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In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point P in space. Its length represents the distance in relation to an arbitrary reference origin O , and its direction represents the angular orientation with respect to given reference axes.
For example, Plücker coordinates are used to determine the position of a line in space. [11] When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function.
In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. [1] The subspace is then called a retract of the original space. A deformation retraction is a mapping that captures the idea of continuously shrinking a space into a ...
The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If v is an absolute value on F, then the function d on F × F, defined by d(a, b) = v(a − b), is a metric and the following are equivalent:
Absolute geometry is an extension of ordered geometry, and thus, all theorems in ordered geometry hold in absolute geometry. The converse is not true. Absolute geometry assumes the first four of Euclid's Axioms (or their equivalents), to be contrasted with affine geometry, which does not assume Euclid's third and fourth axioms. Ordered geometry ...
A simple example is a volume (how big an object occupies a space) as a measure. In mathematics , the concept of a measure is a generalization and formalization of geometrical measures ( length , area , volume ) and other common notions, such as magnitude, mass , and probability of events.
A coordinate system in mathematics is a facet of geometry or of algebra, [9] [10] in particular, a property of manifolds (for example, in physics, configuration spaces or phase spaces). [11] [12] The coordinates of a point r in an n-dimensional space are simply an ordered set of n numbers: [13] [14]
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