Solid Angles in Arbitrary Dimensions
The solid angle subtended by the full surface of the unit n-sphere (in the geometer's sense) can be defined in any number of dimensions . One often needs this solid angle factor in calculations with spherical symmetry. It is given by the formula
where is the Gamma function. When is an integer, the Gamma function can be computed explicitly. It follows that
This gives the expected results of 2π rad for the 2D circumference and 4π sr for the 3D sphere. It also throws the slightly less obvious 2 for the 1D case, in which the origin-centered unit "sphere" is the set, which indeed has a measure of 2.
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Famous quotes containing the words solid, arbitrary and/or dimensions:
“Every winter the liquid and trembling surface of the pond, which was so sensitive to every breath, and reflected every light and shadow, becomes solid to the depth of a foot or a foot and a half, so that it will support the heaviest teams, and perchance the snow covers it to an equal depth, and it is not to be distinguished from any level field. Like the marmots in the surrounding hills, it closes its eyelids and becomes dormant for three months or more.”
—Henry David Thoreau (1817–1862)
“Pity on the person who has become accustomed to seeing in necessity something arbitrary, who ascribes to the arbitrary some sort of reason, and even claims that following that sort of reason has religious value.”
—Johann Wolfgang Von Goethe (1749–1832)
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—Henry David Thoreau (1817–1862)