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:

    “Miss Western: Tell me, child, what objections can you have to the young gentleman?
    Sophie: A very solid objection, in my opinion. I hate him.
    Miss Western: Well, I have known many couples who have entirely disliked each other, lead very comfortable, genteel lives.”
    —John Osborne (1929–1994)

    “What causes adolescents to rebel is not the assertion of authority but the arbitrary use of power, with little explanation of the rules and no involvement in decision-making. . . . Involving the adolescent in decisions doesn’t mean that you are giving up your authority. It means acknowledging that the teenager is growing up and has the right to participate in decisions that affect his or her life.”
    —Laurence Steinberg (20th century)

    “Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
    —Ralph Waldo Emerson (1803–1882)