Inverse oblate spheroidal
3D Coordinate Systems
Description
The Inverse oblate spheroidal is three-dimensional coordinate system.
References
Object definitions
Mapping
- TeX
- MathML
- Mathematica input
- Maple input
\left\{x\to \frac{a \cosh (u) \sin (v) \cos (w)}{\cosh ^2(u)-\cos ^2(v)},y\to \frac{a \cosh (u) \sin (v) \sin (w)}{\cosh ^2(u)-\cos ^2(v)},z\to \frac{a \sinh (u) \cos (v)}{\cosh ^2(u)-\cos ^2(v)}\right\}
<math>
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<mi>x</mi>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mfrac>
<mrow>
<mi>a</mi>
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<mrow>
<mi>cosh</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>sin</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>cos</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>w</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>cosh</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
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<mo>)</mo>
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</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mrow>
<mi>y</mi>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mfrac>
<mrow>
<mi>a</mi>
<mo>⁢</mo>
<mrow>
<mi>cosh</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>sin</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>sin</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>w</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>cosh</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>,</mo>
<mrow>
<mi>z</mi>
<semantics>
<mo>→</mo>
<annotation encoding='Mathematica'>"\[Rule]"</annotation>
</semantics>
<mfrac>
<mrow>
<mi>a</mi>
<mo>⁢</mo>
<mrow>
<mi>sinh</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>⁢</mo>
<mrow>
<mi>cos</mi>
<mo>⁡</mo>
<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>cosh</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<mi>u</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
<mo>(</mo>
<mi>v</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math>
{x -> (a*Cos[w]*Cosh[u]*Sin[v])/(-Cos[v]^2 + Cosh[u]^2), y -> (a*Cosh[u]*Sin[v]*Sin[w])/(-Cos[v]^2 + Cosh[u]^2), z -> (a*Cos[v]*Sinh[u])/(-Cos[v]^2 + Cosh[u]^2)}
[x = a*cos(w)*cosh(u)*sin(v)/(-cos(v)^2+cosh(u)^2), y = a*cosh(u)*sin(v)*sin(w)/(-cos(v)^2+cosh(u)^2), z = a*cos(v)*sinh(u)/(-cos(v)^2+cosh(u)^2)]
Constants
- TeX
- MathML
- Mathematica input
- Maple input
\{a\}
<math>
<mrow>
<mo>{</mo>
<mi>a</mi>
<mo>}</mo>
</mrow>
</math>
{a}
[a]
Cite this as:
3D Coordinate Systems: Inverse oblate spheroidal from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/3DCoordinateSystems/Inverse-Oblate-Spheroidal.php
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