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- W2387017771 abstract "It is unsatisfactory to develop the projective and affine geometries by means of vector spaces in the traditional form. For affine geometry, we specified a point as the origin to form position vectors and regarded them as geometrical points. For projective geometry, we made use of homogeneous vectors and regarded them as points. Both kinds of vectors are not so convenient as the free vectors. In this paper, we develop it in the particle space(another name of free vector space). We obtain the projective geometry when we interpret two dependent particles as “at the same position”. We obtain affine geometry when we specify in the space a certain linear form as the mass function. Finally we obtain Euclidean geometry when we introduce the length function of a vector. To denote the positions in the space, we make use of the frame of reference instead of the base itself. We discuss in detail the position quantities, position formulas and position transformations and their position preserved coefficient. In particular, we introduce the representative transformations of a given geometric one. We have really two traditional projective transformations, the one that preserves collineality would have semi linear transformation as its representative, while the other that preserve cross rarios would have linear transformation as its representative. We show also that in the spaces of dimensions over two,the position transformation preserving addition must possess a universal position preserved coefficient, and hence it must also be semi linear. Any transformation which preserves the cross ratios according to isomorphism would have the semi linear transformations as its representative. In the affine space, the frame of reference containing at least one mass point may be replaced by the base in which every base point is of the same mass.In such space, we have the mass preserving transformations f in which mf(X)=p(mX) for some p. Every mass preserving transformation must have universal position preserved coefficients. Finally, we show that the affine transformations, which were defined as the point point transformations in affine space preserving collinearity, may now, in particle space, be defined as the mass preserving semi linear transformations." @default.
- W2387017771 created "2016-06-24" @default.
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- W2387017771 date "2000-01-01" @default.
- W2387017771 modified "2023-09-24" @default.
- W2387017771 title "GEOMETRIES IN PARTICLE SPACES" @default.
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