A231983 - OEIS

A231983

Decimal expansion of the solid angle (in sr) of a spherical square having sides of one degree.

3, 0, 4, 6, 0, 9, 6, 8, 7, 5, 1, 1, 9, 3, 6, 6, 6, 3, 7, 8, 2, 5, 9, 8, 3, 2, 1, 0, 3, 5, 0, 7, 4, 7, 2, 9, 1, 6, 2, 5, 4, 5, 6, 1, 8, 1, 6, 2, 4, 4, 8, 9, 3, 5, 7, 0, 2, 7, 0, 7, 7, 0, 7, 4, 8, 4, 4, 1, 3, 2, 2, 9, 2, 6, 7, 3, 0, 4, 1, 8, 5, 0, 5, 3, 8, 6, 2, 6, 1, 6, 8, 5, 5, 2, 9, 6, 1, 3, 2, 0, 1, 8, 9, 5, 5

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

-3,1

COMMENTS

Arc-lengths are planar angles, expressed either in radians (rad) or in degrees (deg), while solid angles are areas subtended on a unit sphere and expressed in steradians (sr) or degree-squares (deg^2). Consequently, the solid angle Omega of a spherical rectangle with sides r, s expressed in degrees, cannot be computed as the product of its sides and then converted into steradians by applying the deg^2/sr conversion factor A231982. Rather, one must use the general formula Omega = 4*arcsin(sin(R/2)sin(S/2)), where R=(Pi/180)r, S=(Pi/180)s are the sides expressed in radians. Due to spherical excess, the result differs slightly, but significantly, from A231982.

REFERENCES

Glen Van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 978-0691148922.

LINKS

Stanislav Sykora, Table of n, a(n) for n = -3..2000

Wikipedia, Solid angle, Section 3.3 (Pyramid).

Wikipedia, Square degree.

Wikipedia, Steradian.

FORMULA

Equals 4*arcsin(sin(R/2)sin(S/2)), where R = S = Pi/180.

EXAMPLE

0.0003046096875119366637825983210350747291625456181624489357027...

MATHEMATICA

RealDigits[4 * ArcSin[Sin[Pi/360]^2], 10, 120][[1]] (* Amiram Eldar, Jun 26 2023 *)

PROG

(PARI)

default(realprecision, 120);

4*asin(sin(Pi/360)^2) \\ Rick L. Shepherd, Jan 28 2014

CROSSREFS

Cf. A000796 (Pi), A072097 (rad/deg), A019685 (deg/rad), A231981 (sr/deg^2), A231982 (deg^2/sr), A231984 (this constant in deg^2), A231986 (same for square with 1 rad side), A231985, A231987.

Sequence in context: A111486 A192878 A126826 * A231982 A198575 A112256

Adjacent sequences: A231980 A231981 A231982 * A231984 A231985 A231986

KEYWORD

nonn,cons,easy

AUTHOR

Stanislav Sykora, Nov 17 2013

STATUS

approved

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