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A treatise of Archimedes: Geometrical solutions derived from by Heiberg J.L. (ed.)

By Heiberg J.L. (ed.)

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1(c) one may interpret the fixing of the last coordinate of the homogeneous vector to 1 as follows: take the line parallel to the first axis, intersecting the second axis in 1. This line represents the real axis on which the Euclidean numbers are fixed. Connecting the origin with on the constructed line yield the projective points. 2: For lower dimensions, we will replace the coordinates ters to avoid overloaded use of indices. with the let- Linear Transformations of Projective Points A linear transformation H from to can be represented as a , such that for and If and the matrix H is regular, we call H a homography.

For an overview of recent work in this area, see MAYER 1999. As mentioned above, a polyhedron is not a building model per se, the semantics of its parts are missing: for example surfaces have to be labeled as roof surfaces. Still a lot of researchers use polyhedra as the underlying geometric model and build their semantic models upon it. Reconstruction of buildings may not only rely on gray scale images, one can also use laser-data as in WEIDNER 1997, BRUNN AND WEIDNER 1998 or 16 1 Introduction radar data cf.

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. A definition of a projective point may be given as follows: Definition 2 (Projective point) A projective point of dimension is an element of the quotient of and the equivalence relation In other words: all vectors are equivalent with respect to and define a projective point x.

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