Wurf
An ordered set of points in an
-dimensional projective space if
, and of four points if
. If
, no
points of the wurf belong to a hyperplane. Two wurfs on a straight line or on a conical section will be equal if the two quadruplets of points of which they are constituted are projective. The operations of addition and multiplication can be defined for wurfs. In so doing it is expedient to use wurfs with three base points
,
,
— the so-called reduced wurfs. In this manner operations over wurfs are reduced to operations over points.
The sum of two points and
(other than
) is the point
which corresponds to
under the hyperbolic involution
that interchanges
and
and leaves
fixed. The operation of addition is both commutative and associative. The point
is the zero element, and to each point
there corresponds an opposite point
:
.
The product of two points and
other than
,
is the point
which, together with
, forms a pair under the elliptic or hyperbolic involution
that interchanges
with
and
with
. The operation of multiplication is both commutative and associative. The point
is the unit element, and for each point
there exists an inverse point
:
.
References
[1] | K.G.C. von Staudt, "Beiträge zur Geometrie der Lage" , 2 , Korn , Nürnberg (1959) pp. 166–194 |
[2] | H.S.M. Coxeter, "The real projective plane" , Springer (1992) pp. Chapt. 11 |
Comments
K. von Staudt used his "Würfe" for the coordinatization of projective geometry. Nowadays his methods are considered obsolete, although ingenious.
The modern methods began with [a1].
If are the points on the projective line with coordinates
,
,
, respectively, and
and
are the points with coordinates
and
, then the sum and product of the wurf equivalence classes
and
are the wurf equivalence classes
and
, where
and
have the coordinates
and
.
References
[a1] | O. Veblen, J.W. Young, "Projective geometry" , 1–2 , Blaisdell (1946) |
Wurf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wurf&oldid=49237