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In Euclidean space, such a dilation is a similarity of the space. [2] Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a congruence has a unique fixed point [3] that is called the center of dilation. [4] Some congruences have fixed points and others do not. [5]
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. [1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
The Alcubierre metric defines the warp-drive spacetime.It is a Lorentzian manifold that, if interpreted in the context of general relativity, allows a warp bubble to appear in previously flat spacetime and move away at effectively faster-than-light speed.
Metric geometry is a branch of geometry with metric spaces as the main object of study. ... Dilation (metric space) Dimension function; Discrete metric; Distance ...
A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence (X n, p n) of pointed metric spaces converges to a pointed metric space (Y, p) if, for each R > 0, the sequence of closed R-balls around p n in X n converges to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense. [10]
This is the metric most commonly used for the study of the metric geometry of Teichmüller space. In particular it is of interest to geometric group theorists. There is a function similarly defined, using the Lipschitz constants of maps between hyperbolic surfaces instead of the quasiconformal dilatations, on T ( S ) × T ( S ) {\displaystyle T ...
Dilation (metric space), a function from a metric space into itself; Dilation (operator theory), a dilation of an operator on a Hilbert space; Dilation (morphology), an operation in mathematical morphology; Scaling (geometry), including: Homogeneous dilation , the scalar multiplication operator on a vector space or affine space; Inhomogeneous ...
If A is an open or closed subset of R n (or even Borel set, see metric space), then A is Lebesgue-measurable. If A is a Lebesgue-measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure. A Lebesgue-measurable set can be "squeezed" between a containing open set and a contained closed set.