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A reflection through an axis. In mathematics, a reflection (also spelled reflexion) [1] 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.
A typical example of glide reflection in everyday life would be the track of footprints left in the sand by a person walking on a beach. Frieze group nr. 6 (glide-reflections, translations and rotations) is generated by a glide reflection and a rotation about a point on the line of reflection. It is isomorphic to a semi-direct product of Z and C 2.
Reflection. Reflections, or mirror isometries, denoted by F c,v, where c is a point in the plane and v is a unit vector in R 2.(F is for "flip".) have the effect of reflecting the point p in the line L that is perpendicular to v and that passes through c.
An xy-Cartesian coordinate system rotated through an angle to an x′y′-Cartesian coordinate system In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel. Thus through transformations students learn about Euclidean plane isometry. For instance, consider reflection in a vertical line and a ...
A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an affine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector): [′ ′ ′] = [] [] where = for some point on the plane, or equivalently, + + + =.
In mathematics, a transformation, transform, or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [2] [3] [4] Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific ...