Modern approaches to applied mathematics and modeling using Geometric Algebra
We believe the conventional mathematical frameworks are bottlenecking inovation in the fields of engineering and applied science. 810 Labs Mathematics Division is working to solve this problem by aiding in the paradigm shift of Geometric Algebra.
Implementation with Clifford
Eventually mathematical results must be applied in the real world, and this requires implementation in software. Since Geometric Algebra introduces new objects and operations, unique computational support is required.
At the heart of our development and implementations is
a BSD licensed python module developed and supported
by 810 Labs.
CGA Transmission Line Theory
Dr. Alex Arsenovic
810 Labs LLC was founded in 2016 by Dr. Alex Arsenovic. We believe the conventional mathematical frameworks are bottlenecking inovation in the applied sciences.
Applications of Conformal Geometric Algebra to Transmission Line Theory
Complex numbers are ubiquitous in science and engineering mainly because they provide a powerful way to represent and manipulate rotations. Despite their usefulness, complex numbers are limited to rotations in two-dimensions. This is a serious drawback given that many problems in physics and engineering are inherently multi-dimensional. A better approach might be to use a high dimensional algebra for high dimensional problems. For it stands to reason that if the ability to efficiently handle rotations in two dimensions has been so successful in science and engineering, then a similar ability in higher dimensions should be even more so.
In this project we present the application of a projective geometry tool known as Conformal Geometric Algebra (CGA) to transmission line theory. Explicit relationships between the Smith Chart, Riemann Sphere, and CGA are developed to illustrate the evolution of projective geometry in transmission line theory. By using CGA, fundamental network operations such as adding impedance, admittance, and changing lines impedance can be implemented with rotations, and are shown to form a group. Additionally, the transformations relating different circuit representations such as impedance, admittance, and reflection coefficient are also related by rotations. Thus, the majority of relationships in transmission line theory are linearized.
Modeling Two Port Networks with Conformal Geometric Algebra
This project employs our Conformal Geometric Algebra transmission line model to analyze two-port networks. By doing so, two-port networks and transmission line problems can be analyzed with a single framework, providing theoretical consolidation.
Clifford is an open-source python module which supports the unique data-types and operations GA introduces. In recent years Python has become a dominant language in the field of scientific and engineering software development, making it a suitable language for a GA implementation. We employ `clifford' as a proof-of-concept computational backend. This module provides a large amount of GA objects and operations, and is continually developed as needed for various project.
scikit-rf (aka skrf)
is an Open Source, BSD-licensed package for RF/Microwave engineering implemented in the Python programming language. It provides a modern, object-oriented library which is both flexible and scalable. Most importantly, scikit-rf allows the vast array of incredible python modules to be integrated with RF/microwave engineering. S-parameter data can be crunched like big data, plots are
one-liners, and web front-ends are whipped up in a weekend.
The Jupyter Notebook is a web application that allows you to create and share documents that contain live code, equations, visualizations and explanatory text. Uses include: data cleaning and transformation, numerical simulation, statistical modeling, machine learning and much more.
More information about scikit-rf can be found at jupyter.org.
THe python ecosystem
The python ecosystem is populated with a wide variety of open-source, profesional quality modules. This allows new scientific applications to be integrated with database systems, local user interfaces, and web-apps seamlessly. Not only does this cut down on development costs and time, but allows for solutions that are otherwise intractable with multi-language approaches.
More information about popular python modules can be found at wiki.python.org.
Alex Arsenovic received a B.S and Ph.D in Electrical Engineering from the University of Virginia in 2007 and 2012, respectively. Alex has worked as an independent consultant in Central Virginia for the past 5 years, with clients such as Virginia Diodes Inc, Nuvotronics, and Plotly. He continues to work closely with the University of Virginia, and has authored and co-authored over 15 technical papers in the field of microwave metrology and geometric algebra. In 2016, he created Eight Ten Labs LLC to continue providing the services for microwave metrology, software development, and applied mathematics. His chief interest is in modernizing the theoretical and computation tools used by electrical engineers and scientists.