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This paper studies the problem of identifying directions of axial symmetry in multivariate distributions. Theoretical results are derived on how the measure or cardinality of the set of symmetry directions relates to spherical symmetry. The…
Quasometry is precision measurement of celestial positions and apparent motion of very distant extragalactic objects, such as quasars, galactic nuclei, and QSOs. We use this term to identify a specific area of research, the methodology of…
We develop a direct geometric method to determine the orbital parameters and mass of a planet, and we then apply the method to Neptune using high-precision data for the other planets in the solar system. The method is direct in the sense…
The spherometer used for measuring radius of curvature of spherical surfaces is explicitly based on a geometric relation unique to circles and spheres. We present an alternate approach using coordinate geometry, which reproduces the…
Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and…
This paper develops a geometric model for coupled two-state quantum systems (qubits), which is formulated using geometric (aka Clifford) algebra. It begins by showing how Euclidean spinors can be interpreted as entities in the geometric…
The geometric calibration of the Planck satellite using the planetary transits is investigated, together with the reconstruction of any offsets from the nominal layout of the focal plane. The methods presented here may be applied to a…
Harmonic analysis is a tool to infer cosmic topology from the measured astrophysical cosmic microwave background CMB radiation. For overall positive curvature, Platonic spherical manifolds are candidates for this analysis. We combine the…
I review the current state of determinations of the Hubble constant, which gives the length scale of the Universe by relating the expansion velocity of objects to their distance. There are two broad categories of measurements. The first…
Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…
We study the geometry of surfaces in $\mathbb R^5$ by relating it to the geometry of regular and singular surfaces in $\mathbb R^4$ obtained by orthogonal projections. In particular, we obtain relations between asymptotic directions, which…
The geometric calibration of the Planck satellite is investigated, specifically those parameters which require the use of the science data for their extraction. Methods for the recovery of these geometric-calibration parameters from point…
This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…
A tutorial of the Mathematica package CGAlgebra, for conformal geometric algebra calculations is presented. Using rule-based programming, the 5-dimensional conformal geometric algebra is implemented and defined functions simplify the…
Algorithms for the computation of geodesics on an ellipsoid of revolution are given. These provide accurate, robust, and fast solutions to the direct and inverse geodesic problems and they allow differential and integral properties of…
The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent…
Geometric (Clifford) algebra provides an efficient mathematical language for describing physical problems. We formulate general relativity in this language. The resulting formalism combines the efficiency of differential forms with the…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
Since the dawn of civilization, humanity has grappled with the big questions of existence and creation. Modern cosmology seeks to answer some of these questions using a combination of mathematics and measurement. The questions people hope…
In normal observation procedures, the position of the observer is specified by GPS and celestial positions of an object will be calculated. But in some situations, like small zenith angle FOVs, GPS doesn't work. Therefore in this study, the…