Projective coordinates
A one-to-one correspondence between the elements of a projective space (projective subspaces
) and the equivalence classes of ordered finite subsets of elements of a skew-field
. Projective coordinates of subspaces
for
(also called Grassmann coordinates) are defined in terms of coordinates of the points (
-dimensional subspaces) lying in
. Therefore it suffices to define the projective coordinates of the points of a projective space.
Suppose that in the collection of rows of elements of a skew-field
that are not simultaneously zero (they are also called homogeneous point coordinates) a left (right) equivalence relation is introduced:
if there is a
such that
(
),
. Then the collection of equivalence classes is in one-to-one correspondence with the collection
of points of the projective space
(respectively,
). If
is interpreted as the set of straight lines of the left (right) vector space
(respectively,
), then the homogeneous coordinates of a point
have the meaning of the coordinates of the vectors belonging to the straight line that represents this point, and the projective coordinates have the meaning of the collection of all such coordinates.
In the general case, projective coordinates of points of a projective space relative to some basis are introduced by purely projective means (under the necessary condition that the Desargues assumption holds in
) as follows:
A set of
independent points
of the space
is called a simplex. In this case the points
,
are also independent and determine a subspace
, which is called a face of this simplex. There exists a point
that lies on none of the faces
. Let
be any permutation of the numbers
. The points
,
, and
turn out to be independent and determine some
. Next, the points
also determine some
, and since the sum of
and
is the entire space
,
and
have exactly one common point
that lies in none of the
-dimensional subspaces determined by the points
,
; in this case the points
are also independent. Thus one obtains
points
, including the points
,
, which constitute a frame of the space
; the simplex
is its skeleton.
On each straight line there are three points
; suppose they play the role of the points
in the definition of the skew-field
of the projective geometry under consideration (see Projective algebra). The skew-fields
and
are isomorphic to one another, and the isomorphism is established by a projective correspondence
between the points of the two lines
and
such that the points
correspond to the points
. The element of the skew-field
corresponding to a point
of the straight line
is called the projective coordinate
of the point
in the scale
. In particular, the projective coordinate of
is always 1, while the projective coordinate of
in the scale
is
.
Let be a point of the space that lies on none of the faces of the simplex
:
that together with some point
forms a frame
. If one uses the point
instead of
in the above construction of the frame, then one obtains a sequence of points
where
lies in the subspace determined by
(but lies in none of the faces of the simplex
formed by these points). Let
be the coordinate of a point
(lying on
) in the scale
. If the
are distinct, then
1) ;
2) .
Let be an arbitrary element of
different from zero, and let
,
,
(in this case it turns out that
). Then the collection of equivalent rows determined by various elements
gives the projective coordinates of the point
with respect to the frame
.
Suppose that lies in the subspace
determined by the points
but in none of the faces of the simplex determined by these points. Let the collection of equivalent rows
be the projective coordinates of a point
with respect to a frame
of the subspace
determined by a simplex
and a point
. Then the projective coordinates of the point
with respect to the frame
are given as follows:
,
;
,
.
Any collection of left (right) equivalent rows constructed by the above method corresponds to one and only one point
of the space
and therefore defines projective coordinates in it.
References
[1] | W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 1 , Cambridge Univ. Press (1947) MR0028055 Zbl 0796.14002 Zbl 0796.14003 Zbl 0796.14001 Zbl 0157.27502 Zbl 0157.27501 Zbl 0055.38705 Zbl 0048.14502 |
Comments
For transformations of projective coordinates see e.g. Collineation; Projective transformation.
Projective coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_coordinates&oldid=11384