In what precedes we have determined the position of $$P$$ by the lengths of its coordinates $$OM = x$$, $$MP = y$$. If $$OP = r$$ and $$MOP = \theta$$, $$\theta$$ being an angle between $$0$$ and $$2\pi$$ (measured in the positive direction), it is evident that $\begin{gathered} x = r\cos\theta,\qquad y = r\sin\theta, \\ r = \sqrt{x^{2} + y^{2}},\quad \cos\theta : \sin\theta : 1 :: x : y : r,\end{gathered}$ and that the position of $$P$$ is equally well determined by a knowledge of $$r$$ and $$\theta$$. We call $$r$$ and $$\theta$$ the polar coordinates of $$P$$. The former, it should be observed, is essentially positive.1

If $$P$$ moves on a locus there will be some relation between $$r$$ and $$\theta$$, say $$r = f(\theta)$$ or $$\theta = F(r)$$. This we call the polar equation of the locus. The polar equation may be deduced from the $$(x, y)$$ equation (or vice versa) by means of the formulae above.

Thus the polar equation of a straight line is of the form $r\cos(\theta – \alpha) = p,$ where $$p$$ and $$\alpha$$ are constants. The equation $$r = 2a\cos\theta$$ represents a circle passing through the origin; and the general equation of a circle is of the form $r^{2} + c^{2} – 2rc\cos(\theta – \alpha) = A^{2},$ where $$A$$, $$c$$, and $$\alpha$$ are constants.

1. Polar coordinates are sometimes defined so that $$r$$ may be positive or negative. In this case two pairs of coordinates— $$(1, 0)$$ and $$(-1, \pi)$$—correspond to the same point. The distinction between the two systems may be illustrated by means of the equation $$l/r = 1 – e\cos\theta$$, where $$l > 0$$, $$e > 1$$. According to our definitions $$r$$ must be positive and therefore $$\cos\theta < 1/e$$: the equation represents one branch only of a hyperbola, the other having the equation $$-l/r = 1 – e\cos\theta$$. With the system of coordinates which admits negative values of $$r$$, the equation represents the whole hyperbola.↩︎