Enneper's Minimal Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Equations

Enneper-Weierstrass

FullSimplify[EnneperWeierstrass[1,z,{z,r,ϕ}]]

ew[r_,ϕ_]:={r Cos[ϕ]-(1/3) (r^3) Cos[3 ϕ],-(1/3) r (3 Sin[ϕ]+(r^2) Sin[3 ϕ]),(r^2) Cos[2 ϕ]}

eqns=Append[Thread[{x,y,z}==TrigExpand[ew[r,ϕ]]],((Cos[ϕ])^2)+((Sin[ϕ])^2)==1]

{eqn}=Factor/@GroebnerBasis[eqns,{x,y,z},{r,Sin[ϕ],Cos[ϕ]},MonomialOrder->EliminationOrder]

eqn=-729 (x^6)+2187 (x^4) (y^2)-2187 (x^2) (y^4)+729 (y^6)+729 (x^4) z-1458 (x^2) (y^2) z+729 (y^4) z-4374 (x^4) (z^2)+4374 (y^4) (z^2)+3888 (x^2) (z^3)-1215 (x^4) (z^3)+3888 (y^2) (z^3)-6318 (x^2) (y^2) (z^3)-1215 (y^4) (z^3)-6480 (x^2) (z^4)+6480 (y^2) (z^4)+5184 (z^5)-3888 (x^2) (z^5)-3888 (y^2) (z^5)-432 (x^2) (z^6)+432 (y^2) (z^6)-1152 (z^7)+64 (z^9);

AlgebraicDegree[eqn,{x,y,z}]

Nordstrand

Numerator[Factor[Expand[((((((y^2)-(x^2))/(2z))+(2/9)(z^2)+(2/3)))^3)-6(((((((y^2)-(x^2)))/(4z))-(1/4)((x^2)+(y^2)+(8/9)(z^2))+(2/9)))^2)]]]

(u,v)

Solve[{r Cos[ϕ]==u,r Sin[ϕ]==v},{r,ϕ}]//FullSimplify

Simplify[TrigExpand[ew[r,ϕ]]/.{r->(Sqrt[(u^2)+(v^2)]),ϕ->ArcCos[(u/(Sqrt[(u^2)+(v^2)]))]},(u|v)∈Reals]

Parametrizations

Parametric

Show[ParametricPlot3D[{u-(u^3/3)+u v^2,v-(v^3/3)+u^2v,u^2-v^2},{u,-2,2},{v,-2,2}, Boxed->False, Axes->False]]

<<Utilities`LiveGraphics3D`

WriteLiveForm["enneperm.m",%]

Weierstrass

ParametricPlot3D[ew[r,ϕ],{r,0,#},{ϕ,0,2π},PlotPoints->100]&/@{1,2,3}

(R. Maeder)

Show[ParametricPlot3D[{
  Re[E^(I*phi)*r - (E^(3*I*phi)*r^3)/3], 
  Re[I*E^(I*phi)*r + I/3*E^(3*I*phi)*r^3],
  Re[E^(2*I*phi)*r^2]},
{r, 0, 1}, {phi, -Pi, Pi}, PlotPoints-> {10, 40},
PlotRange -> All, Axes-> False, Boxed -> False],opts]

Curvature

Curvatures[ew[r,ϕ],{r,ϕ},FullSimplify[#,r>0]&]

Simplify[PrincipalCurvatures[ew[r,ϕ],{r,ϕ}],r>0]

Curvatures[{u-u^3/3+u v^2,-v-u^2 v+v^3/3,u^2-v^2},{u,v}]

FullSimplify[PrincipalCurvatures[{u-u^3/3+u v^2,-v-u^2 v+v^3/3,u^2-v^2},{u,v}],(u|v)∈Reals]

Gaussian

ByteCount/@(k=GaussianCurvature[eqn,{x,y,z}])//Timing

FullSimplify[k[[1]]]

Mean Curvature

Numerator[MeanCurvature[eqn,{x,y,z}]]//Simplify

FullSimplify[MeanCurvature[eqn,{x,y,z}]]

Metric Properties

surf={r Cos[ϕ]-(1/3) (r^3) Cos[3 ϕ],-(1/3) r (3 Sin[ϕ]+(r^2) Sin[3 ϕ]),(r^2) Cos[2 ϕ]};

Metric Tensor

MetricTensor[surf,{r,ϕ}]//FullSimplify

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf,{r,ϕ}],r>0]

Vector Properties

eqn={r Cos[ϕ]-1/3 r^3 Cos[3 ϕ],-(1/3) r (3 Sin[ϕ]+r^2 Sin[3 ϕ]),r^2 Cos[2 ϕ]};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&r>0&&0<ϕ<2π;

Vector Length

Assuming[assum,FullSimplify[Norm[eqn]/.rules]]

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{r,ϕ}]/.rules]]