PROJECTION BY GRASSMANN ALGEBRA OF E4
Emil MOLNA'R,
Budapest University of Technology and Economics, Department of Geometry
emolnar@mail.bme.hu
In the analogy of Plücker's line coordinates a general method will be
suggested for describing geometry of k-spaces in a d-dimensional real
projective metric space and for the projection of Ed (as special case
of Pd) onto a 2-dimensional screen E2 .
Although the idea is classical, no traces appeared in the references
about the topic (as far as I know). Indeed, the computer will be necessary
to these computations, based on the Grassmann algebra of the projective
spherical space PS(Vd+1, Vd+1 , R), endowed by the usual antisymmetric and
multilinear wedge (exterior) product. This machinery gives a general
framework for treating projective structure of k-spaces in Pd . E.g. the
projection from an s-plane onto a d-s-1-plane can be defined, then various
collineations with s-centre and d-s-1-axis can be discussed, etc.
We introduce a projective metric in Pd by giving a metric polarity
as a symmetric linear map (* ) , or equivalently by a scalar product. Thus
we associate a point A = (a*) = (a) as the pole for the polar
hyperplane a = (a) . Thus we can approach to Euclidean, hyperbolic,
spherical and other geometries uniformly, just by the same machinery.
Linear transforms, leaving the metric polarity invariant, define the groups
of these geometries, respectively. Then metric invariants can be calculated
as distance, angle, volume, etc. These and other applications will be
illustrated by computer pictures of Istv'n PROK.