Dual Algebra Representations for Volumes Determination. an Overview
Dual Algebra Representations for Volumes Determination. an Overview

Summary/Abstract: The volume, traditionally, as a mathematical model, is a geometric property encoded using linear algebra through determinants and matrix operations. This paper presents a method to calculate volumes directly using dual algebra, bypassing coordinate-dependent discretization and leveraging its inherent capacity to represent surfaces, intersections, and enclosed spaces through concise, error-aware computations. The purpose of this article is to demonstrate the application of dual algebra as a unified computational framework for volume determination, highlighting its mathematical advantages over traditional methods reliant on localized coordinate systems and iterative geometric approximations. Dual algebra, which inherently integrates geometric transformations and error propagation within its algebraic structure, offers a streamlined approach to modelling complex surfaces and calculating enclosed volumes without the need for intermediate coordinate transformations or ad hoc adjustments. Dual algebra operates directly on geometric primitives (e.g., planes, points, and vectors) using a coordinate-free formalism. This enables precise, closed-form solutions for volume calculations, even in irregular geometries. Dual algebra emerges as a mathematically alternative for volume determination, with broad applicability in geomatics, engineering, and environmental modelling.

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