Affine geometry

All solutions of Einstein s equations are conditioned by the need of some ad hoc assumption about the geometry of space-time. The only indisputably valid assumption is that space-time is of absolute non-euclidean geometry. It is interesting to note that chiral space-time, probably demanded by the existence of antimatter and other chiral forms of matter, rules out the possibility of affine geometry, the standard assumption of modern TGR [7]. 

In the projective plane there is only one kind of conic. The familiar distinction between hyperbola, parabola and ellipse belongs to affine geometry only. 

The separation of matter and anti-matter can only happen in chiral space-time. This requirement rules out affine geometry. 

Modern geometry developed from the work of Euclid which gave rise to two self-contained geometries, known as absolute geometry and affine geometry. Euclidean geometry depends on hve postulates ... 

Barycentric coordinates can be referred to any given triangle with vertices (1,0,0), (0,1,0), (0,0,1) and imit point (1,1,1), the centroid. In contrast, projective coordinates can be applied to any quadrangle Take three of the four vertices to determine a system of barycentric coordinates and suppose that the fourth vertex is (/Ui,/U2> Ms)- Converted to projective coordinates the fourth vertex becomes (1,1,1). Whereas all triangles are alike in affine geometry, ali quadrangles are seen to be alike in projective geometry. 

By this means the theory of the underlying space may be reduced to the simultaneous affine geometry of this set of affine spaces. The tensors provide a suitable aid for treatment of this simultaneous-affine theory. As first example we take contravariant vectors or contravariant tensors of first rank. That is a geometrical object that contains four components in each coordinate system... 

The calculation is exactly the same as for the corresponding problem of affine geometry (Bibl. 1927, 22 1932, 10). 

The effect of (12) on the vector (p is that gauge transformations, that leave (15) invariant, exist, we no longer have a projective geometry, but only an affine geometry. Because of the tensor gij this affine geometry is metrical. 

Veblen, O. Projective and affine geometry of paths. Proc. Nat. Acad. Sci. 

Veblen, O., u. J.M. Thomas Projective invariants of affine geometry of paths. 

Zur allgemeinen projektiven Diherential-Geometrie I. Einordnung in die Affin-Geometrie. II. mit eingliedriger Gruppe. Proc. Akad. Wetensch. Amsterd. 

The affine geometry of chromaticity diagrams endows all of them with a number of useful properties. Most fundamental is that an additive mixture of any two lights will fall along a straight line connecting the chromaticities of the... 

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