AFFINE AND PROJECTIVE PLANES CONSTRUCTED FROM RINGS
Flamure SADIKI, Alit IBRAIMI, Miranda XHAFERI, Merita BAJRAMI, Mirlinda SHAQIRI
Abstract
In this paper, firstly, we show that there can be constructed an affine plane Α(F) from ternary ring in natural way. Firstly, we present the basic properties of affine and projective planes including their completion with each other, respectively, then we continue with their definition over a skew-field. Considering that not all affine planes are of the formA^2 (F), we use the Desargues properties to characterize them. Mathematically projective geometry if even more natural than its affine version.
The work continues by obtaining the projective planes Ρ(F) by “completing” the plane constructed from ternary system , by means of projective completion and then constructing affine planes from projective planes by means of affine restriction. One should add a new point “at infinity” for each direction, there will also be a line “at infinity”. Affine lines l are too short, we must force the projective line to contain the direction l ̂=l∪{[l]}. The concepts are equivalent, if you have got one, you have got the other. In the end we show the process of affinization and projectivization of the projective and affine plane.
Affinization of projectivization of an affine plane Α(F) may depend on the choise of line removed from Α ̂(F), and need not be isomorfiphic to Α(F).
Pages:
392 - 397