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.

- 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.↩︎

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