Cornucopia

Author

Eric W. Weisstein
June 20, 2013

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

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

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

Initialization

Definition

surf[a_,b_][u_,v_]:={
Exp[b v]Cos[v]+Exp[a v]Cos[u]Cos[v],
Exp[b v]Sin[v]+Exp[a v]Cos[u]Sin[v],
Exp[a v]Sin[u]
}

Surface

V5

corn=ParametricPlot3D[
,{u,0,2Pi},{v,-5,2.5},Ticks->None,PlotPoints->{20,30},Axes->None,PlotRange->All, Boxed->False]

V6

ParametricPlot3D[surf[Sequence@@#][u,v],{u,0,2Pi},{v,-5,2.5},Axes->None, Boxed->False,PlotRange->All]&/@{{1,1},{.3,0},{0,.5}}

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["cornucop.m",corn]

Properties

Curvatures[surf[a,b][u,v],{u,v},FullSimplify[#,a>0&&b>0]&]

Curvatures[surf[1,0][u,v],{u,v},FullSimplify[#,(u|v)∈Reals]&]

Curvatures[surf[1,1][u,v],{u,v},FullSimplify[#,(u|v)∈Reals]&]

Simplify[PrincipalCurvatures[surf[a,b][u,v],{u,v}],Assumptions->0<u<2π&&v∈Reals&&a>0&&b>0]

FullSimplify[%Assumptions->0<u<2π&&v∈Reals&&a>0&&b>0]

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf[a,b][u,v],{u,v}]]

Vector Properties

eqn={Exp[b v]Cos[v]+Exp[a v]Cos[u]Cos[v],Exp[b v]Sin[v]+Exp[a v]Cos[u]Sin[v],Exp[a v]Sin[u]};

Vector Length

FullSimplify[Norm[eqn]/.Abs[x_]^2:>x^2,(a|b|u|v)∈Reals]

Normal Vector

FullSimplify[NormalVector[eqn,{u,v}]/.Abs[x_]^2:>x^2,(u|v|a|b)∈Reals]