Universität Stuttgart
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Item Open Access Photogrammetry and projective geometry : an historical survey(1993) Buchanan, ThomasGeneral Jean-Victor Poncelet published his treatise on projective geometry in 1822. This was the start of an enormous development in geometry in the 19th century. During this period geometry in the plane and in 3-dimensional space was studied in particular detail. The development culminated in the publishing of the Encyclopedia of Mathematics, which appeared in irregular installments from 1900 to 1934. Photogrammetry - the use of photographic images for surveying, mapping and reconnaissance - began in the second half of the 19th century. By the 1890's substantial theoretical contributions were made by Sebastian Finsterwalder. Finsterwalder reported on his foundational work in a keynote address to the German Mathematical Society in 1897; he also contributed an article on photogrammetry to the Encyclopedia of Mathematics. Among other things Finsterwalder observed that Rudolf Sturm's analysis of the "homography problem" (1869) can be used to solve the problem of 3D-reconstruction using point matches in two images. Subsequently, important theoretical advances were made by mathematicians at the Technical University of Vienna. An excellent reference for geometry and its relationship to photogrammetry is a book of Emil Muller on constructive geometry, which appeared in 1923. Muller's assistent and successor Erwin Kruppa established the "structure-from-motion" theorem in 1913. This theorem was rediscovered by Shimon Ullman in 1977.Item Open Access Critical sets for 3D reconstruction using lines(1992) Buchanan, ThomasThis paper describes the geometrical limitations of algorithms for 3D reconstruction which use corresponding line tokens. In addition to announcing a description of the general critical set, we analyse the configurations defeating the Liu-Huang algorithm and study the relations between these sets.Item Open Access Singularities of non-rotationally symmetric solutions of boundary value problems for the Lamé equations in a 3 dimensional domain with conical points(1992) Sändig, Anna-Margarete; Sändig, RainerIt is well known that singularities are present in solutions of boundary value problems for the Lamé equations in conical domains. It follows from the general theory that the solutions consist of singular terms of the form r α (ln r) q F(α, φ, θ) (r is the distance to the vertex of the cone φ, and θ are the spherical angles) and α more regular term. Rotationally symmetric solutions of the Lamé equations under zero boundary displacements or stress free boundary conditions are investigated in (1, 2), where the values of α and q have been computed. Here we are concerned with the more general case, namely that the volume and surface forces of our problems are non rotationally symmetric. That means that the solutions depend not only on r and θ, but on the polar angle φ too. Using a monotonicity principle of Kozlov, Maz'ja und Schwab one can get regularity results for polyhedral domains too.Item Open Access Coefficient formulae for asymptotic expansions of solutions of elliptic boundary value problems near conical points(1991) Sändig, Anna-MargareteIt is well known that singularities are present in solutions of elliptic boundary value problems in domains with conical boundary points. The solution consists of singular terms, which appear in a neighbourhood of a conical point, and a more regular term. The coefficients of the singular terms, the so-called stress intensity factors, are especially of interest for applications. We describe a method, how some of them may be calculated, if the right hand sides are from standard Sobolev spaces. In some cases the coefficients are unstable and a stabilization procedure is necessary.We handle as examples boundary value problems for the Laplace equation in two and three dimensional domains.