Difference between revisions of "Projective coordinates"
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| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''1''' , Cambridge Univ. Press (1947) {{MR|0028055}} {{ZBL|0796.14002}} {{ZBL|0796.14003}} {{ZBL|0796.14001}} {{ZBL|0157.27502}} {{ZBL|0157.27501}} {{ZBL|0055.38705}} {{ZBL|0048.14502}} </TD></TR></table> |
Revision as of 21:55, 30 March 2012
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