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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is ...
Translation T is a direct isometry: a rigid motion. [1] In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. [a] The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning ...
For translational invariant functions : it is () = (+).The Lebesgue measure is an example for such a function.. In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation (without rotation).
Combining two equal glide plane operations gives a pure translation with a translation vector that is twice that of the glide reflection, so the even powers of the glide reflection form a translation group. In the case of glide-reflection symmetry, the symmetry group of an object contains a glide reflection and the group generated by it. For ...
Metric spaces are also studied in their own right in metric geometry [2] and analysis on metric spaces. [ 3 ] Many of the basic notions of mathematical analysis , including balls , completeness , as well as uniform , Lipschitz , and Hölder continuity , can be defined in the setting of metric spaces.
In mathematics, a translation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x'y'-Cartesian coordinate system in which the x' axis is parallel to the x axis and k units away, and the y' axis is parallel to the y axis and h units away.
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a). [1] In time series analysis, the shift operator is called the lag operator.
It is known, for instance, that every continuous translation invariant continuous linear operator on L 1 is the convolution with a finite Borel measure. More generally, every continuous translation invariant continuous linear operator on L p for 1 ≤ p < ∞ is the convolution with a tempered distribution whose Fourier transform is bounded.