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 dimensionn, 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.
Improper rotationIn geometry, an improper rotation (also called rotation-reflection, rotoreflection,rotary reflection, or rotoinversion) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.
Chasles' theorem (kinematics)In kinematics, Chasles' theorem, or Mozzi–Chasles' theorem, says that the most general rigid body displacement can be produced by a screw displacement or screw motion. A direct Euclidean isometry in three dimensions involves a translation and a rotation. The screw displacement representation of the isometry decomposes the translation into two components, one parallel to the axis of rotation associated with the isometry and the other component perpendicular to that axis.
Rotation (mathematics)Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.
Translation (geometry)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 an isometry. A translation is an isometry that displaces the original figure according to a direction, a sense, and a length (vector).
Euclidean distanceIn mathematics, the Euclidean distance between two points in a Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient Greek mathematiciansEuclid and Pythagoras.
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.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, R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} – such that the function is bijective so that its inverse exists.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))} .
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
OrientabilityIn mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". It generalizes the concept of curve orientation, which for a plane simple closed curve is defined based on whether the curve interior is to the left or to the right of the curve. A space is orientable if such a consistent definition exists.*As an Amazon Associate I earn from qualifying purchases.