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A time–distance diagram is a chart with two axes: one for time, the other for location. The units on either axis depend on the type of project: time can be expressed in minutes (for overnight construction of railroad modification projects such as the installation of switches) or years (for large construction projects); the location can be (kilo)meters, or other distinct units (such as ...
When a coordinate system is chosen in the Euclidean space, this defines coordinates on : the coordinates of a point of are defined as the coordinates of (). The pair formed by a chart and such a coordinate system is called a local coordinate system , coordinate chart , coordinate patch , coordinate map , or local frame .
Coordinate charts are mathematical objects of topological manifolds, and they have multiple applications in theoretical and applied mathematics. When a differentiable structure and a metric are defined, greater structure exists, and this allows the definition of constructs such as integration and geodesics .
A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions [clarification needed]: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate ...
The radial distance ρ is the Euclidean distance from the z-axis to the point P. The azimuth φ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of P on the plane. The axial coordinate or height z is the signed distance from the chosen plane to the point P.
Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic ...
The Earth-centered, Earth-fixed coordinate system (acronym ECEF), also known as the geocentric coordinate system, is a cartesian spatial reference system that represents locations in the vicinity of the Earth (including its surface, interior, atmosphere, and surrounding outer space) as X, Y, and Z measurements from its center of mass.
In this case, for two events which are simultaneous according to the cosmological time coordinate, the value of the cosmological proper distance is not equal to the value of the proper length between these same events, [9] which would just be the distance along a straight line between the events in a Minkowski diagram (and a straight line is a ...