polárkoordináta

A polárkoordináta, or polar coordinate, is a two-dimensional coordinate system in which each point is identified by a distance from a fixed origin point and an angle from a fixed ray. The origin is typically denoted as O, and the ray as the polar axis. A point in polar coordinates is represented by an ordered pair (r, θ), where r is the radial distance from the origin and θ is the angle measured counterclockwise from the polar axis. The distance r is a non-negative real number, while the angle θ can be any real number, usually expressed in radians or degrees.

This system provides an alternative to the Cartesian coordinate system, where points are defined by their perpendicular distances from two perpendicular axes. Polar coordinates are particularly useful for describing phenomena that exhibit radial symmetry, such as circles, spirals, and rotation. For instance, a circle centered at the origin with radius R is simply represented by r = R in polar coordinates, regardless of the angle θ. In contrast, describing a circle in Cartesian coordinates requires the equation x² + y² = R².

The relationship between polar coordinates (r, θ) and Cartesian coordinates (x, y) is given by the equations x = r cos(θ) and y = r sin(θ). Conversely, r = √(x² + y²) and θ = atan2(y, x). The atan2 function is used for calculating the angle to ensure the correct quadrant is identified. While unique in Cartesian coordinates, a point can have multiple polar coordinate representations due to the periodic nature of angles. For example, (r, θ) is equivalent to (r, θ + 2πn) and (-r, θ + (2n+1)π) for any integer n.