Information Technology Reference
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].