Geometry

   

Solid Angle of Torus on Normal Geometrical Axis Using [a] Method of Concentric Cones

Authors: Harish Chandra Rajpoot

An analytical method is developed for the exact evaluation of the solid angle subtended by a torus at a point lying on its axis of symmetry, i.e., on the line perpendicular to the mid-plane of the torus and passing through its centre. The method is based on a geometric enclosure of the torus between two coaxial right circular cones with a common apex at the observation point and with axes coincident with the torus axis. It is shown that the solid angle associated with the torus equals the algebraic sum i.e. the difference between the solid angles subtended by the outer and inner bounding cones at the apex [1,2,3]. This construction leads to closed-form expressions for the solid angle as a function of the torus radii and the axial distance of the observation point, without recourse to surface integration. The resulting formulation provides a concise geometric characterization of toroidal visibility and is well suited for applications in geometric modeling, where exact angular measures are required, and in photometry and radiative transfer, particularly for the evaluation of irradiance, flux, and angular response of axially symmetric toroidal sources and apertures.

Comments: 5 Pages. (Note by ai.viXra.org Admin: Please don't name title, equation/formula etc after the author's name) 3 Figures.

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Submission history

[v1] 2026-02-03 21:00:09

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