Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...

, of special interest are

involution Involution may refer to: * Involute, a construction in the differential geometry of curves * '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...

s which are

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

or affine transformations over the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

R^''n''. Such involutions are easy to characterize and they can be described geometrically.

Linear involutions

To give a linear involution is the same as giving an

involutory matrix In mathematics, an involutory matrix is a square matrix that is its own inverse. That is, multiplication by the matrix A is an involution if and only if A2 = I, where I is the ''n'' × ''n'' identity matrix. Involutory matric ...

, a square matrix ''A'' such that :

A^2=I \quad\quad\quad\quad (1)

where ''I'' is the identity matrix. It is a quick check that a square matrix ''D'' whose elements are all zero off the main diagonal and ±1 on the diagonal, that is, a

signature matrix In mathematics, a signature matrix is a diagonal matrix whose diagonal elements are plus or minus 1, that is, any matrix of the form:. :A=\begin \pm 1 & 0 & \cdots & 0 & 0 \\ 0 & \pm 1 & \cdots & 0 & 0 \\ \vdo ...

of the form :

D=\begin
\pm 1   & 0       & \cdots & 0       & 0      \\
0       & \pm 1   & \cdots & 0       & 0      \\
\vdots  & \vdots  & \ddots & \vdots  & \vdots \\
0       & 0       & \cdots & \pm 1   & 0      \\
0       & 0       & \cdots & 0       & \pm 1  
\end

satisfies (1), i.e. is the matrix of a linear involution. It turns out that all the matrices satisfying (1) are of the form :''A''=''U''⁻¹''DU'', where ''U'' is invertible and ''D'' is as above. That is to say, the matrix of any linear involution is of the form ''D'' up to a

matrix similarity In linear algebra, two ''n''-by-''n'' matrices and are called similar if there exists an invertible ''n''-by-''n'' matrix such that B = P^ A P . Similar matrices represent the same linear map under two (possibly) different bases, with being ...

. Geometrically this means that any linear involution can be obtained by taking

oblique reflection In Euclidean geometry, oblique reflections generalize ordinary reflections by not requiring that reflection be done using perpendiculars. If two points are oblique reflections of each other, they will still stay so under affine transformations. ...

s against any number from 0 through ''n'' hyperplanes going through the origin. (The term ''oblique reflection'' as used here includes ordinary reflections.) One can easily verify that ''A'' represents a linear involution if and only if ''A'' has the form :''A = ±(2P - I)'' for a linear projection ''P''.

Affine involutions

If ''A'' represents a linear involution, then ''x''→''A''(''x''−''b'')+''b'' is an

affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a relative by marriage in law and anthropology * Affine cipher, a special case of the more general substitution cipher * Affine comb ...

involution. One can check that any affine involution in fact has this form. Geometrically this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through ''n'' hyperplanes going through a point ''b''. Affine involutions can be categorized by the dimension of the

affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

of fixed points; this corresponds to the number of values 1 on the diagonal of the similar matrix ''D'' (see above), i.e., the dimension of the eigenspace for

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

1. The affine involutions in 3D are: * the identity * the oblique reflection in respect to a plane * the oblique reflection in respect to a line * the reflection in respect to a point.

Isometric involutions

In the case that the eigenspace for eigenvalue 1 is the

orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...

of that for eigenvalue −1, i.e., every eigenvector with eigenvalue 1 is orthogonal to every eigenvector with eigenvalue −1, such an affine involution is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

. The two extreme cases for which this always applies are the

identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...

and

inversion in a point In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

. The other involutive isometries are inversion in a line (in 2D, 3D, and up; this is in 2D a

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

, and in 3D a rotation about the line by 180°), inversion in a plane (in 3D and up; in 3D this is a reflection in a plane), inversion in a 3D space (in 3D: the identity), etc.

References

{{DEFAULTSORT:Affine Involution Affine geometry