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Introduction of Reflection

In mathematics, a Reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of figure by a reflection is its mirror image in the axis or plane of reflection. For illustration the mirror image of the small Latin letter 'a' for a reflection with respect to a vertical axis would look like b. Its image by reflection in a horizontal axis would look like p. A reflection is an involution: when applied twice in succession every point returns to its original location and every geometrical object is restored to its original state.

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The term "reflection" is sometimes used for a larger class of mappings from a Euclidean space to itself namely the non-identity isometries that are involutions. This type of isometries has a set of fixed points (the "mirror") that is an affine subspace, but possibly smaller than a hyperplane. For example a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it would look like a d. This operation is also known as central inversion (Coxeter 1969, §7.2), and exhibits Euclidean space as a symmetric space. In Euclidean vector space the reflection in the point situated at the origin is the same as vector negation. Other illustrations include reflections in a line in three dimensional spaces. Usually, however, unqualified use of the term "reflection" means reflection in a hyperplane.

Reflection across a line in the plane

It can be defined by the formula:

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Where v denotes the vector being reflected and l denotes any vector in the line being reflected in and v·l denotes the dot product of v with l.

Note: This formula can also be described as

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Where the reflection of line l on a is equal to 2 times the projection of v on line l minus v. Reflections in line have the eigenvalues of 1 and -1.

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