Gömbkoordináták

Gömbkoordináták, often translated as spherical coordinates, form a coordinate system that uses three values to uniquely determine the position of a point in three-dimensional space. Unlike the more familiar Cartesian coordinate system which uses distances along three perpendicular axes (x, y, z), spherical coordinates utilize a radial distance and two angles. The first value is the radial distance, typically denoted by 'r', which represents the straight-line distance from the origin to the point. The second value is the polar angle, often denoted by 'θ' (theta), which measures the angle from the positive z-axis down to the point. This angle is usually restricted to the range [0, π] or [0°, 180°]. The third value is the azimuthal angle, commonly denoted by 'φ' (phi), which measures the angle of the projection of the point onto the xy-plane from the positive x-axis. This angle is typically restricted to the range [0, 2π) or [0°, 360°).

The conversion between Cartesian coordinates (x, y, z) and spherical coordinates (r, θ, φ) follows specific formulas. To convert from spherical to Cartesian, x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ). To convert from Cartesian to spherical, r = sqrt(x² + y² + z²), θ = arccos(z/r), and φ = arctan2(y, x). The 'arctan2' function is used for φ to correctly handle all quadrants. Gömbkoordináták are particularly useful in physics and engineering for problems with spherical symmetry, such as describing the motion of planets, electromagnetic fields, or the properties of certain materials. They simplify calculations in these contexts by reducing the complexity of the equations.