedal equations 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 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 From this all the positive and negative pedals can be computed easily if the pedal equation of the curve is known. From parametric equations Let be the vector for R to P and write , the tangential and normal components of with respect to the curve. Then is the vector from R to X from which the position of X can be computed. Specifically, if c is a parametrization of the curve then parametrises the pedal curve (disregarding points where c' is zero or undefined). For