Multiplanes And Multispheres
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Multiplanes and Multispheres
This book is a collection of notes exploring multiplanes and multispheres using Grassmann algebra with Mathematica. A multiplane is an m-dimensional generalization of the notions of point, line, plane and hyperplane. A multisphere is an m-dimensional generalization of the notions of point-pair, circle, sphere and hypersphere. Grassmann algebra is a generalization of the notions of scalars, vectors and vector spaces. Mathematica is a system for doing mathematics on a computer.Grassmann algebra has now emerged as one of the more important tools for exploring multidimensional geometry and mathematical physics. It not only generalizes the classic vector algebra to enable construction of (unlocated) bivectors, trivectors and multivectors, it is also an algebra par excellence for working with located entities such as points, lines, planes and multiplanes.But multiplanes are not alone in their space. To every multiplane corresponds a docked multisphere and vice versa. (A docked multisphere passes through the origin.) Corresponding points on a multiplane-multisphere pair are inverses. And because we can easily dock a multisphere by adding a displacement vector to its points, we can work with multispheres by operating on their corresponding multiplanes. For example: we can intersect two multispheres, or a multisphere and a multiplane; construct the best-fit multisphere to a system of points; compute the complex of circles for a Clifford circle theorem, or generate the in-multisphere of a simplex.
Hybrid Animation
Think outside the 2D or 3D box and seemlessly integrate 2D and 3D animation mediums into your projects. Develop your entire pipeline from start to finish with Hybrid Animation: Integrating 2D and 3D Assets.
Cybernetics and Control Theory in Systems
Addressing key issues in modern cybernetics and informatics, this book presents vital research within networks and systems. It offers an extensive overview of the latest methods, algorithms, and design innovations. This book compiles the meticulously reviewed proceedings of the Networks and Systems in Cybernetics session of the 13th Computer Science Online Conference 2024 (CSOC 2024), held virtually in April 2024.