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.↩︎

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