Angular distance

Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors – typically in three-dimensional space – or the central angle subtended by the radii through two points on a sphere. When the rays are lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation. Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque.

Angular distance or angular separation is the measure of the angle between the orientation of two straight lines, rays, or vectors – typically in three-dimensional space – or the central angle subtended by the radii through two points on a sphere. When the rays are lines of sight from an observer to two points in space, it is known as the apparent distance or apparent separation. Angular distance appears in mathematics (in particular geometry and trigonometry) and all natural sciences (e.g., kinematics, astronomy, and geophysics). In the classical mechanics of rotating objects, it appears alongside angular velocity, angular acceleration, angular momentum, moment of inertia and torque. ==Measurement== As an angle, the angular separation is measured using angular units such as degrees or radians, using instruments such as goniometers or optical instruments specially designed to point in well-defined directions and record the corresponding angles (such as telescopes). \mathbf{OA}\cdot\mathbf{OB}= R^2 \cos\theta which is equivalent to: \mathbf{n_A} \cdot \mathbf{n_B} = \cos\theta In the (x,y,z) frame, the two unitary vectors are decomposed into: \mathbf{n_A} = \begin{pmatrix} \cos\delta_A \cos\alpha_A\\ \cos\delta_A \sin\alpha_A\\ \sin\delta_A \end{pmatrix} \mathrm{\qquad and \qquad } \mathbf{n_B} = \begin{pmatrix} \cos\delta_B \cos\alpha_B\\ \cos\delta_B \sin\alpha_B\\ \sin\delta_B \end{pmatrix} . Therefore, \mathbf{n_A} \cdot \mathbf{n_B} = \cos\delta_A \cos\alpha_A \cos\delta_B \cos\alpha_B + \cos\delta_A \sin\alpha_A \cos\delta_B \sin\alpha_B + \sin\delta_A \sin\delta_B \equiv \cos\theta then: \theta = \cos^{-1}\left[\sin\delta_A \sin\delta_B + \cos\delta_A \cos\delta_B \cos(\alpha_A - \alpha_B)\right] where \delta x = (\alpha_A - \alpha_B)\cos\delta_A and \delta y=\delta_A-\delta_B . Note that the y -axis is equal to the declination, whereas the x -axis is the right ascension modulated by \cos\delta_A because the section of a sphere of radius R at declination (latitude) \delta is R' = R \cos\delta_A (see Figure). ==See also== Milliradian Gradian Hour angle Central angle Angle of rotation Angular diameter Angular displacement Great-circle distance ==References== CASTOR, author Michael A. Earl. "The Spherical Trigonometry vs. Vector Analysis".

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