Der Ausgangspunkt ffir die hier dargestellten Untersuchungen war die Frage, ob es lokalkompakte topologische Laguerreebenen gibt, in denen die topologische Dimension des Punktraums gr613er ist als vier. Irn Zusammenhang damit galt es zu kl~iren, ob es in kompakten topologischen projektiven Ebenen einer Dimension gr6Ber als vier Ovalelgeben kann, die in topologischer Hinsicht gutartig sind. In dieser Arbeit (3.5,3.6)werden wir beide Fragen verneinend entscheiden (wobei wir allerdings die betrachteten Ebenen als endlichdimensional voraussetzen); dabei geniigt als topologische Voraussetzung an die Ovale ihre Abgeschlossenheit im Punktraum.Dies zeigt ein weiteres Mal die einschneidende Wirkung topologischer Annahmen, da in jeder unendlichen projektiven Ebene Ovale schlechthin zu finden sind: nach einem Verfahren von Mazurkiewicz [29] kann man zum Beispiel eine 2-Kurve konstruieren und diese dann um einen Punkt vermindern, um ein Oval zu erhalten; siehe auch [2], [15].In projektiven Riiumen liegen ~ihnliche Verh~iltnisse vor: Aus unseren Aussagen fiber Ovale k6nnen wir folgern, dab in den projektiven R/iumen h6herer Dimension ( >1 3) fiber den komplexen Zahlen oder den Quaternionen keine abgeschlossenen Ovoide existieren (3.4). Weiter ergibt sich, dab der Punktraum einer endlichdimensionalen lokalkompakten zusammenh~ingen-den M6biusebene nut zweidimensional sein kann (3.2); unter der zus~itzlichen Voraussetzung, dab die Kreise lokal euklidisch sind, ist dies schon lfmger bekannt [18].Grundlage ffir diese Reihe von negativen Resultaten sind verschiedene positive Ergebnisse fiber abgeschlossene Ovale in endlichdimensionalen kompakten zusammenh~ingenden projektiven Ebenen: Die Abgeschlossenheit eines Ovals O impliziert, dab es ein topologisches Oval ist in dem Sinne, dab die Menge der Schnittpunkte einer Geraden L mit O stetig von L abhgngt, auch falls L sich gegen eine Tangente bewegt (2.

We prove that in general the critical set for photogrammetric reconstruction using lines in P3(C) is a line congruence F of order 3 and class 6; F has 10 singular points and no singular planes. The general hyperplane sections of F (ruled surfaces formed by intersecting F with linear line complexes) have genus 5. F can be found in FanG's classification of congruences of order 3, and further properties of F can be found in the literature. THE CRITICAL LOCUS FOR RECONSTRUCTION WITH POINTSThe approach to studying the critical set for reconstruction with lines, which is used here, is similar to the determination of the critical locus for reconstruction with points. It is instructional to discuss this critical locus Geometriae Dedicata 44: 223-232, 1992.

We 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. 0 1988 Academic Press. Inc.

A b s t r a c t . This 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. I n t r o d u c t i o nThe problem of 3D reconstruction is to determine the geometry of a three-dimensional scene on the basis of two-dimensional images. In computer vision it is of utmost importance to develop robust algorithms for solving this problem. It is also of importance to understand the limitations of the algorithms, which are presently available, because knowledge of such limitations guides their improvement or demonstrates their optimality.From a theoretical point of view there are two types of limitations. The first type involves sets of images, where there exist more than one essentially distinct 3D scene, each giving rise to the images. The superfluous reconstructions in this case can be thought of as "optical illusions". This type of limitation describes the absolute "bottom line" of the problem, because it involves scenes where the most optimal algorithm breaks down.The second type of limitation is specific to a given not necessarily optimal algorithm. It describes those scenes which "defeat" that particular algorithm.Currently, algorithms for 3D reconstruction are of two types. One type assumes a correspondence between sets of points in the images. We use projective geometry throughout the paper. Configurations in 3-space are considered to be distinct if they cannot be transformed into one another by a projective linear transformation. The use of the projective standpoint can be thought of as preliminary to studying the situation in euclidean space. But the projective situation is of interest in its own right, because some algorithms operate essentially within the projective setting. Generally, algorithms using a projective setting are easier to analyse and implement than algorithms which fully exploit the euclidean situation. This paper is organized as follows. In section 2 we collect some standard definitions from line geometry, which will allow us to describe the line sets in Sections 3 and 4. In Section 3 we describe line sets gr in 3-space and images of ~ which give rise to ambiguous reconstructions. In Section 4 we describe line sets F in 3-space which defeat the algorithm introduced in [7]. Essential properties of F were first noted in [10, p. 106] in the context of constructive geometry. In Section 5 we discuss the relationship between ~ and F.

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