Polar Coordinate System

The polar coordinate system provides a natural way to describe points in the plane when distances and directions are more meaningful than horizontal and vertical displacements. It is especially useful for modeling non-rectangular regions such as circles and spirals, where symmetry about a center point is easier to express than it is in a rectangular grid. A familiar example is a ship’s plan position indicator, which marks detected targets as dots positioned relative to the ship at the display’s center.

In polar form, the ship’s location is treated as the pole (the fixed reference point). A chosen fixed direction from the pole is the polar axis, analogous to the positive x-axis in Cartesian coordinates. Any target location P is specified by an ordered pair (r, θ), where r is the radius (the directed distance from the pole to the point), and θ is the angle measured from the polar axis to the line segment joining the pole to P. Angles are typically measured in radians in mathematical applications, though the geometric interpretation is the same in degrees.To convert (r, θ) to Cartesian coordinates (x, y), form a right triangle with hypotenuse r and acute angle θ at the pole. Trigonometric definitions then give x = r cos⁡θ, y = r sin⁡θ.

Polar coordinates are not unique. A point opposite P(r, θ) along the same line of sight can be expressed as (r, π + θ) or equivalently as (−r, θ), where a negative radius indicates measuring the distance in the opposite direction while keeping the same angular reference.