I. BASIC TOOL: RATIONAL PARAMETRIZATION Algebraic geometry is a generalization of analytic geometry the familiar study of lines, planes, circles, parabolas, ellipses, hyperbolas, and their 3 dimensional versions: spheres, cones, hyperboloids, ellipsoids, and hyperbolic surfaces. The essential common property these all have is that they are defined by polynomials. This is the defining characteri...

De computer is niet de steen maar de slijpsteen der wijzen. (The computer is not the philosopher's stone but the philosopher's whetstone.) Hugo Battus, Rekenen op taal (1989) Contents 1 Preface 2 Introduction by pictures 3 Some problems in algebraic geometry 4 Some global algorithms 5 Singularities and standard bases 6 Some local algorithms 7 Computer algebra solutions to singularity problems 8...

Deciding how to teach geometry remains a demanding task with one of major arguments being about how to combine the intuitive and deductive aspects of geometry into an effective teaching design. In order to try to obtain an insight into tackling this issue, this paper reports an analysis of innovative geometry textbooks which were published in the early part of the 20th Century, a time when sign...

The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following subjects are discussed via the coherent states: the geodesics, the conjugate locus and the cut locus; the divisors; the Calabi's diastasis and its domain of defi...

Supernova can be polarized by an asymmetry in the explosion process, an off-center source of illumination, scattering in an envelope distorted by rotation or a binary companion, or scattering by the circumstellar dust. Careful polarimetry is the most powerful tool to study the 3-D geometry of supernovae. A deep understanding of the 3-D geometry of SNe is essential in using them as calibrated di...

This paper describes "From Dots to Shapes" (FDTS), an auditory platform composed by three classic games (“Simon”, “Point Connecting” and “concentration game”) for blind and visually impaired pupils. Each game was adapted to work on a concept of the Euclidean geometry (e.g. ) The tool, , is based on sonic and haptic interaction, and therefore could be used by special educators as a help for teac...

A lot of natural structures in industry applications can be hardly described by conventional methods – mostly statistic tools, because they are complex and irregular. A relatively new approach is the application of the fractal geometry that is successfully used in science, but an application in industry is sporadic and experimental only. However, the fractal geometry can be used as a useful too...

In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff metric. This is the first part of our series work, which we try to establish some facts about the counterpart of Gromov-Hausdorff metric in the non-Archimedean spaces. One of the motivation of this work is to find some implied relations between this geometry and number theory via p-adic analysis, so that we can use the for...

Recent studies have shown that fractal geometry, a vocabulary of irregular shapes, can be useful for describing the pathological architecture of tumors and, perhaps more surprisingly, for yielding insights into the mechanisms of tumor growth and angiogenesis that complement those obtained by modern molecular methods. This article outlines the basic methods of fractal geometry and discusses the ...

A characteristic-based constrained interpolation profile (CIP) method for solving three-dimensional, time-dependent Maxwell’s equations is successfully developed. It is utilized to solve one-dimensional wave equations in the formulation of the Maxwell’s equations. Calculation procedure of the CIP method for three-dimensional scattering analysis is described in details. Update equations for boun...