Osnovna definicija
uredi
Eliptične koordinate
(
μ
,
ν
)
{\displaystyle (\mu ,\nu )}
x
=
a
cosh
μ
cos
ν
{\displaystyle x=a\ \cosh \mu \ \cos \nu }
y
=
a
sinh
μ
sin
ν
{\displaystyle y=a\ \sinh \mu \ \sin \nu }
gde je
μ
{\displaystyle \mu }
ν
∈
[
0
,
2
π
]
.
{\displaystyle \nu \in [0,2\pi ].}
μ
{\displaystyle \mu }
x
2
a
2
cosh
2
μ
+
y
2
a
2
sinh
2
μ
=
cos
2
ν
+
sin
2
ν
=
1
{\displaystyle {\frac {x^{2}}{a^{2}\cosh ^{2}\mu }}+{\frac {y^{2}}{a^{2}\sinh ^{2}\mu }}=\cos ^{2}\nu +\sin ^{2}\nu =1}
S druge strane drugom jednačinom određuje se familija hiperbola konstantnoga
ν
{\displaystyle \nu }
x
2
a
2
cos
2
ν
−
y
2
a
2
sin
2
ν
=
cosh
2
μ
−
sinh
2
μ
=
1
{\displaystyle {\frac {x^{2}}{a^{2}\cos ^{2}\nu }}-{\frac {y^{2}}{a^{2}\sin ^{2}\nu }}=\cosh ^{2}\mu -\sinh ^{2}\mu =1}
Lameovi koeficijenti
uredi
U ortogonalnom koordinatnom sistemu dužine vektora baze poznate su kao faktori skaliranja ili kao Lameovi koeficijenti , koji su za eliptične koordinate:
H
μ
=
H
ν
=
a
s
h
2
μ
+
sin
2
ν
.
{\displaystyle H_{\mu }=H_{\nu }=a{\sqrt {\mathrm {sh} ^{2}\,\mu +\sin ^{2}\nu }}.}
a nakon sređivanja kao:
H
μ
=
H
ν
=
a
1
2
(
c
h
2
μ
−
cos
2
ν
)
.
{\displaystyle H_{\mu }=H_{\nu }=a{\sqrt {{\frac {1}{2}}(\mathrm {ch} \,2\mu -\cos 2\nu }}).}
Elemenat površine dat je sa:
d
S
=
a
2
(
s
h
2
μ
+
sin
2
ν
)
d
μ
d
ν
,
{\displaystyle dS=a^{2}(\mathrm {sh} ^{2}\,\mu +\sin ^{2}\nu )\,d\mu \,d\nu ,}
a Laplasijan :
∇
2
Φ
=
1
a
2
(
s
h
2
μ
+
sin
2
ν
)
(
∂
2
Φ
∂
μ
2
+
∂
2
Φ
∂
ν
2
)
.
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}(\mathrm {sh} ^{2}\,\mu +\sin ^{2}\nu )}}\left({\frac {\partial ^{2}\Phi }{\partial \mu ^{2}}}+{\frac {\partial ^{2}\Phi }{\partial \nu ^{2}}}\right).}
Alternativna definicija
uredi
Ponekad se koristi i alternativna definicija eliptičnih koordinata
(
σ
,
τ
)
{\displaystyle (\sigma ,\;\tau )}
σ
=
c
h
μ
,
{\displaystyle \sigma =\mathrm {ch} \,\mu ,}
τ
=
cos
ν
.
{\displaystyle \tau =\cos \nu .}
Koordinate
(
σ
,
τ
)
{\displaystyle (\sigma ,\;\tau )}
d
1
,
d
2
{\displaystyle d_{1},d_{2}}
F
1
{\displaystyle F_{1}}
F
2
{\displaystyle F_{2}}
d
1
+
d
2
=
2
a
σ
,
{\displaystyle d_{1}+d_{2}=2a\sigma ,}
d
1
−
d
2
=
2
a
τ
,
{\displaystyle d_{1}-d_{2}=2a\tau ,}
Na taj način dobija se i:
d
1
=
a
(
σ
+
τ
)
;
{\displaystyle d_{1}=a(\sigma +\tau );}
d
2
=
a
(
σ
−
τ
)
.
{\displaystyle d_{2}=a(\sigma -\tau ).}
Taj koordinatni sistem ima nedostatak da koordinate (x,y) i (x,-y) imaju isti
(
σ
,
τ
)
{\displaystyle (\sigma ,\tau )}
x
=
a
σ
τ
;
{\displaystyle x=a\sigma \tau ;}
y
2
=
a
2
(
σ
2
−
1
)
(
1
−
τ
2
)
.
{\displaystyle y^{2}=a^{2}(\sigma ^{2}-1)(1-\tau ^{2}).}
Lameovi koeficijenti alternativne verzije
uredi
Lameovi koeficijenti alternativnih eliptičnih koordinata
(
σ
,
τ
)
{\displaystyle (\sigma ,\tau )}
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}}}}.}
Elemenat površine dat je sa:
d
A
=
a
2
σ
2
−
τ
2
(
σ
2
−
1
)
(
1
−
τ
2
)
d
σ
d
τ
{\displaystyle dA=a^{2}{\frac {\sigma ^{2}-\tau ^{2}}{\sqrt {\left(\sigma ^{2}-1\right)\left(1-\tau ^{2}\right)}}}d\sigma d\tau }
a Laplasijan je:
∇
2
Φ
=
1
a
2
(
σ
2
−
τ
2
)
[
σ
2
−
1
∂
∂
σ
(
σ
2
−
1
∂
Φ
∂
σ
)
+
1
−
τ
2
∂
∂
τ
(
1
−
τ
2
∂
Φ
∂
τ
)
]
.
{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].}
Literatura
uredi
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 . 
![{\displaystyle \nu \in [0,2\pi ].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad13894fea18f0bc355ad1eca69f2378bef70659)
![{\displaystyle \nabla ^{2}\Phi ={\frac {1}{a^{2}\left(\sigma ^{2}-\tau ^{2}\right)}}\left[{\sqrt {\sigma ^{2}-1}}{\frac {\partial }{\partial \sigma }}\left({\sqrt {\sigma ^{2}-1}}{\frac {\partial \Phi }{\partial \sigma }}\right)+{\sqrt {1-\tau ^{2}}}{\frac {\partial }{\partial \tau }}\left({\sqrt {1-\tau ^{2}}}{\frac {\partial \Phi }{\partial \tau }}\right)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/387a412b798ac56009b9b3afe86bff2995fc2ba2)