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In-Depth Information
3.1
Volume of Hyperspheres
The volume of a n -dimensional hypersphere with radius r can be calculated as
follows :
π n/ 2
Γ 2 +1
V ( n, r )= r n
·
where
Γ ( n +1)= n !for n
N
and
·
·
·
·
...
·
(2 n
π
Γ ( n + 2 )= 1
3
5
7
1)
for half-integer arguments.
2 n
We briefly show the construction idea 3 behind the the volume calculation of
hyperspheres. For a in-depth description see [11], where the complete construc-
tion and a proof is shown.
The volume V ( n )ofa n -dimensional unit sphere can be constructed inductively
V (2) = π
4
3 π
V (3) =
.
π n/ 2
( n/ 2)!
,n even
V ( n )=
2 n π ( n− 1) / 2 (( n
1) / 2)!
,n odd
n !
Given a 2-dimensional unit circle
C 2 =
2
x 1 + x 2
{
( x 1 ,x 2 )
R
|
1
}
The volume V ( C 2 ) can be calculated as a summation of infinitely thin “stripes”.
1
1
V ( C 2 )=2
·
x 2 dx 2
1
1
π
=2
·
cos 2 ( t )sin( t ) dt
0
π
sin 2 ( t ) dt
=2
·
0
=
π
dt = π
0
3 Taken from [11].
 
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