Henneberg's Minimal Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Equation

Enneper-Weierstrass

FullSimplify[EnneperWeierstrass[ 2-2/z^4,z, {z,r,ϕ}]]

ew[r_,ϕ_]:={((2 (-1+(r^2)) Cos[ϕ])/r)-((2 (-1+(r^6)) Cos[3 ϕ])/(3 (r^3))),((2/r)-2 r) Sin[ϕ]-((2 (-1+(r^6)) Sin[3 ϕ])/(3 (r^3))),((2 (1+(r^4)) Cos[2 ϕ])/(r^2))}

eqn=Append[TrigExpand[Numerator[Together[{x,y,z}-ew[r,ϕ]]]],Sin[ϕ]^2+Cos[ϕ]^2-1]

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

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

eqn=-4194304 (x^6)-884736 (x^8)+77760 (x^10)-1458 (x^12)+12582912 (x^4) (y^2)+1769472 (x^6) (y^2)-637632 (x^8) (y^2)-12582912 (x^2) (y^4)+1524096 (x^6) (y^4)+4374 (x^8) (y^4)+4194304 (y^6)-1769472 (x^2) (y^6)-1524096 (x^4) (y^6)+884736 (y^8)+637632 (x^2) (y^8)-4374 (x^4) (y^8)-77760 (y^10)+1458 (y^12)-995328 (x^8) z+34992 (x^10) z+3981312 (x^6) (y^2) z-104976 (x^8) (y^2) z-5971968 (x^4) (y^4) z+69984 (x^6) (y^4) z+3981312 (x^2) (y^6) z+69984 (x^4) (y^6) z-995328 (y^8) z-104976 (x^2) (y^8) z+34992 (y^10) z-25165824 (x^4) (z^2)-4423680 (x^6) (z^2)+248832 (x^8) (z^2)-8748 (x^10) (z^2)-7962624 (x^4) (y^2) (z^2)-1990656 (x^6) (y^2) (z^2)-26244 (x^8) (y^2) (z^2)+25165824 (y^4) (z^2)+7962624 (x^2) (y^4) (z^2)-17496 (x^6) (y^4) (z^2)+4423680 (y^6) (z^2)+1990656 (x^2) (y^6) (z^2)+17496 (x^4) (y^6) (z^2)-248832 (y^8) (z^2)+26244 (x^2) (y^8) (z^2)+8748 (y^10) (z^2)-3145728 (x^4) (z^3)-6660096 (x^6) (z^3)+276480 (x^8) (z^3)-1215 (x^10) (z^3)-18874368 (x^2) (y^2) (z^3)+1351680 (x^4) (y^2) (z^3)-9963 (x^8) (y^2) (z^3)-3145728 (y^4) (z^3)+1351680 (x^2) (y^4) (z^3)-2045952 (x^4) (y^4) (z^3)-23814 (x^6) (y^4) (z^3)-6660096 (y^6) (z^3)-23814 (x^4) (y^6) (z^3)+276480 (y^8) (z^3)-9963 (x^2) (y^8) (z^3)-1215 (y^10) (z^3)-37748736 (x^2) (z^4)-2752512 (x^4) (z^4)-1437696 (x^6) (z^4)+15552 (x^8) (z^4)+37748736 (y^2) (z^4)-9179136 (x^4) (y^2) (z^4)+124416 (x^6) (y^2) (z^4)+2752512 (y^4) (z^4)+9179136 (x^2) (y^4) (z^4)+1437696 (y^6) (z^4)-124416 (x^2) (y^6) (z^4)-15552 (y^8) (z^4)-9437184 (x^2) (z^5)-10420224 (x^4) (z^5)+359424 (x^6) (z^5)-3888 (x^8) (z^5)-9437184 (y^2) (z^5)+17694720 (x^2) (y^2) (z^5)-2294784 (x^4) (y^2) (z^5)-15552 (x^6) (y^2) (z^5)-10420224 (y^4) (z^5)-2294784 (x^2) (y^4) (z^5)-23328 (x^4) (y^4) (z^5)+359424 (y^6) (z^5)-15552 (x^2) (y^6) (z^5)-3888 (y^8) (z^5)+7077888 (x^2) (z^6)-4767744 (x^4) (z^6)+104064 (x^6) (z^6)-216 (x^8) (z^6)-7077888 (y^2) (z^6)+21120 (x^4) (y^2) (z^6)-432 (x^6) (y^2) (z^6)+4767744 (y^4) (z^6)-21120 (x^2) (y^4) (z^6)-104064 (y^6) (z^6)+432 (x^2) (y^6) (z^6)+216 (y^8) (z^6)+1769472 (x^2) (z^7)-651264 (x^4) (z^7)+4320 (x^6) (z^7)+1769472 (y^2) (z^7)-1105920 (x^2) (y^2) (z^7)-7776 (x^4) (y^2) (z^7)-651264 (y^4) (z^7)-7776 (x^2) (y^4) (z^7)+4320 (y^6) (z^7)-442368 (x^2) (z^8)-10752 (x^4) (z^8)+442368 (y^2) (z^8)+10752 (y^4) (z^8)-110592 (x^2) (z^9)-768 (x^4) (z^9)+16 (x^6) (z^9)-110592 (y^2) (z^9)+4608 (x^2) (y^2) (z^9)+48 (x^4) (y^2) (z^9)-768 (y^4) (z^9)+48 (x^2) (y^4) (z^9)+16 (y^6) (z^9)+9216 (x^2) (z^10)-384 (x^4) (z^10)-9216 (y^2) (z^10)+384 (y^4) (z^10)+2304 (x^2) (z^11)+2304 (y^2) (z^11);

Parametric

surf[u_,v_]:={
2 Sinh[u]Cos[v]-2/3 Sinh[3u]Cos[3v],
2Sinh[u]Sin[v]+2/3Sinh[3u]Sin[3v],
2 Cosh[2u]Cos[2v]
}

eqn=Join[Thread[{x,y,z}==TrigExpand[surf[u,v]]],{Cos[v]^2+Sin[v]^2==1,Cosh[u]^2-Sinh[u]^2==1}]

{eqn}=Factor/@GroebnerBasis[eqn,{x,y,z},{Sin[v],Cos[v],Sinh[u],Cosh[u]},MonomialOrder->EliminationOrder]

Plot

EnneperWeierstrass[ 2-2/z^4,z, {z,r,ϕ}]

V5

WriteLiveForm["henneber.m",EnneperWeierstrass[ 2-2/z^4, z, z,{0.35, 0.85}, {-Pi, Pi},
	PlotPoints->{12, 48},Axes->False,PlotRange->All,
	opts]]

V6

ParametricPlot3D[surf[u,v],{u,-1,1},{v,0,π},PlotPoints->25]

ParametricPlot3D[ew[r,ϕ],{r,0,1},{ϕ,0,2π},PlotPoints->100,RegionFunction->(#1^2+#2^2+#3^2<10^2&)]

Curvature

(r,ϕ)

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

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

FullSimplify[{(((r^4) (Sqrt[(1/(1+(r^8)-2 (r^4) Cos[4 ϕ]))]))/(((1+(r^2)))^2)),-(((r^4) (Sqrt[(1/(1+(r^8)-2 (r^4) Cos[4 ϕ]))]))/(((1+(r^2)))^2))},r>0&&ϕ>0]

(u,v)

Curvatures[surf[u,v],{u,v},FullSimplify[#,(u|v)∈Reals]&]

FullSimplify[PrincipalCurvatures[surf[u,v],{u,v}],(u|v)∈Reals]

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

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf[u,v],{u,v}],a>0]

Vector Properties

eqn={2 Sinh[u]Cos[v]-2/3 Sinh[3u]Cos[3v],2Sinh[u]Sin[v]+2/3Sinh[3u]Sin[3v],2 Cosh[2u]Cos[2v]};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&u∈Reals&&0<v<π;

Vector Length

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

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{u,v}]/.rules]]