Metric spaceIn mathematics, a metric space is a set together with a notion of distance between its points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance.
Euclidean spaceEuclidean 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 dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.
Motion (geometry)In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distancemetric is a metric space in which a mapping associating congruent figures is a motion. Motions can be divided into direct (also known as proper or rigid) and indirect (or improper) motions. Direct motions include translations and rotations, which preserve the orientation of a chiralshape.
Reflection (mathematics)In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as the set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection.
IsomorphismIn mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them, and this is often denoted as A ≅ B {\displaystyle A\cong B} . The word is derived from Ancient Greek ἴσος (isos) 'equal' and μορφή (morphe) 'form, shape'.
Geometric transformationIn mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning, such as preserving distances, angles, or ratios (scale). More specifically, it is a function whose domain and range are sets of points – most often a real coordinate space, R2{\displaystyle \mathbb {R} ^{2}} or R3{\displaystyle \mathbb {R} ^{3}} – such that the function is bijective so that its inverse exists.
Unitary operatorIn functional analysis, a unitary operator is a surjectivebounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. Unitary operators are usually taken as operating on a Hilbert space, but the same notion serves to define the concept of isomorphismbetween Hilbert spaces. DefinitionDefinition 1.Function compositionIn mathematics, the composition operator ∘ {\displaystyle \circ } takes two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘ g {\displaystyle f\circ g} (pronounced " f {\displaystyle f} of g {\displaystyle g} ") is evaluated at an input x {\displaystyle x} , the result is ( f ∘ g ) ( x ) = f ( g ( x ) ) {\displaystyle (f\circ g)(x)=f(g(x))} .
BijectionIn mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Given a function f:A→B{\displaystyle f:A\to B}, the image of an elementa∈A{\displaystyle a\in A} is the element f(a)∈B{\displaystyle f(a)\in B} in the codomain.
DistanceDistance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over").
Transformation (function)In mathematics, a transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations.
Ancient GreekAncient Greek (Ἑλληνική, Hellēnikḗ [hellɛːnikɛ̌ː]) includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (c. 1400 – c. 1200 BC), Dark Ages (c. 1200 – c. 800 BC), the Archaic or Homeric period (c. 800 – c. 500 BC), and the Classical period (c. 500 – c. 300 BC).*As an Amazon Associate I earn from qualifying purchases.