Category: Geometric Algebra
Two elementary courses common to modern physics, mathematics, and engineering curricula are Linear Algebra and Vector Calculus. Geometric Algebra is a natural and powerful extension of linear algebra and Geometric Calculus is even more so for vector calculus.
In this post, I interview Dr. Alan Macdonald who talks about his experience with both subjects and how he contributes in educating young undergraduate students about them through his books and online videos.
In this post, I interview Dr. Charles Gunn, a mathematician, movie-maker, and teacher whose mathematical activities center around projective geometry and its connection to the human being and the world. He is particularly excited by his recent work on Projective Geometric Algebra (PGA), which he believes has a great role to play in bringing the benefits of GA to a wider audience.
The area of Scientific Visualization (SciViz) is an interdisciplinary branch of science. According to Friendly, it is “primarily concerned with the visualization of three-dimensional phenomena (architectural, meteorological, medical, biological, etc.), where the emphasis is on realistic renderings of volumes, surfaces, illumination sources, and so forth, perhaps with a dynamic (time) component”. It is also considered a subset of computer graphics, a branch of computer science. The purpose of scientific visualization is to graphically illustrate scientific data to enable scientists to understand, illustrate, and glean insight from their data.
In this post, I interview Dr. Werner Benger who describes his views on SciViz using Geometric Algebra and provides valuable insights about the use of SciViz in Big Data applications.
Planning for the next generation of GMac began in August 2011. I started to design the new version of GMac from scratch by reading significant parts of Terence Parr’s book “Language Implementation Patterns”, Robert W. Sebesta’s classic book “Concepts of Programming Languages”, and the second edition of the bestseller Dragon Book “Compilers: Principles, Techniques, and Tools”. I had learned many lessons during developing the first GMac prototype. These books provided a solid conceptual framework for designing the new version of GMac containing all the lessons I’d learned before.
In the previous post, I talked about the first part of my journey developing GMac, the fascinating discoveries I made, and the difficulties I faced along the way. In this final part, I explain the design decisions I made for GMac and how I came to select them, in addition to the developments I hope to make in the future.
In the summer of 2003, I was almost at the end of my M.Sc. thesis. One day I was surfing the web searching for some references to add to the thesis I’d been writing. I ran into Mikael Nilsson’s interesting M.Sc. thesis “Geometric Algebra with Conzilla: Building a Conceptual Web of Mathematics”. His work contains a nice introduction to Geometric Algebra, with accompanying UML-based conceptual models. It got me very curious about GA. I made some more searches finding a few papers, books, and tutorials. After reading some of them I was suddenly transferred into a new world of algebraic abstractions. For the first time in my life as a student, engineer, and software developer I could hope to express, explore, and manipulate geometric abstractions with such clarity. I then decided that my Ph.D. will be about exploring Geometric Algebra using the best way I know: developing software.
Any useful mathematical structure consists of some integrating abstract elements. The mathematical structure of Geometric Algebra is sophisticated but very elegant and easy to understand. In my view as a software engineer, I could identify 10 main elements of the mathematical structure of GA. Some of these elements are well studied mathematical disciplines by their own right. The integration of the 10 elements, however, produces a rich mathematical language capable of expressing much more than the mere sum of its parts. In this post, I will describe each component and talk a little about its significance and varieties without delving into any mathematical details. The interested reader can find many tutorials and links to explain in full details the mathematics of Geometric Algebra on my GA Online Resources page. The information in this post can be useful for someone starting to study GA and wanting a clear roadmap for understanding and relating its main concepts and algebraic tools.
A Cyberneticist, or cybernetician, is a person practicing Cybernetics: a transdisciplinary approach for exploring regulatory systems; their structures, constraints, and possibilities. Professor Eduardo Bayro-Corrochano is one such leading academic who uses Geometric Algebra to handle the diverse fields of theoretical knowledge and practical application he needs. Such fields include Robotics, Neural Computing, Computer Vision, and Lie Algebras. In this post, I interview Prof. Eduardo Bayro who tells us about how using GA in his work can simplify dealing with such diverse fields, and how can GA relate, generalize, and unify ideas from these fields together in his mind and the minds of his students.
One of the most important fields of application for Geometric algebra can be found in Computer Science. In this post, I interview 3 key researchers who apply Geometric Algebra in their work to share their valuable experience and insights. Their applied research spans many applications in computer science including Computer Graphics, Robotics, Computer Vision, Image Processing, Neural Computing, and more.
Foundations of modern Geometric Algebra and Geometric Calculus were laid down by Prof. David Hestenes and Dr. Garret Sobczyk 50 years ago. In this post, I interview Dr. Garret Sobczyk who tells us about his fascinating life journey with Prof. Hestenes. Their journey eventually inspired many researchers to follow their lead in learning, developing, and applying Geometric Algebra and Geometric Calculus to many fields of science.
In this post, I interview Dr. Yu Zhaoyuan who explains the importance of Geographical Information Science (GIS) and the potential of Geometric Algebra as a mathematical modeling tool in this fascinating field of study. Dr. Zhaoyuan talks about the deep relations between Geometric Computing and GIS. He then tells us about the difficulties facing the creation of good GIS simulators and how GA can help in this regard. Finally, he talks about the recent GAGIS conference and the important research presented there.