Showing papers on "Pencil (mathematics) published in 1995"

NURB curves and surfaces: from projective geometry to practical use

Abstract: NURBS (Non-uniform rational B-splines) have become a de facto standard for geometric definition in CAD/CAM and computer graphics. This book covers NURBS from their geometric beginnings to their industrial applications. The text begins with an introduction to projective geometry for which only an elementary background in linear algebra is necessary. Conics are then treated in terms of projective geometry as well as rational quadratic NURBS. A similar treatment is given to the general case of NURBS curves and surfaces. Each chapter concludes with a set of problems.

The Geometry of Projective Reconstruction I: Matching Constraints and the Joint Image

Abstract: This paper studies the geometry of perspective projection into multiple images and the matching constraints that this induces between the images. The combined projections produce a 3D subspace of the space of combined image coordinates called the joint image. This is a complete projective replica of the 3D world defined entirely in terms of image coordinates, up to an arbitrary choice of certain scale factors. Projective reconstruction is a canonical process in the joint image requiring only the rescaling of image coordinates. The matching constraints tell whether a set of image points is the projection of a single world point. In 3D there are only three types of matching constraint: the fundamental matrix, Shashua's trilinear tensor, and a new quadrilinear 4 image tensor. All of these fit into a single geometric object, the joint image Grassmannian tensor. This encodes exactly the information needed for reconstruction: the location of the joint image in the space of combined image coordinates.

Divisors on the blow-up of the projective plane

Abstract: We study the ampleness of divisors on the blow-up of the complex projective plane atk generic points, and give an affirmative answer to a question of Fujita.

The case of Cabri-géomètre: learning geometry in a computer based environment

Abstract: One of the reasons teachers may resist the use of computers in the mathematics classrooms is the change that this introduction implies about the kind of problems fostering the acquisition of knowledge by the pupil. Some old problems become uninteresting in the new environment. Conversely computers allow the design of some new problems that are impossible in a paper and pencil environment. In our paper we consider the computer as part of the “milieu didactique” : as such it offers specific representations of knowledge, specific facilities and specific feedback to the pupils. We will discuss the role of these characteristics on the way mathematical objects may be perceived on the interface by the pupils and on the solving processes of the pupils when they are confronted to problem situations on the computer, using the geometry program Cabri-geometre.

Some recent results on the projective evolution of 2-D curves

Abstract: In this paper, we begin to explore the evolution of curves of the projective plane according to a family of intrinsic equations generalizing a "projective heat equation". This is motivated by previous work for the Euclidean and the affine case, as well as by applications in the perception of two-dimensional shapes. We establish the projective arclength evolution and the projective curvature evolution. Among this family of equations, we point out the ones preserving an important property of the Euclidean and affine heat equations that was not preserved in the projective case: a curve with constant curvature should remain such a curve during its evolution.

Free pencils on divisors

Abstract: Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the restriction of a base point free pencil on $X$. This result is then extended to families of pencils and to morphisms to arbitrary smooth curves. Serrano had already studied this problem in the case n=2 and 3, and Reider had then attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's instability theorem on a surface (char(k)=0). The argument given here is based on Bogomolov's theorem on an n-dimensional variety, and on its recent adaptations to the setting of prime charachterstic (due to Shepherd-Barron and Moriwaki).

Shape as a Perturbation to Projective Mapping

Abstract: In the classical computer-graphics approach to three-dimensional rendering, a scene is described in terms of its geometry and surface properties. Given this representation, the rendering task can be thought of as a physical simulation problem. Recently, there has been an increased interest in alternate rendering paradigms, most of which fall into a category that we call image-based rendering. In these alternate approaches, reference images, rather than geometry and surface properties, are used as the primary scene description. In this paper we describe a simple extension to the well known results of projective geometry, resulting in an efficient foundation for the construction of image-based rendering systems.

The features for recognition of projectively deformed point sets

Abstract: The paper deals with features of a point set which are invariant with respect to the projective transform. First, some properties of the area of a triangle under a projective transform and the cross-ratio are discussed. Two types of projective triangular invariants of point sets are derived. One of them is a quotient of two different products of the areas of triangles formed by the vertices of the polygon. It is invariant to the labeling of the vertices. The number of the features is discussed and a method for its reduction is presented. The other type of features is expressed as functions of a five-point cross-ratios and it is invariant to the choice of labeling of the points. The possibility of the normalization of these invariants is shown. The performance of the features is demonstrated by numerical experiments.

Projective geometry in engineering

Abstract: After a historical introduction some applications of projective geometry are indicated by the modern calculus of linear algebra

Quest-Sort: A Paper-and-Pencil Alternative to Card-Sorting Q Samples

Abstract: Under conditions that might make a traditional card-sorting task infeasible, the Quest-Sort alternative presented here could be useful. A sample of 31 students demonstrates reasonable comparability of results between the formats, with the Quest-Sort about one-third faster to complete that the card-sort. A second sample of employed students demonstrates the internal validity of Quest-Sort rankings relative to both card-sorted and paired-comparisons rankings. Second-order factor analysis also suggests that the card-sort and the Quest-Sort elicit conceptually similar structures.

Projective Structures and Contact Forms

Abstract: In this paper we show that a projective structure on a real simply connected smooth manifold M determines a family of local Lie algebras on M (i.e., a family of Lie brackets on C ~ ( M ) that satisfy the localization condition). As was shown in Kirillov's paper [4], a nondegenerate local Lie algebra is determined by a contact or a symplectic structure on the manifold. The relation between projective and contact geometries has already been mentioned by Cartan (see [9] and also [2]). Many problems of projective differential geometry lead to contact structures. It turns out that this relationship is absolutely direct. Seemingly, the following elementary (and curious) fact remains unnoticed. T h e o r e m 1. Let M be a simply connected manifold (possibly noncompact) of dimension 2k 1. For a projective structure on M to exist it is necessary and sufficient that there be functions f l , . . . , f2k e C ~ ( M ) such that the 1-form k o~ = E ( f 2 i 1 d f 2 i f2idf2i-1) (1) i=1 is contact. Thus, a locally projective manifold of odd dimension is contact. The corresponding local Lie algebra is nondegenerate and is determined by a Lagrange bracket. The choice of the form (1) is not unique. An even-dimensional manifold with a projective structure must not be symplectic. However, this holds if the projective structure is affine. From this viewpoint, there is a sharp difference between even-dimensionai and odd-dimensional projective structures, while there is a strict analogy between "projective-contact" and "affine-symplectic" geometries. T h e o r e m 1'. For a simply connected manifold M of dimension 2k to possess an affine structure it is necessary and sufficient that there be functions f l , . . . , f2k E C¢~(M) such that the 2-form

Controllability, zeros, and filtrations for singular systems

Abstract: The global zero module of a matrix pencil measures controllability and uncontrollability of an associated singular linear system. If the global zero module vanishes, then the polynomial filtration on the Wedderburn-Forney space of the pencil kernel corresponds to the global controllability filtration of the system. This correspondence gives a new structural interpretation of a matrix pencil's Kronecker indices.

4-Dimensional compact projective planes with small nilradical

Abstract: We consider 4-dimensional compact projective planes with a solvable 6-dimensional collineation group Σ and with orbit type ≥ (2, 1), i.e. Σ fixes a flagv eW, acts transitively onLυ\{W} and fixes no point in the setW\{v}. We We prove a series of lemmas concerning the action of invariant subgroups of Σ. These lemmas are applied to prove that the maximal connected nilpotent invariant subgroup of Σ has dimension at least 4.

A Fully Projective Error Model for Visual Reconstruction

Abstract: Measurement uncertainty is a recurrent concern in visual reconstruction. Image formation and 3D structure recovery are essentially projective processes that do not quite fit into the classical framework of affine least squares, so intrinsically projective error models must be developed. This paper describes initial theoretical work on a fully projective generalization of affine least squares. The result is simple and projectively natural and works for a wide variety of projective objects (points, lines, hyperplanes, and so on). The affine theory is contained as a special case, and there is also a canonical probabilistic interpretation along the lines of the classical leastsquares/ Gaussian/approximate log-likelihood connection. Standard linear algebra often suffices for practical calculations.

Line testing apparatus

Abstract: The utility model discloses a line testing device, composed of an electricity testing pencil and a testing pencil or two testing pencils. The electricity testing pencil and the testing pencil or two testing pencils are connected through conducting wires in the middle. A testing contact head of the electricity testing pencil and the testing probe head end of the testing pencil form a needle shape. The utility model has the advantages of convenient operation, and low cost. The utility model is safe. The utility model can test the make-and-break of the fire wire and the zero wire. The utility model is an ideal tool for testing lines.