极坐标系

The polar coordinate system describes points in a plane using distance and direction from a fixed reference point, called the pole. 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 in a rectangular grid. A familiar application is a ship's plan position indicator, which marks detected targets relative to the ship's center.

To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), form a right triangle with hypotenuse r and acute angle θ at the pole. Using trigonometric definitions, x equals r cosine θ and y equals r sine θ. This conversion allows you to translate between the two coordinate systems.

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), use the Pythagorean theorem to find r squared equals x squared plus y squared, and use the tangent formula to find tangent θ equals y over x. These relationships allow you to determine the radius and angle from horizontal and vertical distances.

A negative radius indicates distance measured in the opposite direction along the same line of sight. For example, a point opposite P(r, θ) can be expressed as (−r, θ), where the negative sign shows the direction reversal while keeping the same angular reference. This non-uniqueness allows multiple representations of the same point.

A polar coordinate pair (r, θ) consists of two components: r is the radius, or 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 the point. The polar axis is analogous to the positive x-axis in Cartesian coordinates.

Polar coordinates are better for circles and spirals because these shapes exhibit symmetry about a center point, which is naturally expressed using distance and angle from a fixed pole. In rectangular grids, such symmetry requires more complex equations. Polar form simplifies the mathematical representation of these non-rectangular regions.

A target location Q directly opposite P(r, θ) can be written as (r, π + θ) or equivalently as (−r, θ). Both representations describe the same point along the same line of sight. This demonstrates the non-uniqueness of polar coordinates and provides flexibility in expressing point locations.