polárkoordinátarendszer

The polárkoordinátarendszer, or polar coordinate system, is a two-dimensional coordinate system where each point is identified by a distance from a central point and an angle with respect to a reference direction. The central point is called the pole, and the reference direction is called the polar axis. A point in the polar coordinate system is represented by an ordered pair (r, θ), where 'r' is the radial distance from the pole and 'θ' is the angle measured counterclockwise from the polar axis.

The radial coordinate 'r' represents the distance of the point from the pole. It can be any real number. If r is positive, the point lies on the ray extending from the pole at an angle θ. If r is negative, the point lies on the ray extending in the opposite direction of the angle θ. The angular coordinate 'θ' is typically measured in radians or degrees. Unlike the Cartesian coordinate system, the representation of a point in polar coordinates is not unique. For example, the point (r, θ) can also be represented as (r, θ + 2πn) or (-r, θ + (2n+1)π) for any integer n.

The polar coordinate system is particularly useful for describing phenomena that exhibit radial symmetry, such as circles, spirals, and certain types of curves. It simplifies the representation and analysis of these shapes compared to their Cartesian equivalents. Conversion between polar and Cartesian coordinates is straightforward. A point with polar coordinates (r, θ) has Cartesian coordinates (x, y) where x = r cos(θ) and y = r sin(θ). Conversely, a point with Cartesian coordinates (x, y) has polar coordinates where r = √(x² + y²) and θ = atan2(y, x). The atan2 function is used to correctly determine the angle in all quadrants.