# Derivations of the solid angle subtended at a point by first- and second-order 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 solid-angle 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 fourth-order equation before specific solid-angle 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 solid-angle 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:
- AC05-96OR22464

- 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 second-order 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 second-order surfaces and volumes as a function of elliptic integrals*. United States.

```
Cramer, S.N. Tue .
"Derivations of the solid angle subtended at a point by first- and second-order 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 second-order 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 solid-angle 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 fourth-order equation before specific solid-angle 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 solid-angle 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 = {Tue Jun 01 00:00:00 EDT 1999},

month = {Tue Jun 01 00:00:00 EDT 1999}

}