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Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. [1] There is also a third system, based on two poles ( biangular coordinates ). The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses , hyperbolas , and Cassini ovals .
Cartesian plane with marked points (signed ordered pairs of coordinates). For any point, the abscissa is the first value (x coordinate), and the ordinate is the second value (y coordinate). In mathematics , the abscissa ( / æ b ˈ s ɪ s . ə / ; plural abscissae or abscissas ) and the ordinate are respectively the first and second coordinate ...
Beyond this, there can be multiple geodesics connecting two points. Suppose we have a Lorentzian manifold with a geodesic congruence. Then, at a conjugate point, the expansion parameter θ in Raychaudhuri's equation becomes negative infinite in a finite amount of proper time, indicating that the geodesics are focusing to a point. This is ...
The coordinate of a point P is defined as the signed distance from O to P, where the signed distance is the distance taken as positive or negative depending on which side of the line P lies. Each point is given a unique coordinate and each real number is the coordinate of a unique point. [4] The number line
A Cartesian coordinate system in two dimensions (also called a rectangular coordinate system or an orthogonal coordinate system [8]) is defined by an ordered pair of perpendicular lines (axes), a single unit of length for both axes, and an orientation for each axis. The point where the axes meet is taken as the origin for both, thus turning ...
Two-point tensors, or double vectors, are tensor-like quantities which transform as Euclidean vectors with respect to each of their indices. They are used in continuum mechanics to transform between reference ("material") and present ("configuration") coordinates. [ 1 ]
These differ in their choice of fundamental plane, which divides the celestial sphere into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, have the same fundamental (x, y) plane and primary (x-axis) direction, such as an axis of rotation. Each coordinate system is named after its choice of fundamental ...
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are the radial distance r along the line connecting the point to a fixed point called the origin; the polar angle θ between this radial line and a given polar axis; [a] and