Catecoid

In the spirit of Self-Conjugate, Family Resemblance, and Gridded, here's the associate family of the catenoid and helicoid (I took a crack at this once before, with Locally Isometric).

The code uses a trick suggested by Jens Noeckel in this Stack Exchange thread to make the mesh lines have variable colors. I didn't really bother optimizing, so it's kind of slow, but here's the code which generated the GIF:

catecoid = With[{
    cols = RGBColor /@ {"#2BCDC1", "#F66095", "#393E46"},
    n = 80},
   ParallelTable[Module[{offset},
       ParametricPlot3D[{Cos[t] Sinh[v] Sin[u] + Sin[t] Cosh[v] Cos[u], 
          -Cos[t] Sinh[v] Cos[u] + Sin[t] Cosh[v] Sin[u], u Cos[t] + v Sin[t]},
        {u, -π, π}, {v, -2, 2},
        ViewPoint -> Front, Boxed -> False, Axes -> None, 
        PlotRange -> 4, ViewAngle -> π/5, 
        MeshFunctions -> 
         Map[Function[{x, y, z}, z - offset] /. offset -> # &, Range[8/(2 n), 8, 8/n]], Mesh -> {{-4}}, PlotPoints -> 50, 
        MeshStyle -> 
         Table[Directive[Thickness[.003], Blend[cols[[;; -2]], Rescale[i, {n/4, 3 n/4}]]], {i, 1, n}], 
        PlotStyle -> None, Background -> cols[[-1]], ImageSize -> 540]],
      {t, 0., 2 π - #, #}] &[2 π/100]
   ];

Export[NotebookDirectory[] <> "catecoid.gif", catecoid, "DisplayDurations" -> {1/24}]