polarvinklen

Polarvinklen is the angle in polar coordinates that specifies the direction of a point from the origin in a plane. In the two-dimensional plane, a point is described by (r, theta), where r is the radial distance from the origin and theta, the polar angle, is measured from the positive x-axis. The angle is commonly expressed in radians, though degrees are also used. Standard conventions place theta in the interval [0, 2π) or (-π, π].

Relationships to Cartesian coordinates are straightforward. The point’s coordinates relate by x = r cos theta and y = r sin theta. Conversely, r = sqrt(x^2 + y^2) and theta = atan2(y, x), which yields the correct angle with respect to the quadrant and handles edge cases on the axes.

In three dimensions, the term polar angle is sometimes used for the angle from the positive z-axis, known as colatitude in spherical coordinates, with the other angle, azimuth, measured in the x-y plane. This usage connects to the idea of describing direction in space, though terminology can vary by field.

Applications and considerations: polarvinklen is used to describe direction, rotations, and vector fields; it facilitates integration and problem setup in polar coordinates and is common in computer graphics for radial layouts and sweeping operations. Numerical handling often requires attention to angle wrap-around and discontinuities at the 0/2π boundary, with normalization to a chosen range as needed.

Examples: a point (1, 1) has theta = π/4; a point (−1, 0) has theta = π; a point (0, −1) can be −π/2 or 3π/2 depending on convention.