Note

This Mathematica notebook is from the web at
http://xahlee.org/surface/gallery.html
Xah Lee, 2005

Helicoid-Catenoid

when b is 0, it's helicoid. When b is Pi/2, it's catenoid.

Manipulate[ParametricPlot3D[{Cos[b] Sinh[v] Sin[u]+Sin[b] Cosh[v] Cos[u],-Cos[b] Sinh[v] Cos[u]+Sin[b] Cosh[v] Sin[u],Cos[b] u+Sin[b] v},{u,0,π 2},{v,-1,1},PlotPoints->{40,20},PlotStyle->Directive[Opacity[0.8]]],{b,0,(π/2)}]

animation. b goes from 0 to Pi/2.

b=0;
vMax=1;
popts={PlotPoints->{40,10},PlotRange->{{-1,1}π,{-1,1}π,{-1,2π}}};
Do[gp1=First@ParametricPlot3D[{Cos[b] Sinh[v] Sin[u]+Sin[b] Cosh[v] Cos[u],-Cos[b] Sinh[v] Cos[u]+Sin[b] Cosh[v] Sin[u],Cos[b] u+Sin[b] v},{u,0,π*2},{v,-0,vMax},Evaluate@popts];
gp2=First@ParametricPlot3D[{Cos[b] Sinh[v] Sin[u]+Sin[b] Cosh[v] Cos[u],-Cos[b] Sinh[v] Cos[u]+Sin[b] Cosh[v] Sin[u],Cos[b] u+Sin[b] v},{u,0,π*2},{v,-vMax,0},Evaluate@popts];
Show[Graphics3D[{gp1,gp2/.Polygon->Line}]],{b,0,π/2,π/2/20}]