Zindler Conoid

Author

Eric W. Weisstein
April 12, 2024

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/ZindlerConoid.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ZindlerConoid.html.

©2024 Wolfram Research, Inc. except for portions noted otherwise

Equation

Wikipedia

https://en.wikipedia.org/wiki/Conoid#Right_circular_conoid

surf={u Cos[v],u Sin[v],a Tan[2v]};

eqn=TrigExpand[Thread[{x,y,z}==surf]]

gbeqn=Append[Numerator[Together[Subtract@@@eqn]]/.{Cos[v]->c,Sin[v]->s},c^2+s^2-1]

GroebnerBasis[gbeqn,{x,y,z},{u,v,c,s}]//FullSimplify

Solve[%[[1]]==0,z]//Collect[#,x]&

Plot

Block[{a=1},ParametricPlot3D[{u Cos[v],u Sin[v],a Tan[2v]},{u,-2,2},{v,0,2Pi},Boxed->False,Axes->False]]

Block[{a=1/2},ParametricPlot3D[{u Cos[v],u Sin[v],a Tan[2v]},{u,-2,2},{v,0,2Pi},Boxed->False,Axes->False]]

Properties

<<MathWorld`Surfaces`
<<MathWorld`CurvilinearCoordinates`

surf={u Cos[v],u Sin[v],a Tan[2v]};
impl=z((x^2)+(y^2))-2 a x y;

assum=u>0&&0≤v<2π&&a>0;

AlgebraicEquation

Subtract@@(impl==0)

AreaElement

FullSimplify[AreaElement[surf,{u,v}],assum]

CartesianEquation

impl==0

Centroid

ChristoffelSymbolOfTheSecondKind

FullSimplify[ChristoffelSymbols[surf,{u,v}],assum]

ChristoffelSymbolOfTheSecondKindArfken

FullSimplify[ChristoffelSymbols2[surf,{u,v}],assum]

FirstFundamentalForm

{ℰ,ℱ,𝒢}=Simplify[FirstFundamentalForm[surf,{u,v}],assum]

GaussianCurvatue

FullSimplify[GaussianCurvature[surf,{u,v}],assum]

ImplicitGaussianCurvatue

FullSimplify[GaussianCurvature[impl,{x,y,z}],assum&&(x|y|z)∈Reals]

ImplicitMeanCurvature

FullSimplify[MeanCurvature[impl,{x,y,z}],assum&&(x|y|z)∈Reals]

InertiaTensor

MeanCurvature

FullSimplify[MeanCurvature[surf,{u,v}],assum]

MetricTensor

MetricTensor[surf,{u,v}]//FullSimplify

NormalVector

Assuming[assum,FullSimplify[NormalVector[surf,{u,v}]]]/.Abs[x_]^2:>x^2//FullSimplify

PrincipalCurvatures

FullSimplify[PrincipalCurvatures[surf,{u,v}],assum]

SecondFundamentalForm

{ℯ,𝒻,ℊ}=FullSimplify[SecondFundamentalForm[surf,{u,v}],assum]

SurfaceArea

{ℰ,ℱ,𝒢}=Simplify[FirstFundamentalForm[surf,{u,v}]]

FullSimplify[(Sqrt[ℰ 𝒢-(ℱ^2)]),assum]

VectorLength

Assuming[assum,FullSimplify[Norm[surf]]]/.Abs[x_]^2:>x^2//FullSimplify

Volume