Perspective
with centre
A mapping of a plane into a plane
under which each point
in
is put into correspondence with the point
of intersection of the straight line
with
(if
is not parallel to
, see Fig.).
Figure: p072410a
More generally, let and
be proper subspaces of identical dimension in a projective space
and let
be a subspace of maximal dimension not having points in common with
or
. Let
be a subspace contained in
, let
be the subspace of minimal dimension containing
and
and let
be the intersection of
and
.
The correspondence by which each subspace contained in
is put into correspondence with the subspace
contained in
is called a perspective mapping from
into
with perspective centre
.
A perspective is a collineation. If the subspaces and
intersect, then each point in the subspace
corresponds to itself.
If projective coordinates are introduced into the spaces and
, then a perspective correspondence can be specified by a linear mapping.
References
[1] | E. Artin, "Geometric algebra" , Interscience (1957) |
[2] | N.A. Glagolev, "Projective geometry" , Moscow (1963) (In Russian) |
Comments
Perspective mappings are also called central projections or perspectivities.
References
[a1] | M. Berger, "Geometry" , I , Springer (1987) |
[a2] | H. Busemann, P.J. Kelly, "Projective geometry and projective metrics" , Acad. Press (1953) |
[a3] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963) |
[a4] | H.S.M. Coxeter, "Projective geometry" , Blaisdell (1964) |
[a5] | H.S.M. Coxeter, "The real projective plane" , McGraw-Hill (1949) |
Perspective. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perspective&oldid=31589