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In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are the point's distance from a reference point called the pole, and; the point's direction from the pole relative to the direction of the polar axis, a ray drawn from the pole. The distance from the pole ...
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 ).
A detailed contemporary collection of constructions for the determination of lengths from a distance using this instrument is known, the Dioptra of Hero of Alexandria (c. 10 –70 AD), which survived in Arabic translation; but the knowledge became lost in Europe until in 1615 Snellius, after the work of Eratosthenes, reworked the technique for ...
The relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then ...
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.
Geodetic latitude and geocentric latitude have different definitions. Geodetic latitude is defined as the angle between the equatorial plane and the surface normal at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure).
Equivalently, in polar coordinates (r, θ) it can be described by the equation = with real number b. Changing the parameter b controls the distance between loops. From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle θ as time elapses.
In a spherical coordinate system, a colatitude is the complementary angle of a given latitude, i.e. the difference between a right angle and the latitude. [1] In geography, Southern latitudes are defined to be negative, and as a result the colatitude is a non-negative quantity, ranging from zero at the North pole to 180° at the South pole.