Computable Coordinate System Objects: Theory and Applications Article Swipe
— We present a unified computational framework for geometry and physics based on the concept of Computable Coordinate Systems. The central idea is to elevate coordinate systems from passive references to first-class algebraic objects, denoted as coord, which support intuitive operations such as multiplication (∗) and division (∕). This algebraic approach replaces cumbersome matrix and tensor calculus with a natural and efficient formalism for hierarchical transformations. The framework extends naturally into differential geometry through the Intrinsic Gradient OperatorGμ=ΔcΔμ∣c-frame,Gμ=ΔμΔcc-frame,which measures the variation of a frame field within its own coordinate system. Curvature is then obtained directly via the Lie bracket [Gu,Gv][Gu,Gv], with a metric normalization that ensures coordinate invariance. Beyond geometry, we unify complex frame transformations, Fourier transforms, and conformal mappings within the same algebraic structure, providing a geometric foundation for path integrals, gauge theories, and quantum field theory. Numerical experiments on canonical surfaces confirm machine-precision accuracy and a 275% speedup over classical methods. This work thus bridges abstract mathematics with practical computation, offering a unified language for graphics, robotics, simulation, and theoretical physics.
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- Type
- preprint
- Language
- en
- Landing Page
- https://doi.org/10.5281/zenodo.17827704
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W7109229477