1. Coordinate Systems

c. Polar Coordinates - 2D

1. 2D Polar Coordinates

Rectangular coordinates \((x,y)\) are one way to specify a point, \(P\), in the plane, but they are not the only way. When we are studying circles, it is useful to use Polar Coordinates \((r,\theta)\) which identify the point, \(P\), using

  • the radius \(r\) and
  • the polar angle \(\theta\).
The plot shows a point P in the first quadrant with a vertical line labeled y going down from P perpendicular to the x axis, a horizontal line labeled x going left from P perpendicular to to the y axis. The plot also shows a line going from P to the origin labeled r with an arc from this line to the x axis which is labeled theta.

Like rectangular coordinates \(P=(x,y)\), the polar coordinates can be written as an ordered pair, \(P=(r,\theta)\). So you need to be careful to say whether this ordered pair is rectangular or polar.

Most often, the radius measures the (positive) distance from the origin, \(O\), to the point, \(P\), and the polar angle measures an angle (usually measured in radians and usually with \(0 \le \theta \lt 2\pi\)) counterclockwise from the positive \(x\)-axis to the ray \(\overrightarrow{OP}\).

However, frequently the angle \(\theta\) is allowed to be bigger than \(2\pi\) or is allowed to be negative, in which case it is measured clockwise from the posiitve \(x\)-axis. Consequently, \(\theta\) is non-unique; we can always add or subtract an arbitrary multiple of \(2\pi\) or \(360^\circ\). Thus the point with polar coordinates \((3,{30^\circ})\) can also be written as \((3,{390^\circ})\), or \((3,{750^\circ})\), or \((3,{-330^\circ})\), or \((3,{-690^\circ})\) all of which are shown in the plot with blue angles positive and red angles negative.

The plot shows a point in the first quadrant with angle 30 degrees with a line going to the origin. There are also with concentric spiral arcs alternating between clockwise and counterclockwise going from the x-axis to the line at 30 degrees. The smallest is the direct counterclockwise 30 degree arc, the next is a clockwise 330 degree arc, next a counterclockwise 390 degree arc, a clockwise 690 degree arc and finally a counterclockwise 750 degree arc.

Further, once in a while, we allow \(r\) to be negative, for example in the context of solving equations or graphing. Then \(r\) is the negative of the distance from \(O\) to \(P\). When \(r\) is negative, the point \((r,\theta)\) is obtained by going a distance \(|r|\) along the ray at the angle \(\theta\pm\pi=\theta\pm180^\circ\). Think of this as going backwards along the ray at the angle \(\theta\). Thus the point with polar coordinates \((-3,{30^\circ})\) is actually the point \((3,{210^\circ})\) or \((3,-150^\circ)\).

The plot shows a point in the third quadrant with angle 210 with a counterclockwise 210 degree arc and a clockwise 150 degree arc and a 30 degree arc which goes to the line opposite to the line from the origin to the point.

Coordinate Curves

When you hold one of the coordinates fixed and let the other one vary, the point \(P=(r,\theta)\) traces out a coordinate curve. The curves are named by the coordinate which is changing.

When \(\theta\) is constant, you get a radial ray (assuming \(r \gt 0\)) called an \(r\)-curve, e.g. here is the \(r\)-curve with \(\theta=\dfrac{\pi}{3}\):

The plot shows a line going from the origin with angle 60 degrees from the x-axis.
\(r\)-curve:   \(\theta=\dfrac{\pi}{3}\)

When \(r\) is constant, you get a circle called a \(\theta\)-curve, e.g. here is the \(\theta\)-curve with \(r=2\):

The plot shows a circle with radius 2 centered at the origin.
\(\theta\)-curve:   \(r=2\)

When you draw several coordinate curves of each type you get a coordinate grid. Here is a polar coordinate grid:

The plot shows circles centered at the origin with radii 1,2 and 3 and lines emanating from the origin with angles that are all the multiples of 30 degrees from 0 to 330.
Polar Coordinate Grid

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Supported in part by NSF Grant #1123255

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