Obseg je v geometriji dolžina zaprte krivulje, po navadi dvorazsežne ravninske krivulje. Največkrat se govori o obsegu pri geometrijskih likih, čeprav pridejo v poštev tudi druge krivulje, kroga, srčnica. V takšnih primerih se še posebej obravnava dolžina loka krivulje.
Mnogokotniki
uredi
Obseg mnogokotnika je vsota dolžin vseh njegovih stranic .
Obseg trikotnika s stranicami dolžin a , b in c je:
o
=
a
+
b
+
c
.
{\displaystyle o=a+b+c\,\!.}
Obseg štirikotnika s stranicami dolžin a , b , c in d je:
o
=
a
+
b
+
c
+
d
.
{\displaystyle o=a+b+c+d\,\!.}
Obseg enakokrakega trikotnika z osnovnico dolžine b in krakoma dolžine a ter pravokotnika s stranicama dolžin a in b je:
o
=
2
a
+
b
,
{\displaystyle o=2a+b\,\!,}
o
=
2
(
a
+
b
)
.
{\displaystyle o=2(a+b)\,\!.}
Obseg pravilnega mnogokotnika z n stranicami dolžine a je:
o
=
n
a
.
{\displaystyle o=na\,\!.}
Obseg enakostraničnega trikotnika in kvadrata s stranicami dolžine a je tako:
o
=
3
a
,
{\displaystyle o=3a\,\!,}
o
=
4
a
.
{\displaystyle o=4a\,\!.}
Približki za obseg elipse z glavnima polosema a in b :
o
≈
2
π
a
b
{\displaystyle o\approx 2\pi {\sqrt {ab}}\,\!}
Kepler , 1609)
o
≈
π
(
a
+
b
)
,
{\displaystyle o\approx \pi (a+b)\,\!,}
o
≈
π
2
(
a
2
+
b
2
)
{\displaystyle o\approx \pi {\sqrt {2(a^{2}+b^{2})}}\,\!}
Euler , 1773)
o
≈
π
[
a
+
b
2
+
a
2
+
b
2
2
]
,
{\displaystyle o\approx \pi \left[{\frac {a+b}{2}}+{\sqrt {\frac {a^{2}+b^{2}}{2}}}\right]\,\!,}
o
≈
π
[
3
2
(
a
+
b
)
−
a
b
]
{\displaystyle o\approx \pi \left[{\frac {3}{2}}(a+b)-{\sqrt {ab}}\right]\,\!}
ali:
o
≈
π
2
(
a
2
+
b
2
)
−
1
2
(
a
−
b
)
2
.
{\displaystyle o\approx \pi {\sqrt {2(a^{2}+b^{2})-{\frac {1}{2}}(a-b)^{2}}}\,\!.}
Vsak približek je točnejši od predhodnega.
Dobra približka je leta 1914 dal Ramanudžan :
o
≈
π
[
3
(
a
+
b
)
−
(
3
a
+
b
)
(
a
+
3
b
)
]
=
π
(
a
+
b
)
[
3
−
4
−
h
]
,
{\displaystyle o\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]=\pi (a+b)\left[3-{\sqrt {4-h}}\right]\,\!,}
o
≈
π
(
a
+
b
)
[
1
+
3
(
a
−
b
a
+
b
)
2
10
+
4
−
3
(
a
−
b
a
+
b
)
2
]
=
π
(
a
+
b
)
[
1
+
3
h
10
+
4
−
3
h
]
.
{\displaystyle o\approx \pi (a+b)\left[1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right]=\pi \left(a+b\right)\left[1+{\frac {3h}{10+{\sqrt {4-3h}}}}\right]\,\!.}
kjer je h parameter:
h
=
λ
2
,
λ
=
a
−
b
a
+
b
.
{\displaystyle h=\lambda ^{2}\,\!,\qquad \lambda ={\frac {a-b}{a+b}}\,\!.}
Tudi tukaj je drugi približek točnejši. Malo manj točen približek je med letoma 1904 in 1920 dal Lindner:
o
≈
π
(
a
+
b
)
[
1
+
h
8
]
2
.
{\displaystyle o\approx \pi (a+b)\left[1+{\frac {h}{8}}\right]^{2}\,\!.}
Obseg elipse s parametrom λ je:
o
=
π
(
a
+
b
)
[
1
+
λ
2
4
+
λ
4
64
+
λ
6
256
+
⋯
]
=
π
(
a
+
b
)
[
1
+
∑
n
=
1
∞
(
(
2
n
−
2
)
!
n
!
(
n
−
1
)
!
2
2
n
−
1
)
2
λ
2
n
]
,
{\displaystyle o=\pi (a+b)\left[1+{\frac {\lambda ^{2}}{4}}+{\frac {\lambda ^{4}}{64}}+{\frac {\lambda ^{6}}{256}}+\cdots \right]=\pi (a+b)\left[1+\sum _{n=1}^{\infty }\left({\frac {(2n-2)!}{n!(n-1)!2^{2n-1}}}\right)^{2}\lambda ^{2n}\right]\,\!,}
oziroma s parametrom h :
o
=
π
(
a
+
b
)
[
1
+
h
4
+
h
2
64
+
h
3
256
+
25
h
4
16384
+
49
h
5
65536
+
⋯
]
=
π
(
a
+
b
)
∑
n
=
0
∞
(
1
2
n
)
2
h
n
,
{\displaystyle o=\pi (a+b)\left[1+{\frac {h}{4}}+{\frac {h^{2}}{64}}+{\frac {h^{3}}{256}}+{\frac {25h^{4}}{16384}}+{\frac {49h^{5}}{65536}}+\cdots \right]=\pi (a+b)\sum _{n=0}^{\infty }{{1 \over 2} \choose n}^{2}h^{n}\,\!,}
približek pa (Hudsonova enačba, 1917):
o
≈
π
(
a
+
b
)
64
−
3
h
2
64
−
16
h
.
{\displaystyle o\approx \pi (a+b){\frac {64-3h^{2}}{64-16h}}\,\!.}
Hudsonovo enačbo po navadi pišejo s parametrom L :
L
=
h
4
=
(
a
−
b
)
2
(
2
(
a
+
b
)
)
2
{\displaystyle L={\frac {h}{4}}={\frac {(a-b)^{2}}{(2(a+b))^{2}}}\,\!}
o
≈
π
4
(
a
+
b
)
[
3
(
1
+
L
)
+
1
1
−
L
]
.
{\displaystyle o\approx {\frac {\pi }{4}}(a+b)\left[3(1+L)+{\frac {1}{1-L}}\right]\,\!.}
Zunanje povezave
uredi
![{\displaystyle o\approx \pi \left[{\frac {a+b}{2}}+{\sqrt {\frac {a^{2}+b^{2}}{2}}}\right]\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abb6324843f8035c4e8abd8e9b22e34b9c2fd731)
![{\displaystyle o\approx \pi \left[{\frac {3}{2}}(a+b)-{\sqrt {ab}}\right]\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cca12f290b84d4e2ed6746bfcff1e669394a0abf)
![{\displaystyle o\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]=\pi (a+b)\left[3-{\sqrt {4-h}}\right]\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/835354480bc9c1bd90572f45d6846ae662a2235e)
![{\displaystyle o\approx \pi (a+b)\left[1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right]=\pi \left(a+b\right)\left[1+{\frac {3h}{10+{\sqrt {4-3h}}}}\right]\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/263256bcbbb9e6485fc8782bcf3435ff8f291018)
![{\displaystyle o\approx \pi (a+b)\left[1+{\frac {h}{8}}\right]^{2}\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfa0b11eb69b023618b39502e62b5b02ceff845)
![{\displaystyle o=\pi (a+b)\left[1+{\frac {\lambda ^{2}}{4}}+{\frac {\lambda ^{4}}{64}}+{\frac {\lambda ^{6}}{256}}+\cdots \right]=\pi (a+b)\left[1+\sum _{n=1}^{\infty }\left({\frac {(2n-2)!}{n!(n-1)!2^{2n-1}}}\right)^{2}\lambda ^{2n}\right]\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/479ca192a53c3afe8a85967c5bbf875e499f861f)
![{\displaystyle o=\pi (a+b)\left[1+{\frac {h}{4}}+{\frac {h^{2}}{64}}+{\frac {h^{3}}{256}}+{\frac {25h^{4}}{16384}}+{\frac {49h^{5}}{65536}}+\cdots \right]=\pi (a+b)\sum _{n=0}^{\infty }{{1 \over 2} \choose n}^{2}h^{n}\,\!,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8daa4652a7ac827658f86c041165fa46f1f0ab)
![{\displaystyle o\approx {\frac {\pi }{4}}(a+b)\left[3(1+L)+{\frac {1}{1-L}}\right]\,\!.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f5e0e5716ef938a88692638c969ba742ed43135)
