Nekaj enotskih sfer v
R
2
{\displaystyle \mathbb {R} ^{2}}
|
|
x
|
|
{\displaystyle ||x||\,}
Enotska sfera je v matematiki množica točk na razdalji 1 od središčne točke, To lahko enostavno povemo tudi, da je enotska sfera tista sfera , ki ima polmer enak 1.
Podobno lahko definiramo, da je enotska krogla množica točk, ki so na razdalji manjši ali enaki 1 od stalne središčne točke. Tako lahko govorimo o enotski sferi (površina) in enotski krogli (telo), Pomen enotske sfere je v tem, da lahko vsako sfero pretvorimo v enotsko sfero z uporabo translacije in skaliranja ,
Naj bo
(
X
,
|
|
.
|
|
)
{\displaystyle (X,||.||)\,}
normirani vektorski prostor . V tem primeru imenujemo množico točk, katerih oddaljenost od ničelne točke je manjša od 1, odprta enotska sfera v
X
{\displaystyle X\,}
B
X
:=
{
x
∈
X
:
‖
x
‖
<
1
}
{\displaystyle B_{X}:=\{x\in X:\|x\|<1\}\,}
Pri tem pa lahko označimo z
B
X
¯
:=
{
x
∈
X
:
‖
x
‖
≤
1
}
{\displaystyle {\overline {B_{X}}}:=\{x\in X:\|x\|\leq 1\}}
zaprto enotsko sfero v
X
{\displaystyle X\,}
∂
B
X
:=
{
x
∈
X
:
‖
x
‖
=
1
}
{\displaystyle \partial B_{X}:=\{x\in X:\|x\|=1\}}
X
{\displaystyle X\,}
V Evklidskem prostoru , ki ima
n
{\displaystyle n\,}
(
x
1
,
…
,
x
n
)
{\displaystyle (x_{1},\dots ,x_{n})\,}
x
1
2
+
x
2
2
+
⋯
+
x
n
2
=
1
{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1}
množica toč, ki pa zadošča neenačbi
x
1
2
+
x
2
2
+
⋯
+
x
n
2
≤
1
,
{\displaystyle x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\leq 1,}
pa je enotska krogla.
Označimo z
V
n
{\displaystyle V_{n}\,}
n
{\displaystyle n\,}
P
n
{\displaystyle P_{n}\,}
Prostornina krogle je enaka
V
n
=
π
n
/
2
Γ
(
1
+
n
/
2
)
=
{
π
n
/
2
/
(
n
/
2
)
!
k
a
d
a
r
j
e
n
≥
0
i
n
p
a
r
e
n
,
π
⌊
n
/
2
⌋
2
⌈
n
/
2
⌉
/
n
!
!
k
a
d
a
r
j
e
n
≥
0
i
n
n
e
p
a
r
e
n
,
{\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma (1+n/2)}}={\begin{cases}{\pi ^{n/2}}/{(n/2)!}&\mathrm {kadar\quad je\quad } n\geq 0\mathrm {\quad } {in\quad paren,}\\~\\{\pi ^{\lfloor n/2\rfloor }2^{\lceil n/2\rceil }}/{n!!}&\mathrm {kadar\quad je} \quad n\geq 0\mathrm {\quad } {in\quad neparen,}\end{cases}}}
kjer je
Hipervolumen
n
−
1
{\displaystyle n-1\,}
n
{\displaystyle n\,}
P
n
{\displaystyle P_{n}\,}
A
n
=
n
V
n
=
n
π
n
/
2
Γ
(
1
+
n
/
2
)
=
2
π
n
/
2
Γ
(
n
/
2
)
,
{\displaystyle A_{n}=nV_{n}={\frac {n\pi ^{n/2}}{\Gamma (1+n/2)}}={\frac {2\pi ^{n/2}}{\Gamma (n/2)}}\,,}
kjer zadnja enačba velja samo za n > 0.
Površine in prostornine za nekatere vrednosti
n
{\displaystyle n\,}
n
{\displaystyle n}
P
n
{\displaystyle P_{n}}
V
n
{\displaystyle V_{n}}
0
0
(
1
/
0
!
)
π
0
{\displaystyle 0(1/0!)\pi ^{0}}
0,000
(
1
/
0
!
)
π
0
{\displaystyle (1/0!)\pi ^{0}}
1,000
1
1
(
2
1
/
1
!
!
)
π
0
{\displaystyle 1(2^{1}/1!!)\pi ^{0}}
2,000
(
2
1
/
1
!
!
)
π
0
{\displaystyle (2^{1}/1!!)\pi ^{0}}
2,000
2
2
(
1
/
1
!
)
π
1
=
2
π
{\displaystyle 2(1/1!)\pi ^{1}=2\pi }
6,283
(
1
/
1
!
)
π
1
=
π
{\displaystyle (1/1!)\pi ^{1}=\pi }
3,142
3
3
(
2
2
/
3
!
!
)
π
1
=
4
π
{\displaystyle 3(2^{2}/3!!)\pi ^{1}=4\pi }
12,57
(
2
2
/
3
!
!
)
π
1
=
(
4
/
3
)
π
{\displaystyle (2^{2}/3!!)\pi ^{1}=(4/3)\pi }
4,189
4
4
(
1
/
2
!
)
π
2
=
2
π
2
{\displaystyle 4(1/2!)\pi ^{2}=2\pi ^{2}}
19,74
(
1
/
2
!
)
π
2
=
(
1
/
2
)
π
2
{\displaystyle (1/2!)\pi ^{2}=(1/2)\pi ^{2}}
4,935
5
5
(
2
3
/
5
!
!
)
π
2
=
(
8
/
3
)
π
2
{\displaystyle 5(2^{3}/5!!)\pi ^{2}=(8/3)\pi ^{2}}
26,32
(
2
3
/
5
!
!
)
π
2
=
(
8
/
15
)
π
2
{\displaystyle (2^{3}/5!!)\pi ^{2}=(8/15)\pi ^{2}}
5,264
6
6
(
1
/
3
!
)
π
3
=
π
3
{\displaystyle 6(1/3!)\pi ^{3}=\pi ^{3}}
31,01
(
1
/
3
!
)
π
3
=
(
1
/
6
)
π
3
{\displaystyle (1/3!)\pi ^{3}=(1/6)\pi ^{3}}
5,168
7
7
(
2
4
/
7
!
!
)
π
3
=
(
16
/
15
)
π
3
{\displaystyle 7(2^{4}/7!!)\pi ^{3}=(16/15)\pi ^{3}}
33,07
(
2
4
/
7
!
!
)
π
3
=
(
16
/
105
)
π
3
{\displaystyle (2^{4}/7!!)\pi ^{3}=(16/105)\pi ^{3}}
4,725
8
8
(
1
/
4
!
)
π
4
=
(
1
/
3
)
π
4
{\displaystyle 8(1/4!)\pi ^{4}=(1/3)\pi ^{4}}
32,47
(
1
/
4
!
)
π
4
=
(
1
/
24
)
π
4
{\displaystyle (1/4!)\pi ^{4}=(1/24)\pi ^{4}}
4,059
9
9
(
2
5
/
9
!
!
)
π
4
=
(
32
/
105
)
π
4
{\displaystyle 9(2^{5}/9!!)\pi ^{4}=(32/105)\pi ^{4}}
29,69
(
2
5
/
9
!
!
)
π
4
=
(
32
/
945
)
π
4
{\displaystyle (2^{5}/9!!)\pi ^{4}=(32/945)\pi ^{4}}
3,299
10
10
(
1
/
5
!
)
π
5
=
(
1
/
12
)
π
5
{\displaystyle 10(1/5!)\pi ^{5}=(1/12)\pi ^{5}}
25,50
(
1
/
5
!
)
π
5
=
(
1
/
120
)
π
5
{\displaystyle (1/5!)\pi ^{5}=(1/120)\pi ^{5}}
2,550
Vrednosti
P
n
{\displaystyle P_{n}\,}
rekurziji
P
0
=
0
{\displaystyle P_{0}=0}
P
1
=
2
{\displaystyle P_{1}=2}
P
2
=
2
π
{\displaystyle P_{2}=2\pi }
P
n
=
2
π
n
−
2
P
n
−
2
{\displaystyle P_{n}={\frac {2\pi }{n-2}}P_{n-2}}
n
>
2
{\displaystyle n>2}
Vrednosti za prostornino
V
n
{\displaystyle V_{n}\,}
V
0
=
1
{\displaystyle V_{0}=1}
V
1
=
2
{\displaystyle V_{1}=2}
V
n
=
2
π
n
V
n
−
2
{\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}
n
>
1
{\displaystyle n>1}
Površina
n
−
1
{\displaystyle n-1\,}
r
{\displaystyle r\,}
A
n
r
n
−
1
{\displaystyle A_{n}r^{n-1}\,}
A
n
{\displaystyle A_{n}\,}
n
{\displaystyle n\,}
r
{\displaystyle r\,}
V
n
r
n
{\displaystyle V_{n}r^{n}\,}
r
{\displaystyle r\,}
A
=
4
π
r
2
{\displaystyle A=4\pi r^{2}\,}
V
=
4
π
r
3
/
3
{\displaystyle V=4\pi r^{3}/3\,}
Enotska krogla v normiranih vektorskih prostorih [ uredi | uredi kodo ] Odprta enotska krogla v normiranem vektorskem prostoru
V
{\displaystyle V\,}
normo
|
|
.
|
|
{\displaystyle ||.||\,}
{
x
∈
V
:
‖
x
‖
<
1
}
{\displaystyle \{x\in V:\|x\|<1\}\,}
Pomeni pa notranjost zaprte enotske krogle, ki pripada (V , ||·||)
{
x
∈
V
:
‖
x
‖
≤
1
}
{\displaystyle \{x\in V:\|x\|\leq 1\}}
To pa sta disjunktni množici te krogle in njene skupne razmejitve z enotsko sfero (V ,||·||)
{
x
∈
V
:
‖
x
‖
=
1
}
{\displaystyle \{x\in V:\|x\|=1\}}
