By John Bonnycastle
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Additional info for An Introduction To Mensuration And Practical Geometry; With Notes, Containing The Reason Of Every Rule
Visualization of dual entities. Left: the dual of a 2D point x is a 2D line: middle: the dual of a 3D point X is a plane; right: the dual of a 3D line L is again a 3D line. the distance of the dual entities is inverse to the distance of the original entities. Dual Lines One question has not been answered yet: what is the dual of a 3D line L? 10). 6). The dot-product is actually an instance of the socalled cap-product, denoted by < U, A >: in general the cap-product is defined as a product of elements of two dual projective subspaces onto More precisely, the cap-product is defined for the two corresponding vector spaces where the sum of the dimensions of the underlying vector subspaces is equal to the dimension of the vector space.
Since we restrict our application to polyhedral objects, we are interested in linear entities, although there exists work on a probabilistic modeling of general curves, see for example BLAKE et al. 1998. As opposed to representing geometric primitives, an example for the uncertain representation of projective invariants (MUNDY 1992) is the work by MAYBANK 1995. He develops a probability density function (pdf) for cross-ratios in images and defines a decision rule based on the pdf. This may be extended to more complicated invariants.
For a detailed and more precise introduction to projective geometry in the context of Computer Vision, see FAUGERAS AND LUONG 2001 or chapter 23 in MUNDY 1992. 1 Projective Space and Homogeneous Coordinates Generally, a projective space is the set of one-dimensional vector subspaces of a given vector space Each one-dimensional vector subspace represents a projective point x with the projective dimension 0. 1 (a) on the facing page: the projective points can be identified with lines going through the origin of the vector space Thus these lines are one-dimensional subspaces of Note that no coordinate system but only the origin is defined here.
An Introduction To Mensuration And Practical Geometry; With Notes, Containing The Reason Of Every Rule by John Bonnycastle