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Polar coordinates provide a number of fundamental tools in Riemannian geometry. The radial coordinate is the most significant: geometrically it represents the geodesic distance to p of nearby points. Gauss's lemma asserts that the gradient of r is simply the partial derivative /. That is, , = for any smooth function ƒ. As a result, the metric ...
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 .
Many hyperbolic lines through point P not intersecting line a in the Beltrami Klein model A hyperbolic triheptagonal tiling in a Beltrami–Klein model projection. In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit ...
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as , see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range ...
The operator can also be written in polar coordinates. Let (t, ξ) be spherical coordinates on the sphere with respect to a particular point p of H n−1 (say, the center of the Poincaré disc). Here t represents the hyperbolic distance from p and ξ a parameter representing the choice of direction of the geodesic in S n−2. Then the ...
In differential geometry, conjugate points or focal points [1] [2] are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing.
They argued in the context of string theory that the coordinate functions of the endpoints of open strings constrained to a D-brane in the presence of a constant Neveu–Schwarz B-field—equivalent to a constant magnetic field on the brane—would satisfy the noncommutative algebra set out above. The implication is that a quantum field theory ...
These are called the coordinates of the point. An n-dimensional differentiable manifold is a generalisation of n-dimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold that can be mapped into n-dimensional Euclidean space.