

A231983


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


5



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

Arclengths 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 degreesquares (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

G. V. Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2012, ISBN 9780691148922.


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

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


EXAMPLE

0.0003046096875119366637825983210350747291625456181624489357027...


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



