Parabolic Conoid

Author

Eric W. Weisstein
April 8, 2024

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

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

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

Equation

Wikipedia

surf

surf={1,u,-u^2}+v{-1,0,u^2};

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

GroebnerBasis[Subtract@@@eqn,{x,y,z},{u,v}]

AlgebraicDegree[eqn,{x,y}]

Ferréol

https://mathcurve.com/surfaces.gb/conoide_parabolique/conoide_parabolique.shtml

(a^2) z==x ((b^2)-(y^2))

surf={u,v,((u((b^2)-(v^2)))/(a^2))};

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

GroebnerBasis[Subtract@@@eqn,{x,y,z},{u,v}]//FullSimplify

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

Plot

ParametricPlot3D[{1,u,-u^2}+v{-1,0,u^2},{u,-2,2},{v,-2,2},PlotRange->1,Boxed->False,Axes->False]

Block[{a=1,b=0},ParametricPlot3D[{u,v,((u((b^2)-(v^2)))/(a^2))},{u,-2,2},{v,-2,2},PlotRange->1,Boxed->False,Axes->False]]

Block[{a=1,b=1},ParametricPlot3D[{u,v,((u((b^2)-(v^2)))/(a^2))},{u,-2,2},{v,-2,2},PlotRange->1,Boxed->False,Axes->False]]

Properties

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

surf={u,v,((u((b^2)-(v^2)))/(a^2))};
impl=(a^2)z-x ((b^2)-(y^2));

assum=(u|v)∈Reals&&a>0&&b>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]

ImplicitMeanCurvature

FullSimplify[MeanCurvature[impl,{x,y,z}],assum]

InertiaTensor

MeanCurvature

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

MetricTensor

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

NormalVector

Assuming[assum,FullSimplify[NormalVector[surf,{u,v}]]]

Numerator[%]

Denominator[%%][[1]]

PrincipalCurvatures

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

Subtract@@%//Simplify

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