Reflection (mathematics)

Last updated March 15, 2024

A reflection through an axis. SimmetriainvOK.svg — 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 a 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. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a vertical reflection) would look like q. Its image by reflection in a horizontal axis (a horizontal reflection) would look like b. 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.

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. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance 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 a central inversion ( Coxeter 1969 , §7.2), and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.

Some mathematicians use "flip" as a synonym for "reflection".^[2]^[3]^[4]

Construction

In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.

To reflect point $P$ through the line $AB$ using compass and straightedge, proceed as follows (see figure):

Step 1 (red): construct a circle with center at $P$ and some fixed radius $r$ to create points $A'$ and $B'$ on the line $AB$ , which will be equidistant from $P$ .
Step 2 (green): construct circles centered at $A'$ and $B'$ having radius $r$ . $P$ and $Q$ will be the points of intersection of these two circles.

Point $Q$ is then the reflection of point $P$ through line $AB$ .

Properties

The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem.

Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. The finite groups generated in this way are examples of Coxeter groups.

Reflection across a line in the plane

Reflection across an arbitrary line through the origin in two dimensions can be described by the following formula

\operatorname {Ref} _{l}(v)=2{\frac {v\cdot l}{l\cdot l}}l-v,

where $v$ denotes the vector being reflected, $l$ denotes any vector in the line across which the reflection is performed, and $v\cdot l$ denotes the dot product of $v$ with $l$ . Note the formula above can also be written as

\operatorname {Ref} _{l}(v)=2\operatorname {Proj} _{l}(v)-v,

saying that a reflection of $v$ across $l$ is equal to 2 times the projection of $v$ on $l$ , minus the vector $v$ . Reflections in a line have the eigenvalues of 1, and −1.

Reflection through a hyperplane in n dimensions

Given a vector $v$ in Euclidean space $\mathbb {R} ^{n}$ , the formula for the reflection in the hyperplane through the origin, orthogonal to $a$ , is given by

\operatorname {Ref} _{a}(v)=v-2{\frac {v\cdot a}{a\cdot a}}a,

where $v\cdot a$ denotes the dot product of $v$ with $a$ . Note that the second term in the above equation is just twice the vector projection of $v$ onto $a$ . One can easily check that

$Ref a (v) = - v$ , if $v$ is parallel to $a$ , and
$Ref a (v) = v$ , if $v$ is perpendicular to $a$ .

Using the geometric product, the formula is

\operatorname {Ref} _{a}(v)=-{\frac {ava}{a^{2}}}.

Since these reflections are isometries of Euclidean space fixing the origin they may be represented by orthogonal matrices. The orthogonal matrix corresponding to the above reflection is the matrix

R=I-2{\frac {aa^{T}}{a^{T}a}},

where $I$ denotes the $n\times n$ identity matrix and $a^{T}$ is the transpose of a. Its entries are

R_{ij}=\delta _{ij}-2{\frac {a_{i}a_{j}}{\left\|a\right\|^{2}}},

where $δ ij$ is the Kronecker delta.

The formula for the reflection in the affine hyperplane $v\cdot a=c$ not through the origin is

\operatorname {Ref} _{a,c}(v)=v-2{\frac {v\cdot a-c}{a\cdot a}}a.

Notes

↑ "Reflexion" is an archaic spelling
↑ Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, p. 251, ISBN 9780387745275
↑ Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning, p. 32, ISBN 978-1285402734
↑ Isaacs, I. Martin (1994), Algebra: A Graduate Course, American Mathematical Society, p. 6, ISBN 9780821847992

Related Research Articles

<span class="mw-page-title-main">Cartesian coordinate system</span> Most common coordinate system (geometry)

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes of the system. The point where they meet is called the origin and has $(0, 0)$ as coordinates.

Euclidean 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. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

In Euclidean geometry, an affine transformation or affinity is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.

In geometry, a normal is an object that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In mathematics, the orthogonal group in dimension $n$ , denoted $O(n)$ , is the group of distance-preserving transformations of a Euclidean space of dimension $n$ that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of $n \times n$ orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space $under the operation of composition.$

In mathematics, an isometry is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος isos meaning "equal", and μέτρον metron meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion.

In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842-3) and Kelvin (1845).

In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space, the affine group consists of those functions from the space to itself such that the image of every line is a line.

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. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire $(n - 1)$ -dimensional flat of fixed points in a $n$ -dimensional space.

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H $, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.$

In Euclidean geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.

In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space $; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n).$

In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types: translations, rotations, reflections, and glide reflections.

In Euclidean geometry, of special interest are involutions which are linear or affine transformations over the Euclidean space Rⁿ. Such involutions are easy to characterize and they can be described geometrically.

In geometry, a three-dimensional space is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called 3-manifolds. The term may also refer colloquially to a subset of space, a three-dimensional region, a solid figure.

<span class="mw-page-title-main">Hyperboloid model</span> Model of n-dimensional hyperbolic geometry

In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S⁺ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S⁺ or by wedge products of m vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space.

In geometry, a point reflection is a transformation of affine space in which every point is reflected across a specific fixed point. When dealing with crystal structures and in the physical sciences the terms inversion symmetry, inversion center or centrosymmetric are more commonly used.

In mathematics, a rigid transformation is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.

References

Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930
Popov, V.L. (2001) [1994], "Reflection", Encyclopedia of Mathematics , EMS Press
Weisstein, Eric W. "Reflection". MathWorld .

External links

Reflection in Line at cut-the-knot
Understanding 2D Reflection and Understanding 3D Reflection by Roger Germundsson, The Wolfram Demonstrations Project.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] "Reflexion" is an archaic spelling

[2] Childs, Lindsay N. (2009), A Concrete Introduction to Higher Algebra (3rd ed.), Springer Science & Business Media, p. 251, ISBN 9780387745275

[3] Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning, p. 32, ISBN 978-1285402734

[4] Isaacs, I. Martin (1994), Algebra: A Graduate Course, American Mathematical Society, p. 6, ISBN 9780821847992

[1]

[2]

[3]

[4]