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)