Universität Stuttgart
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Item Open Access Topologische Ovale(1981) Buchanan, Thomas; Hähl, Hermann; Löwen, RainerDer Ausgangspunkt für die hier dargestellten Untersuchungen war die Frage, ob es lokalkompakte topologische Laguerreebenen gibt, in denen die topologische Dimension des Punktraums größer ist als vier. Im Zusammenhang damit galt es zu klären, ob es in kompakten topologischen projektiven Ebenen einer Dimension größer als vier Ovale geben kann, die in topologischer Hinsicht gutartig sind. In dieser Arbeit werden wir beide Fragen verneinend entscheiden (wobei wir allerdings die betrachteten Ebenen als endlichdimensional voraussetzen); dabei genügt als topologische Voraussetzung an die Ovale ihre Abgeschlossenheit im Punktraum.Item Open Access Ovale und Kegelschnitte in der komplexen projektiven Ebene(1979) Buchanan, ThomasDie Kegelschnitte der komplexen projektiven Ebene werden in dieser Arbeit durch ihre geometrischen und toplogischen Eigenschaften charakterisiert - genauer, durch die Ovaleigenschaft und die topologische Abgeschlossenheit.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 The topology of the flag space of a topological projective plane with 2-spheres as point rows(1979) Buchanan, Thomas-Item Open Access The transposition of locally compact, connected translation planes(1978) Buchanan, Thomas; Hähl, HermannThis note deals with the transposition of translation planes in the topological context. We show that a topological congruence C of the real vector space R 2n has the property that every hyperplane of R 2n contains a component of C. This makes it possible to define the transposeP tau of the topological translation planeP associated with C; it is proved that the translation plane P τ is topological also. The relationship between collineation groups and the relationship between coordinatizing quasifields of P and P τ are also discussed.Item Open Access On the critical set for photogrammetric reconstruction using line tokens in P3(C)(1992) Buchanan, ThomasWe prove that in general the critical set for photogrammetric reconstruction using lines in P 3 (C) is a line congruence Γ of order 3 and class 6; Γ has 10 singular points and no singular planes. The general hyperplane sections of Γ (ruled surfaces formed by intersecting Γ with linear line complexes) have genus 5. Γ can be found in Fano's classification of congruences of order 3, and further properties of Γ can be found in the literature.Item Open Access The twisted cubic and camera calibration(1988) Buchanan, ThomasWe state a uniqueness theorem for camera calibration in terms of the twisted cubic. The theorem assumes the general linear model and is essentially a reformulation of Seydewitz's star generation theorem.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 Zur Topologie der projektiven Ebenen über reellen Divisionsalgebren(1979) Buchanan, ThomasDas Ziel dieser Arbeit ist der Beweis des folgenden Satzes: Alle Punkträume von projektiven Ebenen über reellen Divisionsalgebren gleicher endlicher Dimension sind zueinander homöomorph. Überdies ist jede Punktreihe einer solchen Ebene genauso im Punktraum eingebettet wie im klassischen Fall.Item Open Access On the kernel and the nuclei of 8-dimensional locally compact quasifields(1977) Buchanan, Thomas; Hähl, HermannIn this note we determine the nuclei of 8-dimensional real nonassociative division algebras and also study the corresponding substructures of 8-dimensional locally compact quasifields.