By Lucien Guillou, Alexis Marin
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Additional info for A la recherche de la topologie perdue: I Du côté de chez Rohlin, II Le côté de Casson
2 Representation of Geometric Entities 25 in 2D and 3D are covered. They are represented using homogeneous vectors, but we will give a Euclidean interpretation for the vectors of every entity. At the end of this section, we will see that the chosen representations are special cases of the Plücker coordinates, a representation of subspaces within a projective space. 1(d) on page 21. g. vertical lines for the slope-intercept form) and some require more parameters than necessary: the determinant form requires two points hence four parameters opposed to the fact that only two parameters are necessary, for example angle and distance.
Note that again the point and its dual line are on opposite sides of the origin and the dual has an inverse distance to the origin with respect to the original entity. To summarize, the possible cap-products in 2D and 3D are: entity 1 dim. of vector subspace dual entity 2 dim. of vector subspace dim. of vector space point x 1 line l 2 1+2=3 point X 1 plane A 3 1+3=4 line L 2 line M 2 2+2=4 cap-product Usually the cap-product is defined in the context of the so-called double or Grassmann-Cayley algebra, where operations such as join and intersection form an algebra on projective subspaces.
24). 3 on page 117. Note that because we have subdivided all homogeneous vectors of geometric entities into a homogeneous and a Euclidean part, the conditioning of geometric entities is very simple and works consistently for homogeneous vectors for points, lines and planes within a coordinate frame. 5 Duality Principle The duality principle is an important feature of projective geometry as all propositions occur twice: taking the dual of a true proposition yields a new proposition, possibly with a different meaning.