Spljošteni sferoidni sistem u trodimenzionalnom prostoru predstavlja ortogonalni koordinatni sistem nastao rotacijom sferoida oko male osi, na kojoj se ne nalaze fokusi .
Spljoštene sferoidne koordinate koriste se da se reše različite parcijalne diferencijalne jednačine, u kojima granični uslovi odgovaraju spljoštenom sferoidu sa dva fokusa na velikoj osi.
Najčešća definicija izduženih sferoidnih koordinata
(
μ
,
ν
,
ϕ
)
{\displaystyle (\mu ,\nu ,\phi )}
x
=
a
cosh
μ
cos
ν
cos
ϕ
{\displaystyle x=a\ \cosh \mu \ \cos \nu \ \cos \phi }
y
=
a
cosh
μ
cos
ν
sin
ϕ
{\displaystyle y=a\ \cosh \mu \ \cos \nu \ \sin \phi }
z
=
a
sinh
μ
sin
ν
{\displaystyle z=a\ \sinh \mu \ \sin \nu }
gde je
μ
{\displaystyle \mu }
ν
∈
[
−
π
/
2
,
π
/
2
]
{\displaystyle \nu \in [-\pi /2,\pi /2]}
Površi konstantnoga
μ
{\displaystyle \mu }
x
2
+
y
2
a
2
cosh
2
μ
+
z
2
a
2
sinh
2
μ
=
cos
2
ν
+
sin
2
ν
=
1
{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {z^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}
One predstavljaju elipse , koje se rotiraju oko z osi, koja razdvaja fokuse. Elipsa u x -z ravni ima veću poluos dužine a cosh μ duž x osi, a manja poluos je a sinh μ duž z osi.
Na sličan način dobija se i sledeća relacija:
x
2
+
y
2
a
2
cos
2
ν
−
z
2
a
2
sin
2
ν
=
cosh
2
μ
−
sinh
2
μ
=
1
{\displaystyle {\frac {x^{2}+y^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {z^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}
iz koje se vidi da površi konstantnoga
ν
{\displaystyle \nu }
Lameovi koeficijenti skaliranja su:
h
μ
=
h
ν
=
a
sinh
2
μ
+
sin
2
ν
{\displaystyle h_{\mu }=h_{\nu }=a{\sqrt {\sinh ^{2}\mu +\sin ^{2}\nu }}}
h
ϕ
=
a
cosh
μ
cos
ν
{\displaystyle h_{\phi }=a\cosh \mu \ \cos \nu }
Infinitezimalni element zapremine je:
d
V
=
a
3
cosh
μ
cos
ν
(
sinh
2
μ
+
sin
2
ν
)
d
μ
d
ν
d
ϕ
{\displaystyle dV=a^{3}\cosh \mu \ \cos \nu \ \left(\sinh ^{2}\mu +\sin ^{2}\nu \right)d\mu d\nu d\phi }
a Laplasijan je:
∇
2
Φ
=
1
a
2
(
sinh
2
μ
+
sin
2
ν
)
[
1
cosh
μ
∂
∂
μ
(
cosh
μ
∂
Φ
∂
μ
)
+
1
cos
ν
∂
∂
ν
(
cos
ν
∂
Φ
∂
ν
)
]
+
1
a
2
(
cosh
2
μ
+
cos
2
ν
)
∂
2
Φ
∂
ϕ
2
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {1}{\cosh \mu }}{\frac {\partial }{\partial \mu }}\left(\cosh \mu {\frac {\partial \Phi }{\partial \mu }}\right)+{\frac {1}{\cos \nu }}{\frac {\partial }{\partial \nu }}\left(\cos \nu {\frac {\partial \Phi }{\partial \nu }}\right)\right]+{\frac {1}{a^{2}\left(\cosh ^{2}\mu +\cos ^{2}\nu \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}
Spljošteni sferoidni sistem može da se parametrizira i sa druge tri koordinate
(
ζ
,
ξ
,
ϕ
)
{\displaystyle (\zeta ,\xi ,\phi )}
x
=
a
(
1
+
ζ
2
)
(
1
−
ξ
2
)
cos
ϕ
{\displaystyle x=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}\,\cos \phi \,}
y
=
a
(
1
+
ζ
2
)
(
1
−
ξ
2
)
sin
ϕ
{\displaystyle y=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}\,\sin \phi \,}
z
=
a
ζ
ξ
{\displaystyle z=a\zeta \xi \,}
Lameovi koeficijenti druge verzije [ uredi | uredi izvor ]
h
ζ
=
a
ζ
2
+
ξ
2
1
+
ζ
2
{\displaystyle h_{\zeta }=a{\sqrt {\frac {\zeta ^{2}+\xi ^{2}}{1+\zeta ^{2}}}}}
h
ξ
=
a
ζ
2
+
ξ
2
1
−
ξ
2
{\displaystyle h_{\xi }=a{\sqrt {\frac {\zeta ^{2}+\xi ^{2}}{1-\xi ^{2}}}}}
h
ϕ
=
a
(
1
+
ζ
2
)
(
1
−
ξ
2
)
{\displaystyle h_{\phi }=a{\sqrt {(1+\zeta ^{2})(1-\xi ^{2})}}}
Infinitezimalni element zapremine je:
d
V
=
a
3
(
ζ
2
+
ξ
2
)
d
ζ
d
ξ
d
ϕ
{\displaystyle dV=a^{3}(\zeta ^{2}+\xi ^{2})\,d\zeta \,d\xi \,d\phi }
a Laplasijan je:
∇
2
V
=
1
a
2
(
ζ
2
+
ξ
2
)
{
∂
∂
ζ
[
(
1
+
ζ
2
)
∂
V
∂
ζ
]
+
∂
∂
ξ
[
(
1
−
ξ
2
)
∂
V
∂
ξ
]
}
+
1
a
2
(
1
+
ζ
2
)
(
1
−
ξ
2
)
∂
2
V
∂
ϕ
2
{\displaystyle \nabla ^{2}V={\frac {1}{a^{2}\left(\zeta ^{2}+\xi ^{2}\right)}}\left\{{\frac {\partial }{\partial \zeta }}\left[\left(1+\zeta ^{2}\right){\frac {\partial V}{\partial \zeta }}\right]+{\frac {\partial }{\partial \xi }}\left[\left(1-\xi ^{2}\right){\frac {\partial V}{\partial \xi }}\right]\right\}+{\frac {1}{a^{2}\left(1+\zeta ^{2}\right)\left(1-\xi ^{2}\right)}}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}}
Treća verzija sistema ima sledeće tri koordinate
(
σ
,
τ
,
ϕ
)
{\displaystyle (\sigma ,\tau ,\phi )}
x
=
a
σ
τ
cos
ϕ
{\displaystyle x=a\sigma \tau \cos \phi \,}
y
=
a
σ
τ
sin
ϕ
{\displaystyle y=a\sigma \tau \sin \phi \,}
z
2
=
a
2
(
σ
2
−
1
)
(
1
−
τ
2
)
{\displaystyle z^{2}=a^{2}\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}
Lameovi koeficijenti skaliranja su:
h
σ
=
a
σ
2
−
τ
2
σ
2
−
1
{\displaystyle h_{\sigma }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{\sigma ^{2}-1}}}}
h
τ
=
a
σ
2
−
τ
2
1
−
τ
2
{\displaystyle h_{\tau }=a{\sqrt {\frac {\sigma ^{2}-\tau ^{2}}{1-\tau ^{2}}}}}
h
ϕ
=
a
σ
τ
{\displaystyle h_{\phi }=a\sigma \tau }
Infinitezimalni element zapremine je:
d
V
=
a
3
σ
τ
σ
2
−
τ
2
(
σ
2
−
1
)
(
1
−
τ
2
)
d
σ
d
τ
d
ϕ
{\displaystyle dV=a^{3}\sigma \tau {\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau d\phi }
a Laplasijan je:
∇
2
Φ
=
1
a
2
(
σ
2
−
τ
2
)
{
σ
2
−
1
σ
∂
∂
σ
[
(
σ
σ
2
−
1
)
∂
Φ
∂
σ
]
+
1
−
τ
2
τ
∂
∂
τ
[
(
τ
1
−
τ
2
)
∂
Φ
∂
τ
]
}
+
1
a
2
σ
2
τ
2
∂
2
Φ
∂
ϕ
2
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left\{{\frac {\sqrt {\sigma ^{2}-1}}{\sigma }}{\frac {\partial }{\partial \sigma }}\left[\left(\sigma {\sqrt {\sigma ^{2}-1}}\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\sqrt {1-\tau ^{2}}}{\tau }}{\frac {\partial }{\partial \tau }}\left[\left(\tau {\sqrt {1-\tau ^{2}}}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}
Spljošteni sferoidni sistem Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers , McGraw-Hill.
Abramowitz, Milton; Stegun, Irene A., eds. (1965), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover. ISBN 978-0-486-61272-0 .
Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 0-07-043316-X 
![{\displaystyle \nu \in [-\pi /2,\pi /2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49b15801b2da923b1276a83e4d221265fdfaca19)
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sinh ^{2}\mu +\sin ^{2}\nu \right)}}\left[{\frac {1}{\cosh \mu }}{\frac {\partial }{\partial \mu }}\left(\cosh \mu {\frac {\partial \Phi }{\partial \mu }}\right)+{\frac {1}{\cos \nu }}{\frac {\partial }{\partial \nu }}\left(\cos \nu {\frac {\partial \Phi }{\partial \nu }}\right)\right]+{\frac {1}{a^{2}\left(\cosh ^{2}\mu +\cos ^{2}\nu \right)}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5d81529bb6d17bca254304cd7481396b325e8c)
![{\displaystyle \nabla ^{2}V={\frac {1}{a^{2}\left(\zeta ^{2}+\xi ^{2}\right)}}\left\{{\frac {\partial }{\partial \zeta }}\left[\left(1+\zeta ^{2}\right){\frac {\partial V}{\partial \zeta }}\right]+{\frac {\partial }{\partial \xi }}\left[\left(1-\xi ^{2}\right){\frac {\partial V}{\partial \xi }}\right]\right\}+{\frac {1}{a^{2}\left(1+\zeta ^{2}\right)\left(1-\xi ^{2}\right)}}{\frac {\partial ^{2}V}{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b357b2c9bd2183b7e40d3f74589b6b2b948608de)
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left\{{\frac {\sqrt {\sigma ^{2}-1}}{\sigma }}{\frac {\partial }{\partial \sigma }}\left[\left(\sigma {\sqrt {\sigma ^{2}-1}}\right){\frac {\partial \Phi }{\partial \sigma }}\right]+{\frac {\sqrt {1-\tau ^{2}}}{\tau }}{\frac {\partial }{\partial \tau }}\left[\left(\tau {\sqrt {1-\tau ^{2}}}\right){\frac {\partial \Phi }{\partial \tau }}\right]\right\}+{\frac {1}{a^{2}\sigma ^{2}\tau ^{2}}}{\frac {\partial ^{2}\Phi }{\partial \phi ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14964682baacf6ce7e88a813cf299106742a402b)