Reflection
A mapping of an
-dimensional simply-connected space
of constant curvature (i.e. of a Euclidean or affine space
, a sphere
or a hyperbolic (Lobachevskii) space
) the set of fixed points
of which is an
-dimensional hyperplane. The set
is called the mirror of the mapping
; in other words,
is a reflection in
. Every reflection is uniquely defined by its mirror. The order (period) of a reflection in the group of all motions of
is equal to 2, i.e.
.
The Euclidean or affine space can be identified with the vector space
of its parallel translations. The mapping
is then a linear orthogonal transformation of
with matrix
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in a certain orthonormal basis, and conversely, every orthogonal transformation of with this matrix in a certain orthonormal basis is a reflection in
. More generally, a linear transformation
of an arbitrary vector space
over a field
, of characteristic other than 2, is called a linear reflection if
and if the rank of the transformation
is equal to
. In this case, the subspace
of fixed vectors relative to
has codimension
in
, the subspace
of eigenvectors with eigenvalue
has dimension
and
. If
is a linear form on
such that
when
, and if
is an element such that
, then
is defined by the formula
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The description of a reflection in an arbitrary simply-connected space of constant curvature can be reduced to the description of linear reflections in the following way. Every such space
can be imbedded as a hypersurface in a real
-dimensional vector space
in such a way that the motions of
can be extended to linear transformations of
. Moreover, in a suitable coordinate system in
the equations of the hypersurface can be written in the following way:
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Every hypersurface in , given this imbedding, is the intersection with
of a certain
-dimensional subspace in
, and every reflection in
is induced by a linear reflection in
.
If, in the definition of a linear reflection, the requirement that is dropped, then the more general concept of a pseudo-reflection is obtained. If
is the field of complex numbers and
is a pseudo-reflection of finite order (not necessarily equal to 2), then
is called a unitary reflection. Every biholomorphic automorphism of finite order of a bounded symmetric domain in a complex space the set of fixed points of which has a complex codimension 1 is also called a unitary reflection.
See also Reflection group.
References
[1] | N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Hermann (1968) pp. Chapts. 4–6 |
[2] | E.B. Vinberg, "Discrete linear groups generated by reflections" Math. USSR Izv. , 35 : 5 (1971) pp. 1083–1119 Izv. Akad. Nauk SSSR Ser. Mat. , 35 : 5 (1971) pp. 1072–1112 |
[3] | E. Gottschling, "Reflections in bounded symmetric domains" Comm. Pure Appl. Math. , 22 (1969) pp. 693–714 |
[4] | B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian) |
Comments
The spelling reflexion also occurs in the literature.
A basic fact is that the reflections generate the orthogonal group; see [a2], Sects. 8.12.12, 13.7.12.
References
[a1] | B.A. [B.A. Rozenfel'd] Rosenfel'd, "A history of non-euclidean geometry" , Springer (1988) (Translated from Russian) |
[a2] | M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French) |
[a3] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |
[a4] | M. Greenberg, "Euclidean and non-euclidean geometry" , Freeman (1980) |
[a5] | B. Artmann, "Lineare Algebra" , Birkhäuser (1986) |
[a6] | P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958) |
Reflection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reflection&oldid=15842