Oriented projective geometry

Abstract: The image points in two images satisfy epipolar constraint. However, not all sets of points satisfying epipolar constraint correspond to any real geometry because there can exist no cameras and scene points projecting to given image points such that all image points have positive depth. Using the cheirability theory due to Hartley and previous work an oriented projective geometry, we give necessary and sufficient conditions for an image point set to correspond to any real geometry. For images from conventional cameras, this condition is simple and given in terms of epipolar lines and epipoles. Surprising, this is not sufficient for central panoramic cameras. Apart from giving the insight to epipolar geometry, among the applications are reducing the search space and ruling out impossible matches in stereo, and ruling out impossible solutions for a fundamental matrix computed from seven points.

Abstract: Well-known matching constraints for points and lines in muliple images are necessary but not sufficient condition for the existence of real structure and cameras, underlying the image correspondences. To obtain sufficient conditions, the following additional constraints must be imposed: positive scales, the existence of a plane at infinity not intersecting the scene, and the existence of handedness preserving cameras. We present modifications of the well-known matching constraints and also some new constraints, taking into account some of this additional knowledge. Not only conventional but also central panoramic cameras are naturally described. To achieve this, we have generalized and simplified Hartley’s ch(e)irality theory by formulating it in the language of oriented projective geometry and Grassmann tensors.

Abstract: It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.

Abstract: The theme of this thesis is dynamic geometry, a new way of exploring classical geometry using interactive computer software. This kind of software allows the user to make geometric constructions on a computer's screen. The constructions might consist of points, lines and conics whose positions have been constrained in various ways. The constraints, which may involve incidences, distances and angles, can be added and removed dynamically. For example, to force a line to always be incident on a point, the user would simply grab the line with the cursor and drop it onto the point. Any object whose position is not completely determined by the constraints can be grabbed and dragged around on the screen. The rest of the objects will then automatically self-adjust in order to keep the constraints satis ed. Dynamic geometry software is primarily used for teaching mathematics, but is useful in any situation where it is important to understand the geometric properties of a dynamic system. Over the last few years, a number of tools for dynamic geometry have been developed. Most of them have focused on elementary Euclidean geometry. In this thesis we present a new software that has been based entirely on projective concepts and thus allows us to illustrate the classical theorems of projective geometry. The software has also extensive support for di erent types of metrics, which makes it possible to explore both Euclidean and non-Euclidean geometry. In fact, the user is given direct access to the absolute elements which de ne the metric. Moreover, the system can handle objects in the complex projective plane, which permits, for example, the circular points in Euclidean geometry to be used in geometric constructions. We discuss how the user interface of a dynamic geometry system should be designed and we identify a number of problems and shortcomings which the user interfaces of all previous systems seem to su er from. Most of these defects are related to the fundamental problem of choosing the right solution of an underdetermined system of constraint equations. We show how this problems can be solved by letting the system automatically add extra constraints if necessary, and by using a richer internal representation based on oriented projective geometry. The thesis is written in English.