Geometrie
"Zwillinge brauchen alles doppelt" - Einführung des Verdoppelns im Rahmen des Mathematikunterrichts
''h''- Cobordism
(3+3+2) Warped-Like Product Manifolds With Spin(7) Holonomy
In the theory of Riemannian holonomy groups there are two exceptional cases, the holonomy group G_2 in 7-dimensional and the holonomy group Spin(7) in 8-dimensional manifolds. In the present work, we investigate the structure of Riemannian manifolds whose holonomy group is a subgroup of Spin(7) for a special case. Manifolds with Spin(7) holonomy are characterized by the existence of a 4-form, called the Bonan form (Cayley form or Fundamental form), which is self-dual in the Hodge sense, Spin(7) invariant and closed. We review two methods for the construction of the Bonan form, based on the octonionic multiplication and the triple vector cross products on octonions. Here we define "(3+3+2) warped-like product manifolds" as a generalization of multiply warped product manifolds, by allowing the fiber metric to be non block diagonal. In this thesis we prove that the fibre spaces of (3+3+2) warped-like product manifolds are isometric to 3-sphere under some global assumptions.
1 + 2 + 4 + 8 + · · ·
1 ¿ 2 + 4 ¿ 8 + · · ·
100 construcciones geométricas manuales e informáticas
120- Cell
3D Reconstruction Of Anatomical Models Based On Geometric Knowledge
Geometrically accurate three-dimensional (3D) model reconstructions of anatomical shapes play a major role in medical sector, because these 3D models are currently required for clinical studies such as visualization, diagnosis, patient follow-up, pre-operative surgery planning, and bio-mechanical applications. Such 3D models are generally extracted from data sets, collected by 3D sensors, like computed tomography (CT) scan and magnetic resonance imaging (MRI). Those methods have various limitations such as high expensive, high radiation dose (for CT), requires a high volume of image data sets, and more time to generate 3D models. To overcome above limitations, and obtain accurate 3D models, many attempts have been taken based on 2D X-Ray imageries. However, the demand for automatic process to create 3D models of anatomical structures using X-ray images is still increasing. As such, the present work is an attempt for introducing a cost and time effective method to reconstruct 3D models of human bones automatically from multiple X-ray images using photogrammetric techniques.
3D Visual Shape Processing based on Discrete Differential Geometry
With the increase in the number of 3D scanned shapes, 3D shape processing is becoming popular in the fields of computer vision, computer graphics, computer aided design (CAD), etc. Digital Geometry Processing (DGP) is a new and much needed application area for the next wave of multimedia data: 3D geometry. Thanks to nearly two-decade efforts of many researchers, much of basic theoretical problems in the field of DGP have been solved until about 2010. This book aims to present the techniques and applications of 3D Visual Shape Processing (VSP) based on discrete differential geometry. The book considers the ¿perception¿ of the geometry in the views both of globality (Part I) and locality (Part II). In Part I, we explore some research work on global shape analysis based on manifold harmonics (i.e. the eigenfunctions of the Laplace-Beltrami operators defined on the surface). In Part II, based on local differential operators, we develop new methods for efficient and reliable tools for mesh deformation, correspondence, segmentation and many other types of DGP operations. This provides a complete mesh processing pipeline to allow for easy and efficient manipulation of 3D models.
3D-FRAKTALE
3D-rekonstrukciq anatomicheskih modelej (na osnowe geometricheskih znanij)
Geometricheski tochnye trehmernye (3D) modeli anatomicheskih form igraüt wazhnuü rol' w medicinskom sektore, poskol'ku äti 3D modeli w nastoqschee wremq neobhodimy dlq klinicheskih issledowanij, takih kak wizualizaciq, diagnostika, nablüdenie za pacientami, planirowanie predoperacionnyh operacij i biomehanicheskie prilozheniq. Takie 3D-modeli obychno izwlekaütsq iz naborow dannyh, sobrannyh 3D-datchikami, takimi kak komp'üternaq tomografiq (KT) i magnitno-rezonansnaq tomografiq (MRT). Jeti metody imeüt razlichnye ogranicheniq, takie kak wysokaq stoimost', wysokaq doza oblucheniq (dlq KT), trebuetsq bol'shoj ob#em naborow dannyh izobrazhenij i bol'she wremeni dlq sozdaniq 3D-modelej. Dlq preodoleniq wysheukazannyh ogranichenij i polucheniq tochnyh 3D modelej bylo predprinqto mnozhestwo popytok na osnowe 2D rentgenowskih izobrazhenij. Odnako potrebnost' w awtomaticheskom processe sozdaniq 3D-modelej anatomicheskih struktur na osnowe rentgenowskih izobrazhenij prodolzhaet rasti. Takim obrazom, nastoqschaq rabota predstawlqet soboj popytku wnedreniq äkonomichnogo i äffektiwnogo metoda awtomaticheskogo wosstanowleniq 3D-modelej kostej cheloweka po neskol'kim rentgenowskim izobrazheniqm s ispol'zowaniem fotogrammetricheskih metodow.
3D-Rekonstruktion anatomischer Modelle (auf der Grundlage geometrischer Kenntnisse)
Geometrisch genaue dreidimensionale (3D) Modellrekonstruktionen anatomischer Formen spielen im medizinischen Bereich eine wichtige Rolle, da diese 3D-Modelle derzeit für klinische Studien wie Visualisierung, Diagnose, Patientennachsorge, präoperative Operationsplanung und biomechanische Anwendungen benötigt werden. Solche 3D-Modelle werden im Allgemeinen aus Datensätzen extrahiert, die von 3D-Sensoren wie Computertomographie (CT) und Magnetresonanztomographie (MRT) erfasst wurden. Diese Methoden haben verschiedene Einschränkungen, wie z. B. hohe Kosten, eine hohe Strahlendosis (bei CT), ein großes Volumen an Bilddatensätzen und mehr Zeit für die Erstellung von 3D-Modellen. Um die oben genannten Einschränkungen zu überwinden und genaue 3D-Modelle zu erhalten, wurden viele Versuche auf der Grundlage von 2D-Röntgenbildern unternommen. Die Nachfrage nach automatischen Verfahren zur Erstellung von 3D-Modellen anatomischer Strukturen anhand von Röntgenbildern steigt jedoch weiter. Die vorliegende Arbeit ist daher ein Versuch, eine kosten- und zeiteffiziente Methode zur automatischen Rekonstruktion von 3D-Modellen menschlicher Knochen aus mehreren Röntgenbildern mit Hilfe photogrammetrischer Verfahren einzuführen.
50 Jahre Bundeswettbewerb Mathematik
5000 Jahre Geometrie
5000 Years of Geometry
A Brief Introduction to Berezin-Toeplitz Operators on Compact Kähler Manifolds
A Brief Introduction to Berezin-Toeplitz Operators on Compact Kähler Manifolds
A Brief Journey from Euclidean to Smarandachean Geometry
The aim of this book is straightforward: to illustrate the true essence of geometry and justify the necessity for the existence of Smarandache geometry. This goal is accomplished by narrating the history of this mathematical field from its origins, akin to a captivating tale featuring characters fixated on depicting spatial forms throughout millennia. The mathematicians portrayed in this narrative exemplify that talent is not solely an intellectual attribute but also demands audacity for transcendence. The first chapter summarizes the book's content, aiming to captivate the reader. The second chapter covers Euclidean Geometry, while the third chapter explores hyperbolic and elliptical geometries, as well as Taxicab and finite geometries. Chapter 4 delves into Smarandache geometries and NeutroGeometries, presenting known and new models. Lastly, the Appendix provides access to Julia language code utilized for generating three-dimensional figures showcased in the book, with the intention of inspiring adept readers to develop software enabling exploration of these spaces virtually. It is hoped that this narrative will engross readers and leave them eager for more at its conclusion.
A Budget of Trisections
It is impossible to trisect angles with straightedge and compass alone, but many people try and think they have succeeded. This book is about angle trisections and the people who attempt them. Its purposes are to collect many trisections in one place, inform about trisectors, to amuse the reader, and, perhaps most importantly, to reduce the number of trisectors. This book includes detailed information about the personalities of trisectors and their constructions. It can be read by anyone who has taken a high school geometry course.
A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague.In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.
A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague.In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed.
A Course in Differential Geometry
A Course in Digital Signal Processing with Examples
Digital Signal Processing (DSP) is a field of acquiring, analyzing, transforming, filtering and enhancing signals, aiming at some application benefits. DSP expertise should learn linear algebra, Fourier analysis, Z-transform, wavelet-transform, filter design and filtering techniques, linear systems and system analysis. DSP expertise should also learn certain scientific and technical computing languages such as Python, Octave, R, Matlab, etc. to use the Fast Fourier Transform (FFT) algorithm and other open source DSP packages which are essential for solving and simulating DSP problems. Since most real-world signals and data are generated by random processes, then learning probability, statistics, statistical analysis and inference in order to process and handle these random signals is an essential subject. In this context, this book provides a first and basic DSP course for university students and those intending to become a DSP expertise. The next step to generate a course in statistical digital signal processing which will be in a separate book.