Wurf

An ordered set of $ n + 2 $ points in an $ n $- dimensional projective space if $ n > 1 $, and of four points if $ n = 1 $. If $ n > 1 $, no $ n + 1 $ 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 $ P _ {0} $, $ P _ {1} $, $ P _ \infty $— the so-called reduced wurfs. In this manner operations over wurfs are reduced to operations over points.

The sum of two points $ A $ and $ B $( other than $ P _ \infty $) is the point $ A + B $ which corresponds to $ P _ {0} $ under the hyperbolic involution $ ( AB)( P _ \infty P _ \infty ) $ that interchanges $ A $ and $ B $ and leaves $ P _ \infty $ fixed. The operation of addition is both commutative and associative. The point $ P _ {0} $ is the zero element, and to each point $ A $ there corresponds an opposite point $ - A $: $ A + (- A) = P _ {0} $.

The product of two points $ A $ and $ B $ other than $ P _ {0} $, $ P _ \infty $ is the point $ A \times B $ which, together with $ P _ {1} $, forms a pair under the elliptic or hyperbolic involution $ ( AB)( P _ {0} P _ \infty ) $ that interchanges $ A $ with $ B $ and $ P _ {0} $ with $ P _ \infty $. The operation of multiplication is both commutative and associative. The point $ P _ {1} $ is the unit element, and for each point $ A $ there exists an inverse point $ A ^ {-} 1 $: $ A \times A ^ {-} 1 = P _ {1} $.

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 $ P _ {0} , P _ {1} , P _ \infty $ are the points on the projective line with coordinates $ ( 0:1) $, $ ( 1:1) $, $ ( 1:0) $, respectively, and $ A $ and $ B $ are the points with coordinates $ ( a:1) $ and $ ( b:1) $, then the sum and product of the wurf equivalence classes $ ( P _ {0} , P _ {1} , P _ \infty , A) $ and $ ( P _ {0} , P _ {1} , P _ \infty , B) $ are the wurf equivalence classes $ ( P _ {0} , P _ {1} , P _ \infty , A+ B) $ and $ ( P _ {0} , P _ {1} , P _ \infty , A \times B) $, where $ A+ B $ and $ A \times B $ have the coordinates $ ( a+ b:1) $ and $ ( ab:1) $.

References

[a1]	O. Veblen, J.W. Young, "Projective geometry" , 1–2 , Blaisdell (1946)