Derivations of the solid angle subtended at a point by first and secondorder surfaces and volumes as a function of elliptic integrals
Abstract
An analytical study of the solid angle subtended at a point by objects of first and second algebraic order has been made. It is shown that the derived solid angle for all such objects is in the form of a general elliptic integral, which can be written as a linear combination of elliptic integrals of the first and third kind and elementary functions. Many common surfaces and volumes have been investigated, including the conic sections and their volumes of revolution. The principal feature of the study is the manipulation of solidangle equations into integral forms that can be matched with those found in handbook tables. These integrals are amenable to computer special function library routine analysis requiring no direct interaction with elliptic integrals by the user. The general case requires the solution of a fourthorder equation before specific solidangle formulations can be made, but for many common geometric objects this equation can be solved by elementary means. Methods for the testing and application of solidangle equations with Monte Carlo rejection and estimation techniques are presented. Approximate and degenerate forms of the equations are shown, and methods for the evaluation of the solid angle of a torus are outlined.
 Authors:
 Oak Ridge National Lab., TN (United States)
 Publication Date:
 Sponsoring Org.:
 USDOE, Washington, DC (United States)
 OSTI Identifier:
 357696
 DOE Contract Number:
 AC0596OR22464
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Nuclear Science and Engineering; Journal Volume: 132; Journal Issue: 2; Other Information: PBD: Jun 1999
 Country of Publication:
 United States
 Language:
 English
 Subject:
 66 PHYSICS; RADIATION TRANSPORT; TRANSPORT THEORY; ANALYTICAL SOLUTION; INTEGRALS; VOLUME; SURFACES
Citation Formats
Cramer, S.N. Derivations of the solid angle subtended at a point by first and secondorder surfaces and volumes as a function of elliptic integrals. United States: N. p., 1999.
Web.
Cramer, S.N. Derivations of the solid angle subtended at a point by first and secondorder surfaces and volumes as a function of elliptic integrals. United States.
Cramer, S.N. 1999.
"Derivations of the solid angle subtended at a point by first and secondorder surfaces and volumes as a function of elliptic integrals". United States.
doi:.
@article{osti_357696,
title = {Derivations of the solid angle subtended at a point by first and secondorder surfaces and volumes as a function of elliptic integrals},
author = {Cramer, S.N.},
abstractNote = {An analytical study of the solid angle subtended at a point by objects of first and second algebraic order has been made. It is shown that the derived solid angle for all such objects is in the form of a general elliptic integral, which can be written as a linear combination of elliptic integrals of the first and third kind and elementary functions. Many common surfaces and volumes have been investigated, including the conic sections and their volumes of revolution. The principal feature of the study is the manipulation of solidangle equations into integral forms that can be matched with those found in handbook tables. These integrals are amenable to computer special function library routine analysis requiring no direct interaction with elliptic integrals by the user. The general case requires the solution of a fourthorder equation before specific solidangle formulations can be made, but for many common geometric objects this equation can be solved by elementary means. Methods for the testing and application of solidangle equations with Monte Carlo rejection and estimation techniques are presented. Approximate and degenerate forms of the equations are shown, and methods for the evaluation of the solid angle of a torus are outlined.},
doi = {},
journal = {Nuclear Science and Engineering},
number = 2,
volume = 132,
place = {United States},
year = 1999,
month = 6
}

TABLES OF SOLID ANGLES AND ACTIVATIONS. I. SOLID ANGLE SUBTENDED BY A CIRCULAR DISC. II. SOLID ANGLE SUBTENDED BY A CYLINDER. III. ACTIVATION OF A CYLINDER BY A POINT SOURCE
The solid angles subtended by discs and cylinders have been calculated for a large range of disc and cylinder dimensions and spacings and are presented in tabular form. The interaction of radiation from a point source on right cylindrical bodies is calculated for a large range of cylinder dimensions and spacing. (W.D.M.) 
The Dirichlet problem for a secondorder elliptic equation with an L{sub p} boundary function
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