Dini's Surface

Author

Eric W. Weisstein
June 20, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equations

Eliminate[{
x==a Cos[u]Sin[v],
y==a Sin[u]Sin[v],
z==a (Cos[v]+Log[Tan[v/2]])+b u},{u,v}]

Graphic

V5.2

g=ParametricPlot3D[Evaluate[{
a Cos[u]Sin[v],
a Sin[u]Sin[v],
a (Cos[v]+Log[Tan[v/2]])+b u}/.{a->1,b->0.2}
],{u,0,4Pi},{v,.001,2},Boxed->False,Axes->False,PlotPoints->{30,15}
]

V6

g=ParametricPlot3D[Evaluate[{
a Cos[u]Sin[v],
a Sin[u]Sin[v],
a (Cos[v]+Log[Tan[v/2]])+b u}/.{a->1,b->0.2}
],{u,0,4Pi},{v,.001,2},Boxed->False,Axes->False
]

Calculate

g=ParametricPlot3D[Evaluate[{
a Cos[u]Sin[v],
a Sin[u]Sin[v],
a (Cos[v]+Log[Tan[v/2]])+b u}/.{a->1,b->0.2}
],{u,0,4Pi},{v,.001,2},Boxed->False,Axes->False,PlotPoints->10,MaxRecursion->Automatic
]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["dini.m",g]

Properties

Curvatures

Curvatures[{a Cos[u]Sin[v],a Sin[u]Sin[v],a (Cos[v]+Log[Tan[v/2]])+b u},{u,v}]

FullSimplify[PrincipalCurvatures[{a Cos[u]Sin[v],a Sin[u]Sin[v],a (Cos[v]+Log[Tan[v/2]])+b u},{u,v}],a>0&&b>0&&(u|v)∈Reals]

PiecewiseExpand[{((Abs[Csc[v]]-Cos[2 v] Csc[v])/(2 (Sqrt[(a^2)+(b^2)]) Abs[Cos[v]])),-((Abs[Csc[v]]+Cos[2 v] Csc[v])/(2 (Sqrt[(a^2)+(b^2)]) Abs[Cos[v]]))},-π<v<π]//FullSimplify//Refine

Line Element

Vector Properties

eqn={a Cos[u] Sin[v],a Sin[u] Sin[v],a (Cos[v]+Log[Tan[v/2]])+b u};

Vector Length

FullSimplify[FullSimplify[Norm[eqn],a>0&&(u|v)∈Reals]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n]

Normal Vector

Assuming[a>0&&(u|v)∈Reals,FullSimplify[NormalVector[eqn,{u,v}]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n]]