Affine involution
Encyclopedia
In Euclidean geometry
, of special interest are involutions which are linear
or affine transformation
s over the Euclidean space
Rn. Such involutions are easy to characterize and they can be described geometrically.
where I is the identity matrix
.
It is a quick check that a square matrix D that has zero off the main diagonal and ±1 on the diagonal, that is, a signature matrix of the form
satisfies (1), i.e. is the matrix of a linear involution. It turns out that all the matrices satisfying (1) are of the form
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 similarity. Geometrically this means that any linear involution can be obtained by taking oblique reflection
s against any number from 0 through n hyperplane
s 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
for a linear projection
P.
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
of fixed point
s; 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 1.
The affine involutions in 3D are:
. The two extreme cases for which this always applies are the identity function
and inversion in a point.
The other involutive isometries are inversion in a line (in 2D, 3D, and up; this is in 2D a reflection
, 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.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
, of special interest are involutions which are linear
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
or affine transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s over the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
Rn. Such involutions are easy to characterize and they can be described geometrically.
Linear involutions
To give a linear involution is the same as giving a square matrix A such thatwhere I is the identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
.
It is a quick check that a square matrix D that has zero off the main diagonal and ±1 on the diagonal, that is, a signature matrix of the form
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 −1DU,
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
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
a similarity. Geometrically this means that any linear involution can be obtained by taking oblique reflection
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 hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s 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
Projection (linear algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged....
P.
Affine involutions
If A represents a linear involution, then x→A(x−b)+b is an affineAffine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
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
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
of fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
s; 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 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 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 isometryIsometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
. The two extreme cases for which this always applies are the identity function
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
and inversion in a point.
The other involutive isometries are inversion in a line (in 2D, 3D, and up; this is in 2D a reflection
Reflection (mathematics)
In mathematics, a reflection 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 or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
, and in 3D a rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. 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.