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)

    “If you should rise from Nowhere up to Somewhere,
    From being No one up to being Someone,
    Be sure to keep repeating to yourself
    You owe it to an arbitrary god....”
    —Robert Frost (1874–1963)

    “Why is it that many contemporary male thinkers, especially men of color, repudiate the imperialist legacy of Columbus but affirm dimensions of that legacy by their refusal to repudiate patriarchy?”
    —bell hooks (b. c. 1955)