image of Paraboloidal 2

Paraboloidal 2

3D Coordinate Systems

Description

The Paraboloidal 2 coordinates are a three-dimensional orthogonal coordinate system (λ,μ,ν) that generalizes the two-dimensional parabolic coordinate system.
Similar to the related ellipsoidal coordinates, the Paraboloidal 2 coordinate system has orthogonal quadratic coordinate surfaces that are not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
Se also Paraboloidal coordinate system.

References

Moon & Spencer
Murray R. Spiegel

Object definitions

Mapping

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Mapping of Paraboloidal 2
\left\{x\to 2 \sqrt{\frac{(u-a) (a-v) (a-w)}{a-b}},y\to 2 \sqrt{\frac{(u-b) (b-v) (b-w)}{a-b}},z\to -a-b+u+v+w\right\}
<math> <mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mn>2</mn> <mo>&#8290;</mo> <msqrt> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>-</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>-</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>-</mo> <mi>w</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mn>2</mn> <mo>&#8290;</mo> <msqrt> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mi>u</mi> <mo>-</mo> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>-</mo> <mi>v</mi> </mrow> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mo>-</mo> <mi>w</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mi>a</mi> <mo>-</mo> <mi>b</mi> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo>,</mo> <mrow> <mi>z</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mrow> <mo>-</mo> <mrow> <mi>a</mi> </mrow> </mrow> <mo>-</mo> <mi>b</mi> <mo>+</mo> <mi>u</mi> <mo>+</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </math>
{x -> 2*Sqrt[((-a + u)*(a - v)*(a - w))/(a - b)], y -> 2*Sqrt[((-b + u)*(b - v)*(b - w))/(a - b)], z -> -a - b + u + v + w}
[x = 2*((-a+u)*(a-v)*(a-w)/(a-b))^(1/2), y = 2*((-b+u)*(b-v)*(b-w)/(a-b))^(1/2), z = -a-b+u+v+w]

Constants

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Constants of Paraboloidal 2
\{a,b\}
<math> <mrow> <mo>{</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mrow> <mo>}</mo> </mrow> </math>
{a, b}
[a, b]

Cite this as:

3D Coordinate Systems: Paraboloidal 2 from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/3DCoordinateSystems/Paraboloidal2.php

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