Calculus/Polar Introduction

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A polar grid with several angles labeled in degrees

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the more familiar Cartesian coordinate system or rectangular coordinate system, such a relationship can only be found through trigonometric formulae.

As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as ) denotes the point's distance from a central point known as the pole (equivalent to the origin in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by or ) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or polar axis (which is equivalent to the positive -axis in the Cartesian coordinate plane).


Plotting points with polar coordinatesEdit

The points (3,60°) and (4,210°) on a polar coordinate system

Each point in the polar coordinate system can be described with the two polar coordinates, which are usually called   (the radial coordinate) and θ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as   or  ). The   coordinate represents the radial distance from the pole, and the θ coordinate represents the anticlockwise (counterclockwise) angle from the   ray (sometimes called the polar axis), known as the positive  -axis on the Cartesian coordinate plane.

For example, the polar coordinates   would be plotted as a point 3 units from the pole on the   ray. The coordinates   would also be plotted at this point because a negative radial distance is measured as a positive distance on the opposite ray (the ray reflected about the origin, which differs from the original ray by  ).

One important aspect of the polar coordinate system, not present in the Cartesian coordinate system, is that a single point can be expressed with an infinite number of different coordinates. This is because any number of multiple revolutions can be made around the central pole without affecting the actual location of the point plotted. In general, the point   can be represented as   or   , where   is any integer.

The arbitrary coordinates   are conventionally used to represent the pole, as regardless of the θ coordinate, a point with radius 0 will always be on the pole. To get a unique representation of a point, it is usual to limit   to negative and non-negative numbers   and   to the interval   or   (or, in radian measure,   or  ).

Angles in polar notation are generally expressed in either degrees or radians, using the conversion   . The choice depends largely on the context. Navigation applications use degree measure, while some physics applications (specifically rotational mechanics) and almost all mathematical literature on calculus use radian measure.

Converting between polar and Cartesian coordinatesEdit

A diagram illustrating the conversion formulae

The two polar coordinates   can be converted to the Cartesian coordinates   by using the trigonometric functions sine and cosine:


while the two Cartesian coordinates   can be converted to polar coordinate   by

  (by a simple application of the Pythagorean theorem).

To determine the angular coordinate   , the following two ideas must be considered:

  • For   ,   can be set to any real value.
  • For   , to get a unique representation for   , it must be limited to an interval of size   . Conventional choices for such an interval are   and   .

To obtain   in the interval   , the following may be used (  denotes the inverse of the tangent function):


To obtain   in the interval   , the following may be used:


One may avoid having to keep track of the numerator and denominator signs by use of the atan2 function, which has separate arguments for the numerator and the denominator.

Polar equationsEdit

The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining   as a function of   . The resulting curve then consists of points of the form   and can be regarded as the graph of the polar function   .

Different forms of symmetry can be deduced from the equation of a polar function   . If   the curve will be symmetrical about the horizontal   ray, if   it will be symmetric about the vertical   ray, and if   it will be rotationally symmetric   counterclockwise about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the polar rose, Archimedean spiral, lemniscate, limaçon, and cardioid.

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.


A circle with equation  

The general equation for a circle with a center at   and radius   is


This can be simplified in various ways, to conform to more specific cases, such as the equation


for a circle with a center at the pole and radius   .


Radial lines (those running through the pole) are represented by the equation


where   is the angle of elevation of the line; that is,   where   is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line   perpendicularly at the point   has the equation


Polar roseEdit

A polar rose with equation  

A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,


for any constant   (including 0). If   is an integer, these equations will produce a  -petaled rose if   is odd, or a  -petaled rose if   is even. If   is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable   represents the length of the petals of the rose.

Archimedean spiralEdit

One arm of an Archimedean spiral with equation   for  

The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation


Changing the parameter   will turn the spiral, while   controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for   and one for   . The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the   line will yield the other arm. This curve is notable as one of the first curves, after the Conic Sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.

Conic sectionsEdit

Ellipse, showing semi-latus rectum

A conic section with one focus on the pole and the other somewhere on the   ray (so that the conic's semi-major axis lies along the polar axis) is given by:


where   is the eccentricity and   is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve).

  1. If   , this equation defines a hyperbola.
  2. If   , it defines a parabola.
  3. If   , it defines an ellipse. The special case   of the latter results in a circle of radius   .