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 . What's more, one often allows negative values of  under the assumption that  is plotted identically to .

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 radialaxes 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 the area element by

(7)

The area enclosed by a polar curve  is

(8)

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

(9)

The angle between the tangent 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.

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)

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.