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Geometry
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
In the real projective plane, consider the six points A= [1:2:1], B = [2:0:1],C = [5:5:1], A′= [2:4:1], B′= [4:0:1], and C′= [10:10:1]. Identifying the projective point with homogenous coordinates [a:b:1] with the ordinary Euclidean point [ab], as in Example 6 above, it can be seen by a simple diagram that the triangles ABC and A′B′C′ are perspective from the origin, since all lines AA′, BB′, and CC′ pass through this point. Thus, in the real projective plane these triangles are perspective from the point [0:0:1]. Desargues’ Theorem, therefore, asserts that the points of intersection of the lines AB and A′B′, AC and A′C′, BC and B′C′ will lie on a line. A simple calculation shows that these three points, in homogenous coordinates, have third component zero, so these points are indeed collinear. This corresponds to the fact that as Euclidean lines there are no points of intersection — the corresponding sides of the triangles are parallel in pairs. The points with third component zero are the “ideal points” that have been added to Euclidean geometry to form the real projective plane.
Geometric theory of topological defects: methodological developments and new trends
Published in Liquid Crystals Reviews, 2021
Sébastien Fumeron, Bertrand Berche, Fernando Moraes
For the isotropic-nematic phase, the order parameter space is given by : the resulting space, called the real projective plane , can be pictured as a 2-sphere having its antipodal points identified. The manifold corresponding to an immersion of the real projective plane in 3D space is called a Boy surface and its topology is encompassed into its first four homotopy groups: for uniaxial nematics in 3D, these are (no domain wall), (existence of linear defects), (existence of point defects) and (existence of textures).