# Search Results for "projective-geometry-from-foundations-to-applications"

## Projective Geometry

*From Foundations to Applications*

**Author**: Albrecht Beutelspacher,Ute Rosenbaum**Publisher:**Cambridge University Press**ISBN:**9780521483643**Category:**Mathematics**Page:**258**View:**1022

A textbook on projective geometry that emphasises applications in modern information and communication science.

## Parallel Coordinates

*Visual Multidimensional Geometry and Its Applications*

**Author**: Alfred Inselberg**Publisher:**Springer Science & Business Media**ISBN:**0387686282**Category:**Mathematics**Page:**554**View:**9562

This is one book that can genuinely be said to be straight from the horse’s mouth. Written by the originator of the technique, it examines parallel coordinates as the leading methodology for multidimensional visualization. Starting from geometric foundations, this is the first systematic and rigorous exposition of the methodology's mathematical and algorithmic components. It covers, among many others, the visualization of multidimensional lines, minimum distances, planes, hyperplanes, and clusters of "near" planes. The last chapter explains in a non-technical way the methodology's application to visual and automatic data mining. The principles of the latter, along with guidelines, strategies and algorithms are illustrated in detail on real high-dimensional datasets.

## Uncertain Projective Geometry

*Statistical Reasoning for Polyhedral Object Reconstruction*

**Author**: Stephan Heuel**Publisher:**Springer Science & Business Media**ISBN:**3540220291**Category:**Mathematics**Page:**210**View:**6108

Algebraic projective geometry, with its multilinear relations and its embedding into Grassmann-Cayley algebra, has become the basic representation of multiple view geometry, resulting in deep insights into the algebraic structure of geometric relations, as well as in efficient and versatile algorithms for computer vision and image analysis. This book provides a coherent integration of algebraic projective geometry and spatial reasoning under uncertainty with applications in computer vision. Beyond systematically introducing the theoretical foundations from geometry and statistics and clear rules for performing geometric reasoning under uncertainty, the author provides a collection of detailed algorithms. The book addresses researchers and advanced students interested in algebraic projective geometry for image analysis, in statistical representation of objects and transformations, or in generic tools for testing and estimating within the context of geometric multiple-view analysis.

## David Hilbert’s Lectures on the Foundations of Geometry 1891–1902

**Author**: Michael Hallett,Ulrich Majer**Publisher:**Springer Science & Business Media**ISBN:**9783540643739**Category:**Mathematics**Page:**661**View:**6310

This volume contains six sets of notes for lectures on the foundations of geometry held by Hilbert in the period 1891-1902. It also reprints the first edition of Hilbert’s celebrated Grundlagen der Geometrie of 1899, together with the important additions which appeared first in the French translation of 1900. The lectures document the emergence of a new approach to foundational study and contain many reflections and investigations which never found their way into print.

## An Essay on the Foundations of Geometry

**Author**: Bertrand Russell**Publisher:**Cosimo, Inc.**ISBN:**1602068585**Category:**Mathematics**Page:**220**View:**357

Bertrand Russell was a prolific writer, revolutionizing philosophy and doing extensive work in the study of logic. This, his first book on mathematics, was originally published in 1897 and later rejected by the author himself because it was unable to support Einstein's work in physics. This evolution makes An Essay on the Foundations of Geometry invaluable in understanding the progression of Russell's philosophical thinking. Despite his rejection of it, Essays continues to be a great work in logic and history, providing readers with an explanation for how Euclidean geometry was replaced by more advanced forms of math. British philosopher and mathematician BERTRAND ARTHUR WILLIAM RUSSELL (1872-1970) won the Nobel Prize for Literature in 1950. Among his many works are Why I Am Not a Christian (1927), Power: A New Social Analysis (1938), and My Philosophical Development (1959).

## Geometric Algebra with Applications in Engineering

**Author**: Christian Perwass**Publisher:**Springer Science & Business Media**ISBN:**3540890688**Category:**Computers**Page:**386**View:**747

The application of geometric algebra to the engineering sciences is a young, active subject of research. The promise of this field is that the mathematical structure of geometric algebra together with its descriptive power will result in intuitive and more robust algorithms. This book examines all aspects essential for a successful application of geometric algebra: the theoretical foundations, the representation of geometric constraints, and the numerical estimation from uncertain data. Formally, the book consists of two parts: theoretical foundations and applications. The first part includes chapters on random variables in geometric algebra, linear estimation methods that incorporate the uncertainty of algebraic elements, and the representation of geometry in Euclidean, projective, conformal and conic space. The second part is dedicated to applications of geometric algebra, which include uncertain geometry and transformations, a generalized camera model, and pose estimation. Graduate students, scientists, researchers and practitioners will benefit from this book. The examples given in the text are mostly recent research results, so practitioners can see how to apply geometric algebra to real tasks, while researchers note starting points for future investigations. Students will profit from the detailed introduction to geometric algebra, while the text is supported by the author's visualization software, CLUCalc, freely available online, and a website that includes downloadable exercises, slides and tutorials.

## Perspectives on Projective Geometry

*A Guided Tour Through Real and Complex Geometry*

**Author**: Jürgen Richter-Gebert**Publisher:**Springer Science & Business Media**ISBN:**9783642172861**Category:**Mathematics**Page:**571**View:**8825

Projective geometry is one of the most fundamental and at the same time most beautiful branches of geometry. It can be considered the common foundation of many other geometric disciplines like Euclidean geometry, hyperbolic and elliptic geometry or even relativistic space-time geometry. This book offers a comprehensive introduction to this fascinating field and its applications. In particular, it explains how metric concepts may be best understood in projective terms. One of the major themes that appears throughout this book is the beauty of the interplay between geometry, algebra and combinatorics. This book can especially be used as a guide that explains how geometric objects and operations may be most elegantly expressed in algebraic terms, making it a valuable resource for mathematicians, as well as for computer scientists and physicists. The book is based on the author’s experience in implementing geometric software and includes hundreds of high-quality illustrations.

## Symmetry and Pattern in Projective Geometry

**Author**: Abby Enger**Publisher:**N.A**ISBN:**9781681176499**Category:****Page:**312**View:**3489

We are all familiar with Euclidean geometry and with the fact that it describes our three dimensional world so well. In Euclidean geometry, the sides of objects have lengths, intersecting lines determine angles between them, and two lines are said to be parallel if they lie in the same plane and never meet. Moreover, these properties do not change when the Euclidean transformations (translation and rotation) are applied. Since Euclidean geometry describes our world so well, it is at first tempting to think that it is the only type of geometry. However, when we consider the imaging process of a camera, it becomes clear that Euclidean geometry is insufficient: Lengths and angles are no longer preserved, and parallel lines may intersect. Euclidean geometry is actually a subset of what is known as projective geometry. Projective geometry exists in any number of dimensions, just like Euclidean geometry. Projective geometry has its origins in the early Italian Renaissance, particularly in the architectural drawings of Filippo Brunelleschi (1377-1446) and Leon Battista Alberti (1404-72), who invented the method of perspective drawing. Projective geometry deals with the relationships between geometric figures and the images, or mappings that result from projecting them onto another surface. Common examples of projections are the shadows cast by opaque objects and motion pictures displayed on a screen.First of all, projective geometry is a jewel of mathematics, one of the outstanding achievements of the nineteenth century, a century of remarkable mathematical achievements such as non-Euclidean geometry, abstract algebra, and the foundations of calculus. Projective geometry is as much a part of a general education in mathematics as differential equations and Galois theory. Moreover, projective geometry is a prerequisite for algebraic geometry, one of today's most vigorous and exciting branches of mathematics. Secondly, for more than fifty years projective geometry has been propelled in a new direction by its combinatorial connections. The challenge of describing a classical geometric structure by its parameters - properties that at first glance might seem superficial - provided much of the impetus for finite geometry, another of today's flourishing branches of mathematics.

## Foundations of Geometric Algebra Computing

**Author**: Dietmar Hildenbrand**Publisher:**Springer Science & Business Media**ISBN:**3642317944**Category:**Computers**Page:**196**View:**1131

The author defines “Geometric Algebra Computing” as the geometrically intuitive development of algorithms using geometric algebra with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric algebra as a powerful, intuitive mathematical language for engineering applications in academia and industry. The related technology is driven by the invention of conformal geometric algebra as a 5D extension of the 4D projective geometric algebra and by the recent progress in parallel processing, and with the specific conformal geometric algebra there is a growing community in recent years applying geometric algebra to applications in computer vision, computer graphics, and robotics. This book is organized into three parts: in Part I the author focuses on the mathematical foundations; in Part II he explains the interactive handling of geometric algebra; and in Part III he deals with computing technology for high-performance implementations based on geometric algebra as a domain-specific language in standard programming languages such as C++ and OpenCL. The book is written in a tutorial style and readers should gain experience with the associated freely available software packages and applications. The book is suitable for students, engineers, and researchers in computer science, computational engineering, and mathematics.