Polar Coordinates

The polar coordinates (the radial coordinate) and (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates by

			(1)
			(2)

where is the radial distance from the origin, and is the counterclockwise angle from the x-axis. In terms of and ,

			(3)
			(4)

(Here, should be interpreted as the two-argument inverse tangent which takes the signs of and into account to determine in which quadrant lies.) It follows immediately that polar coordinates aren't inherently unique; in particular, will be precisely the same polar point as for any integer . A common convention also allows negative radial coordinates. In that convention, and represent the same point for every integer .

The expression of a point as an ordered pair is known as polar notation, the equation of a curve expressed in polar coordinates is known as a polar equation, and a plot of a curve in polar coordinates is known as a polar plot.

In much the same way that Cartesian curves can be plotted on rectilinear axes, polar plots can be drawn on radial axes such as those shown in the figure above.

The arc length of a polar curve given by is

(5)

The line element is given by

(6)

and, using the usual convention , the area element by

(7)

For a simple closed polar curve traced once as increases from to , the enclosed area is

(8)

The slope of a polar function at the point is given by

(9)

where the denominator is nonzero. The angle between the tangent line and radial line at the point is

(10)

A polar curve is symmetric about the x-axis if replacing by in its equation produces an equivalent equation, symmetric about the y-axis if replacing by in its equation produces an equivalent equation, and symmetric about the origin if replacing by in its equation produces an equivalent equation.

The following vector formulas use the usual convention . In Cartesian coordinates, the radius vector is

(11)

giving derivative

(12)

Its unit vector is

(13)

giving derivative

(14)

In polar coordinates, the radius vector is given by

(15)

giving derivatives

			(16)
			(17)
			(18)
			(19)

For the usual convention , the unit vectors are

			(20)
			(21)

giving derivatives

			(22)
			(23)

By way of the Euler formula, the graphical representation of a complex number in terms of its complex modulus and its complex argument is closely related to polar coordinates. Indeed, the Argand diagram of such a is easily seen to be analogous to the usual polar plot. In particular, is the polar form of .