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The 3-sphere is the boundary of a -ball in four-dimensional space. The ( n − 1 ) {\displaystyle (n-1)} -sphere is the boundary of an n {\displaystyle n} -ball. Given a Cartesian coordinate system , the unit n {\displaystyle n} -sphere of radius 1 {\displaystyle 1} can be defined as:
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space.
Topography in a narrow sense involves the recording of relief or terrain, the three-dimensional quality of the surface, and the identification of specific landforms; this is also known as geomorphometry. In modern usage, this involves generation of elevation data in digital form .
The boundary of a cross-section in three-dimensional space that is parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line; for example, if a plane cuts through mountains of a raised-relief map parallel to the ground, the result is a contour line in two-dimensional space ...
The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds. A Warning on terminology: Our two-sphere is defined in three-dimensional space, where it is the boundary of a three-dimensional ball. This terminology is standard among mathematicians, but not among physicists.
A global geometry is a local geometry plus a topology. It follows that a topology alone does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries. As stated in the introduction, investigations within the study of the global structure of the universe include:
Consider the projection centered at S = (0, 0, −1) on the unit sphere, which is the set of points (x, y, z) in three-dimensional space R 3 such that x 2 + y 2 + z 2 = 1. In Cartesian coordinates (x, y, z) on the sphere and (X, Y) on the plane, the projection and its inverse are then described by
A point in three-dimensional Euclidean space can be located by three coordinates. Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer ...