Empowering Novel Geometric Algebra for Graphics and Engineering: 7th International Workshop, ENGAGE 2022, Virtual Event, September 12, 2022, Proceedings

Description

This book constitutes the proceedings of the Workshop Empowering Novel Geometric Algebra for Graphics and Engineering, ENGAGE 2022, held in conjunction with Computer Graphics International conference, CGI 2022, which took place virtually, in September 2022. The 10 full papers included in this volume were carefully reviewed and selected from 12 submissions. The workshop focused specifically on important aspects of geometric algebra including algebraic foundations, digitized transformations, orientation, conic fitting, protein modelling, digital twinning, and multidimensional signal processing. Preface Organization Contents Foundations of Geometric Algebra The Supergeometric Algebra: The Square Root of the Geometric Algebra 1 Introduction 2 Spinors as a Bitcode 3 Spinor Metric 4 Column Spinors and Row Spinors 5 The GA Is the Square of the SGA 6 Conjugation 7 Supersymmetry References Calculation of the Exponential in Arbitrary Clp,q Clifford Algebra 1 Introduction and Notation 2 MV Characteristic Polynomial and Equation 3 MV Exponentials in Clp,q Algebra 3.1 Exponential of MV in Coordinate (Orthogonal Basis) Form 3.2 Exponential in Basis-Free Form 3.3 Making the Answer Real 4 Elementary Functions of MV 5 Conclusion A Minimal Polynomial of MV References On Noncommutative Vieta Theorem in Geometric Algebras 1 Introduction 2 Generalized Vieta's Formulas in Geometric Algebras 2.1 The Case n=1 2.2 The Case n=2 2.3 The Case n=3 3 On Gelfand – Retakh Noncommutative Vieta Theorem 4 Application of Noncommutative Vieta Theorem to Geometric Algebras 4.1 The Case n=1 4.2 The Case n=2 4.3 The Case n=3 4.4 The Cases n4 5 Conclusions References Transformations, Orientation and Fitting Conjecture on Characterisation of Bijective 3D Digitized Reflections and Rotations 1 Introduction 2 Digitized Reflections and Rotations via Geometric Algebra 2.1 Reflections 2.2 Rotations 2.3 Geometric Algebra Rotations and Quaternions 2.4 Cubic Grids and Cells and Digitized Reflections 2.5 Bijectivity Condition of Digitized Reflections and Characterization in 2D 3 Conjecture on the Characterization in 3D 3.1 Certification of Bijective Reflections Through Lipshitz Quaternions 3.2 Bijective Digitized Reflections on Base Planes 1,2,3 3.3 Conjecture on Bijectivity for Digitized Reflections and Rotations 4 Approximation with a Bijective Digitized Reflection 5 Conclusion References Complementary Orientations in Geometric Algebras 1 Oriented Geometry 2 Two Complementary Types of Orientation 3 A Representational Paradox 4 Dualization 5 Complementary Orientation by Hodge Dualization 6 Visualization of Oriented 3D DGA Primitives 7 Visualization of Oriented 3D PGA Primitives 8 The Dual Join 9 The Four Sibling Relationships 10 Computing with Complementary Orientations 11 Conclusion References On Proper and Improper Points in Geometric Algebra for Conics and Conic Fitting Through Given Waypoints 1 Introduction 2 Fitting in GAC – Without Given Waypoints 3 Proper and Improper Points in GAC 3.1 Homogeneous Coordinates 3.2 Projectivisation of GAC 4 Fitting in GAC - With Given Waypoints 4.1 Implementation 4.2 Number of Waypoints and Degrees of Freedom of the Conic 5 Experimental Results 6 Conclusion References Modelling Proteins and Cities Using a Graph Transformer Network to Predict 3D Coordinates of Proteins via Geometric Algebra Modelling 1 Introduction 2 Modelling Proteins 2.1 Proteins as Rigid Bodies 2.2 Proteins as Graphs 3 Architecture 3.1 Graph Transformer 3.2 3D Projector 3.3 Training Details 4 Results 5 Conclusions References How Does Geometric Algebra Support Digital Twin—A Case Study with the Passive Infrared Sensor Scene 1 Introduction 2 Basic Idea 3 Construction of Geometric Algebra-based Digital Twin 3.1 Geometric Algebraic Systems Definition 3.2 Object Representation Based on Geometric Algebra 3.3 Geometric Algebra Operator Definition 4 Digital Twinning of PIR Scene 4.1 Data and Framework 4.2 Digital Twinning for Interaction of PIR scene 4.3 Digital Twinning for Data Mining of PIR Scene 5 Conclusion and Discussion References Signal Processing with Octonions Beurling's Theorem Associated with Octonion Algebra Valued Signals 1 Introduction 2 On Octonion and Clifford Algebra C 0,3 2.1 Octonion Algebra 2.2 Clifford Algebra C 0,3 and Its Clifford-Fourier Transform 3 Octonion Fourier Transform 4 Uncertainty Relations for the Octonion Fourier Transform 4.1 Beurling's up 5 Conclusion References Embedding of Octonion Fourier Transform in Geometric Algebra of R3 and Polar Representations of Octonion Analytic Signals 1 Introduction 2 Octonions 3 Embedding of Octonions in Clifford Geometric Algebra of Three-Dimensional Euclidean Space 4 Octonion Fourier Transform 4.1 Hypercomplex Analytic Signal 5 Embedding the OFT in Clifford Geometric Algebra of Three-Dimensional Euclidean Space 5.1 Embedding of Octonion Analytic Signal in Geometric Algebra Cl(3,0) 6 Polar Representation of Embedded Octonion Analytic Signal 7 Conclusion A Computation of Example 1 References Author Index