The Power of Geometric Algebra Computing For Engineering and Quantum Computing

Foreword

Preface

Acknowledgements

Introduction
1.1 GEOMETRIC ALGEBRA 
1.2 GEOMETRIC ALGEBRA COMPUTING 
1.3 OUTLINE 

Geometric Algebras for Engineering 
2.1 THE BASICS OF GEOMETRIC ALGEBRA 
2.2 CONFORMAL GEOMETRIC ALGEBRA (CGA) 
2.2.1 Geometric Objects of Conformal Geometric Algebra 
2.2.2 Angles and Distances in 3D 
2.2.3 3D Transformations 
2.3 COMPASS RULER ALGEBRA (CRA) 
2.3.1 Geometric objects 
2.3.2 Angles and Distances 
2.3.3 Transformations 
2.4 PROJECTIVE GEOMETRIC ALGEBRA (PGA) WITH GANJA 
2.4.1 2D PGA 
2.4.2 3D PGA 

GAALOP 
3.1 INSTALLATION 26
3.2 GAALOPSCRIPT 28
3.2.1 The main notations 28
3.2.2 Macros and Pragmas 28
3.2.3 Bisector Example 29
3.2.4 Line-Sphere Example 30


GAALOPWeb
4.1 THE WEB INTERFACE 
4.2 THE WORKFLOW 
4.3 GAALOPWEB VISUALIZATIONS 
4.3.1 Visualization of the Bisector Example 
4.3.2 Visualization of the Rotation of a Circle 
4.3.3 Visualization of the Line-Sphere Example 
4.3.4 Visualization of a Sphere Of Four Points 
4.3.5 Sliders 

GAALOPWeb for C/C++ 
5.1 GAALOPWEB HANDLING 
5.2 CODE GENERATION AND RUNTIME PERFORMANCE
BASED ON GAALOPWEB 

GAALOPWeb for Python 
6.1 THE WEB INTERFACE 
6.2 THE PYTHON CONNECTOR FOR GAALOPWEB 
6.3 CLIFFORD/PYGANJA 
6.4 GAALOPWEB INTEGRATION INTO CLIFFORD/PYGANJA 
6.5 USING PYTHON TO GENERATE CODE NOT SUPPORTED BY GAALOPWEB 

Molecular Distance Application using GAALOPWeb
for Mathematica 
7.1 DISTANCE GEOMETRY EXAMPLE 
7.2 GAALOPWEB FOR MATHEMATICA 
7.2.1 Mathematica code generation 
7.2.2 The Web-Interface 
7.3 COMPUTATIONAL RESULTS 

Robot Kinematics based on GAALOPWeb for Matlab 
8.1 THE MANIPULATOR MODEL 
8.2 KINEMATICS OF A SERIAL ROBOT ARM 
8.3 MATLAB TOOLBOX IMPLEMENTATION 
8.4 THE GAALOP IMPLEMENTATION 
8.5 GAALOPWEB FOR MATLAB 
8.6 COMPARISON OF RUNTIME PERFORMANCE

The Power of highdimensional Geometric Algebras
9.1 GAALOP DEFINITION 
9.2 VISUALIZATION 

GAALOPWeb for Conics 
10.1 GAALOP DEFINITION 
10.1.1 definition.csv 
10.1.2 macros.clu 
10.2 GAC OBJECTS 
10.3 GAC TRANSFORMATIONS 
10.4 INTERSECTIONS 

Double Conformal Geometric Algebra 
11.1 GAALOP DEFINITION OF DCGA 
11.2 THE DCGA OBJECTS 
11.2.1 Ellipsoid, Toroid and Sphere 
11.2.2 Planes and Lines 
11.2.3 Cylinders 
11.2.4 Cones 
11.2.5 Paraboloids 
11.2.6 Hyperboloids 
11.2.7 Parabolic and Hyperbolic Cylinders 
11.2.8 Specific Planes
11.2.9 Cyclides 
11.3 THE DCGA TRANSFORMATIONS 
11.4 INTERSECTIONS 
11.5 REFLECTIONS AND PROJECTIONS 
11.6 INVERSIONS

Geometric Algebra for Cubics 
12.1 GAALOP DEFINITION 
12.2 CUBIC CURVES 

GAALOPWeb for GAPP 
13.1 THE REFLECTOR EXAMPLE 
13.2 THE WEB INTERFACE 1
13.3 GAPP CODE GENERATION

GAALOPWeb for GAPPCO 
14.1 GAPPCO IN GENERAL 
14.2 GAPPCO I 
14.2.1 GAPPCO I architecture 
14.2.2 The Compilation Process 
14.2.3 Configuration Phase 
14.2.4 Runtime Phase 
14.3 THE WEB INTERFACE 

GAPPCO II 
15.1 THE PRINCIPLE 
15.2 EXAMPLE 
15.3 IMPLEMENTATION ISSUES 

Introduction to Quantum Computing 
16.1 COMPARING CLASSIC COMPUTERS WITH QUANTUM COMPUTERS 
16.2 DESCRIPTION OF QUANTUM BITS 
16.3 QUANTUM REGISTER 
16.4 COMPUTING STEPS IN QUANTUM COMPUTING 
16.4.1 The NOT-operation 
16.4.2 The Hadamard transform 
16.4.3 The CNOT operation 
CHAPTER 17 □ GAALOPWeb as a qubit calculator 
17.1 QUBIT ALGEBRA QBA 
17.2 GAALOPWEB FOR QUBITS 
17.3 THE NOTOPERATION ON A QUBIT 
17.4 THE 2QUBIT ALGEBRA QBA2 

Appendix

Index 

Biography

Dietmar Hildenbrand is a lecturer in Geometric Algebra at TU Darmstadt.