Solid Angle - Solid Angles in Arbitrary Dimensions

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

\Omega_{d} = \frac{2\pi^{d/2}}{\Gamma(\frac{d}{2})}

where is the Gamma function. When is an integer, the Gamma function can be computed explicitly. It follows that

\Omega_{d} = \begin{cases} \frac{2\pi^{d/2}}{ \left (\frac{d}{2}-1 \right )!} & d\text{ even} \\ \frac{2^d\left (\frac{d-1}{2} \right )!}{(d-1)!} \pi^{(d-1)/2} & d\text{ odd} \end{cases}

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.

Read more about this topic:  Solid Angle

Famous quotes containing the words solid, arbitrary and/or dimensions:

    “Manuel showed her his open hand: “Look at this finger, how meager it seems, and this one even weaker, and this other one no stronger, and this one all by himself and on his own.”
    Then he made a fist: “But now, is it strong enough, big enough, solid enough? It seems so doesn’t it?””
    —Jacques Roumain (1907–1945)

    “... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
    —Angelina Grimké (1805–1879)

    “The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.”
    —Thomas Jefferson (1743–1826)