Projective Geometry

In a recent paper, two of the authors used polarities in PG(2d−1,p) (p⩾2 prime, d⩾2) to construct non-geometric designs having the same parameters and the same p-rank as the geometric design PGd(2d,p) having as blocks the d-subspaces in the projective space PG(2d,p), hence providing the first known infinite family of examples where projective geometry designs are not characterized by their

We address the problem of estimating three-dimensional motion, and structure from motion with an uncalibrated moving camera. We show that point correspondences between three images, and the fundamental matrices computed from these point correspondences, are sufficient to recover the internal orientation of the camera (its calibration), the motion parameters, and to compute coherent perspective projection matrices which enable us to reconstruct 3-D structure up to a similarity. In contrast with other methods, no calibration object with a known 3-D shape is needed, and no limitations are put upon the unknown motions to be performed or the parameters to be recovered, as long as they define a projective camera. The theory of the method, which is based on the constraint that the observed points are part of a static scene, thus allowing us to link the intrinsic parameters and the fundamental matrix via the absolute conic, is first detailed. Several algorithms are then presented, and their performances compared by means of extensive simulations and illustrated by several experiments with real images.

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more ...

We introduce the notion of an n-section and reformulate a number of standard Euclidean results regarding angles in terms of 2-sections (with proof). Using 6-sections, we define the notion of reciprocal (complete) qua drangles and derive some properties of such quadrangles. While classical geometry is still admired as a model for mathematical reasoning, it is only fair to admit that following through an argument in Euclidean geometry in its full generality can be rather cumbersome. More often than not, a discussion of all manner of special cases is required. Specifically, Euc lid's notion of an angle is highly unsatisfactory. With the rise of projective geometry in the 19th century, some of these issues (such as the role of points at infinity) we re addressed. The need to resolve any of the difficulties connected with the not ion of an angle was simply obviated by (largely) avoiding any direct appeal to the concept. By the end of the 19th century, as projective geometry and metr...

Geometry and in particular projective geometry (and its corresponding invariant theory) deals a lot with structural properties of geometric objects and their interrelations. This papers describes how concepts of tensor calculus can be used to express geometric invariants and how, in particular, diagrammatic notation can be used to deal with invariants in a highly intuitive way. In particular we explain how geometries like euclidean or spherical geometry can be dealt with in this framework.

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of

Ce texte a pour but de décrire les contextes mathématiques qui permirent l'émergence d'un des groupes dits classiques, à savoir le groupe symplectique, par lequel le terme même de “symplectique”, fit son apparition en mathématiques sous la plume d'Hermann Weyl. Nous montrerons plus précisément, que ce groupe possède une double origine, l'une dans le cadre de la géométrie projective avec l'étude des complexes de droites, et l'autre dans des questions relatives aux transformations des intégrales abéliennes.The goal of this paper is the description of the mathematical contexts which allowed the emergence of one of the so called classical groups, namely the symplectic group, by which the word “symplectic” appeared in mathematics. We show that it has a double origin, one in the line complexes of the projective geometry and the other one, in some questions about transformations of abelian integrals.