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To define a spherical coordinate system, one must designate an origin point in space, O, and two orthogonal directions: the zenith reference direction and the azimuth reference direction. These choices determine a reference plane that is typically defined as containing the point of origin and the x– and y–axes , either of which may be ...
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
The azimuth is the angle formed between a reference direction (in this example north) and a line from the observer to a point of interest projected on the same plane as the reference direction orthogonal to the zenith. An azimuth (/ ˈ æ z ə m ə θ / ⓘ; from Arabic: اَلسُّمُوت, romanized: as-sumūt, lit.
In stellar navigation, the reference direction is that of the North Star, Polaris. In satellite broadcasting, a bearing is the combination of antenna azimuth and elevation required to point (aim) a satellite dish antenna in a given direction. The bearing for geostationary satellites is constant. The bearing for polar-orbiting satellites varies ...
Difference in longitude of the points on the auxiliary sphere; α 1, α 2: forward azimuths at the points; α: forward azimuth of the geodesic at the equator, if it were extended that far; s: ellipsoidal distance between the two points; σ: angular separation between points: σ 1: angular separation between the point and the equator: σ m
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates v = (x, y), it should be written as a column vector, and multiplied by the matrix R:
A point is chosen as the pole and a ray from this point is taken as the polar axis. For a given angle θ, there is a single line through the pole whose angle with the polar axis is θ (measured counterclockwise from the axis to the line). Then there is a unique point on this line whose signed distance from the origin is r for given number r.