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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 mathematics a translation surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used to model billiards, and in Teichmüller theory.
In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object.
Any equivalence relation is the negation of an apartness relation, though the converse statement only holds in classical mathematics (as opposed to constructive mathematics), since it is equivalent to the law of excluded middle. Each relation that is both reflexive and left (or right) Euclidean is also an equivalence relation.
Formal equivalence is often more goal than reality, if only because one language may contain a word for a concept which has no direct equivalent in another language. In such cases, a more dynamic translation may be used or a neologism may be created in the target language to represent the concept (sometimes by borrowing a word from the source ...
Similar figures. In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other.More precisely, one can be obtained from the other by uniformly scaling (enlarging or reducing), possibly with additional translation, rotation and reflection.
This is a glide reflection, except in the special case that the translation is perpendicular to the line of reflection, in which case the combination is itself just a reflection in a parallel line. The identity isometry, defined by I(p) = p for all points p is a special case of a translation, and also a special case of a rotation. It is the ...
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics.