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Concept
Volume of an n-ball
Summary
In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An n-ball is a ball in an n-dimensional Euclidean space. The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n-ball of radius R is where is the volume of the unit n-ball, the n-ball of radius 1.
The real number can be expressed via a two-dimension recurrence relation.
Closed-form expressions involve the gamma, factorial, or double factorial function.
The volume can also be expressed in terms of , the area of the unit n-sphere.
The first volumes are as follows:
The n-dimensional volume of a Euclidean ball of radius R in n-dimensional Euclidean space is:
where Γ is Euler's gamma function. The gamma function is offset from but otherwise extends the factorial function to non-integer arguments. It satisfies Γ(n) = (n − 1)! if n is a positive integer and Γ(n + 1/2) = (n − 1/2) · (n − 3/2) · ... · 1/2 · pi1/2 if n is a non-negative integer.
The volume can be computed without use of the Gamma function. As is proved below using a vector-calculus double integral in polar coordinates, the volume V of an n-ball of radius R can be expressed recursively in terms of the volume of an (n − 2)-ball, via the interleaved recurrence relation:
This allows computation of V_n(R) in approximately n / 2 steps.
The volume can also be expressed in terms of an (n − 1)-ball using the one-dimension recurrence relation:
Inverting the above, the radius of an n-ball of volume V can be expressed recursively in terms of the radius of an (n − 2)- or (n − 1)-ball:
Using explicit formulas for particular values of the gamma function at the integers and half-integers gives formulas for the volume of a Euclidean ball in terms of factorials. For non-negative integer k, these are:
The volume can also be expressed in terms of double factorials.
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