Polar Coordinates

1. Core Concept Overview & Intuition

The Cartesian coordinate system locates a point by two perpendicular signed distances from a fixed origin. The polar coordinate system locates the same point by a directed distance from a fixed point together with an angle measured from a fixed line. The fixed point is the pole, denoted $O$, and the fixed line is the initial line, denoted $OX$. The distance from the pole to the point is denoted $r$, and the angle from the initial line to the line joining the pole to the point is denoted $\theta$. The polar coordinates of a point are written as the ordered pair $(r,\theta)$.

The angle $\theta$ is measured in an anticlockwise direction from the fixed line $OX$. A negative value of $\theta$ means the angle is measured clockwise from the fixed line. The distance $r$ is always positive as long as it is measured from $O$ to the point. Points at various positions arise when $r$ and $\theta$ take negative values, so the sign conventions for both quantities must be handled with care.

The central idea of the topic is that the Cartesian and polar descriptions of a point are two encodings of the same geometric location. A point on the $xy$-plane with Cartesian coordinates $(x,y)$ can be re-expressed as a polar pair $(r,\theta)$, and conversely. The same correspondence extends from points to equations: a Cartesian equation can be rewritten as a polar equation, and a polar equation can be rewritten in Cartesian form. Because the circle was already studied as the locus of a point at constant distance from a fixed centre, the polar description of a circle follows naturally from the meaning of $r$ as a distance from the pole.

2. Mathematical Formalization & Core Definitions

2.1 The polar coordinate system

Let the fixed point be $O$. Suppose a point $P$ is at a distance $r$ from $O$, and let the angle $XOP$ be $\theta$. Then the polar coordinates of $P$ are given as the ordered pair $(r,\theta)$.

The angle $\theta$ is measured anticlockwise from the fixed line $OX$; a negative value of $\theta$ corresponds to a clockwise measurement. The distance $r$ is positive when measured from $O$ to $P$. Allowing negative values of $r$ and $\theta$ produces the various possible positions of a point in the plane.

2.2 Relationship between polar and rectangular coordinates

Consider a point whose polar coordinates are $(r,\theta)$. From the geometry of the right triangle formed with the initial line, the relationships are

$$
x=r\cos\theta,\qquad y=r\sin\theta.
$$

These relationships transform polar coordinates into Cartesian coordinates. The reverse transformation uses

$$
r^{2}=x^{2}+y^{2},\qquad \tan\theta=\frac{y}{x}.
$$

These relationships transform Cartesian coordinates into polar coordinates and vice versa.

2.3 Converting a Cartesian equation to a polar equation

A Cartesian equation in $x$ and $y$ is converted to polar form by substituting $x=r\cos\theta$ and $y=r\sin\theta$, then simplifying. A Cartesian equation in $x$ and $y$ may also be converted by using $x^{2}+y^{2}=r^{2}$ where a sum of squares appears.

2.4 Converting a polar equation to a Cartesian equation

A polar equation in $r$ and $\theta$ is converted to Cartesian form by using $x=r\cos\theta$, $y=r\sin\theta$, and $r^{2}=x^{2}+y^{2}$, together with the identity $\tan\theta=\dfrac{y}{x}$ where an angle appears explicitly.

2.5 Polar equations of lines and circles

A line through the pole has a polar equation in which $\theta$ is constant, since every point on such a line makes the same angle with the initial line. A circle with centre at the pole and radius $a$ has the polar equation $r=a$, since every point on the circle is at distance $a$ from the pole. More general lines and circles are obtained by transforming their Cartesian equations into polar form using the substitutions above.