image of Astroidal ellipsoid

Astroidal ellipsoid

Surfaces

Description

The Astroidal ellipsoid is a three-dimensional surface which is the inverse of the ellipsoid in the sense that it "goes in" where the ellipsoid "goes out."

Object definitions

Mapping

Mapping of Astroidal ellipsoid
\left\{x\to a \cos ^3(u) \cos ^3(v),y\to b \sin ^3(u) \cos ^3(v),z\to c \sin ^3(v)\right\}
<math> <mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mrow> <msup> <mi>cos</mi> <mn>3</mn> </msup> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <msup> <mi>cos</mi> <mn>3</mn> </msup> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>b</mi> <mo>&#8290;</mo> <mrow> <msup> <mi>sin</mi> <mn>3</mn> </msup> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <msup> <mi>cos</mi> <mn>3</mn> </msup> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mrow> <mi>z</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mi>c</mi> <mo>&#8290;</mo> <mrow> <msup> <mi>sin</mi> <mn>3</mn> </msup> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> </mrow> <mo>}</mo> </mrow> </math>
{x -> a*Cos[u]^3*Cos[v]^3, y -> b*Cos[v]^3*Sin[u]^3, z -> c*Sin[v]^3}
[x = a*cos(u)^3*cos(v)^3, y = b*cos(v)^3*sin(u)^3, z = c*sin(v)^3]

Constants

Constants of Astroidal ellipsoid
\{a,b,c\}
<math> <mrow> <mo>{</mo> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mrow> <mo>}</mo> </mrow> </math>
{a, b, c}
[a, b, c]

Cite this as:

Surfaces: Astroidal ellipsoid from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/Surfaces/Astroidal-Ellipsoid.php

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