Trinoid

Author

Eric W. Weisstein
June 11, 2014

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

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

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

Equation

EnneperWeierstrass[ 1/(z^3-1)^2,z^2, {z,r,ϕ}]

FullSimplify[%,r>0&&0<ϕ<2π]

Trinoid[r_,ϕ_]:={(1/18) (((6 r (r+(1+(r^2)) Cos[ϕ]))/(1+(r^2)+(r^4)+2 (r+(r^3)) Cos[ϕ]+2 (r^2) Cos[2 ϕ]))-4 Log[1+(r^2)-2 r Cos[ϕ]]+2 Log[1+(r^2)+(r^4)+2 (r+(r^3)) Cos[ϕ]+2 (r^2) Cos[2 ϕ]]),(1/9) (-2 (Sqrt[3]) ArcTanh[(((Sqrt[3]) r Sin[ϕ])/(1+(r^2)+r Cos[ϕ]))]-((3 r (1+(r^4)+2 (r+(r^3)) Cos[ϕ]) Sin[ϕ])/(1+(r^6)-2 (r^3) Cos[3 ϕ]))),(1/(3+((3-3 (r^6))/(-2+2 (r^3) Cos[3 ϕ]))))}

Trinoid[r,ϕ]//Column

Plots

Second Form

ParametricPlot3D[Trinoid[r,t],{t,0,2Pi},{r,0,4},Boxed->False,Axes->False,RegionFunction->(#1^2+#2^2+#3^2<3^2&)]

Curvatures

Curvatures[Trinoid[r,ϕ],{r,ϕ},Simplify[#,r>0&&0<ϕ<2π]&]

FullSimplify[{((((1+(r^4)))^2)/(((1+(r^6)-2 (r^3) Cos[3 ϕ]))^2)),0,((((r+(r^5)))^2)/(((1+(r^6)-2 (r^3) Cos[3 ϕ]))^2))},r>0]

FullSimplify[{-((4 r (-2 (r^3)+(1+(r^6)) Cos[3 ϕ]))/(((1+(r^6)-2 (r^3) Cos[3 ϕ]))^2)),-((4 (r^2) (-1+(r^2)) (1+(r^2)+(r^4)) (1+2 Cos[2 ϕ]) Sin[ϕ])/(((1+(r^6)-2 (r^3) Cos[3 ϕ]))^2)),((4 (r^3) (-2 (r^3)+(1+(r^6)) Cos[3 ϕ]))/(((1+(r^6)-2 (r^3) Cos[3 ϕ]))^2))},r>0]

Principal Curvatures

Simplify[PrincipalCurvatures[Trinoid[r,ϕ],{r,ϕ}],r>0&&0<ϕ<2π]

Mean Curvature

Table[MeanCurvature[Trinoid[r,ϕ],{r,ϕ}]/.{r->RandomReal[{0,5}],ϕ->RandomReal[{0,2π}]},{20}]//Chop//Union

Simplify[MeanCurvature[Trinoid[r,ϕ],{r,ϕ}],r>0]//Timing

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

Simplify[ChristoffelSymbols[Trinoid[r,ϕ],{r,ϕ}],r>0]

...ran overnight...

Vector Properties

eqn={1/18 ((6 r (r+(1+r^2) Cos[phi]))/(1+r^2+r^4+2 (r+r^3) Cos[phi]+2 r^2 Cos[2 phi])-4 Log[1+r^2-2 r Cos[phi]]+2 Log[1+r^2+r^4+2 (r+r^3) Cos[phi]+2 r^2 Cos[2 phi]]),1/9 (-2 Sqrt[3] ArcTanh[(Sqrt[3] r Sin[phi])/(1+r^2+r Cos[phi])]-(3 r (1+r^4+2 (r+r^3) Cos[phi]) Sin[phi])/(1+r^6-2 r^3 Cos[3 phi])),1/(3+(3-3 r^6)/(-2+2 r^3 Cos[3 phi]))};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&r>0&&0<phi<2π;

Vector Length

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

Normal Vector

Assuming[assum,Simplify[NormalVector[eqn,{r,phi}]/.rules]]//Timing