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The azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the radial line segment OP on the reference plane. The sign of the azimuth is determined by designating the rotation that is the positive sense of turning about the zenith. This choice is arbitrary, and is part of ...
Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector | |, and a, b, c are the direction angles of the vector v. The direction angles a, b, c are acute or obtuse angles, i.e., 0 ≤ a ≤ π, 0 ≤ b ≤ π and 0 ≤ c ≤ π, and they denote the angles formed between v and the unit basis vectors e x, e y, e z.
A two-dimensional direction can also be represented by its angle, measured from some reference direction, the angular component of polar coordinates (ignoring or normalizing the radial component). A three-dimensional direction can be represented using a polar angle relative to a fixed polar axis and an azimuthal angle about the polar axis: the ...
where for every direction in the base space, S n, the fiber over it in the total space, SO(n + 1), is a copy of the fiber space, SO(n), namely the rotations that keep that direction fixed. Thus we can build an n × n rotation matrix by starting with a 2 × 2 matrix, aiming its fixed axis on S 2 (the ordinary sphere in three-dimensional space ...
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 ...
Each point corresponds to its signed distance from the origin (a number with an absolute value equal to the distance and a + or − sign chosen based on direction). A geometric transformation of the line can be represented by a function of a real variable , for example translation of the line corresponds to addition, and scaling the line ...
Direction determination refers to the ways in which a cardinal direction or compass point can be determined in navigation and wayfinding.The most direct method is using a compass (magnetic compass or gyrocompass), but indirect methods exist, based on the Sun path (unaided or by using a watch or sundial), the stars, and satellite navigation.
[56] [57] Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude modulo 1 / 2 turn, 180°, or π radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a ...