The Power of Geometric Algebra 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