polárkoordinátarendszerben

The polar coordinate system is a two-dimensional coordinate system in which each point is identified by a distance from a fixed origin and an angle from a fixed ray. The origin is called the pole, and the ray is called the polar axis. In the polar coordinate system, a point is represented by an ordered pair (r, θ), where r is the radial distance from the pole and θ is the angular position from the polar axis. The angle θ is usually measured in radians or degrees.

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 = sqrt(x^2 + y^2) and θ = arctan(y/x). This means that any point in the Cartesian plane can be uniquely represented in polar coordinates, and vice versa.

Polar coordinates are particularly useful for describing circular or spiral shapes. For example, a circle centered at the origin with radius a can be described by the equation r = a. A spiral can be described by an equation where r is a function of θ, such as r = aθ. Many scientific and engineering applications, including navigation, astronomy, and robotics, utilize polar coordinates due to their ability to simplify the representation of certain types of motion and geometric figures. The concept of polar coordinates was first introduced by Jakob Iversen in 1691 and later further developed by Leonhard Euler.