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Monday, April 13, 2015

Math Gems: the Snail Shell

Previous post: the Breather

Gem


Snail Shell faceted (facets have been post processed with Blender, not Mathematica output)


Snail shell smooth


The pearl material has been based on Kaluura's Post on Blenderartists

Math

The snail shell is given by (see e.g. virtualmathmuseum.org)


The shape above is the result of setting the factors to:
  • aa:=0.1
  • bb:=0.1
  • cc:=0.2
  • dd:=0
  • ee:=-1 
The corresponding Mathematica script looks like this:

aa := 0.1;
bb := 0.1;
cc := 0.2;
dd := 0;
ee := -1;
r := s (aa + bb Cos[u]);
vv := v + (v + ee)^2/16;
s := Exp[\[Minus]cc*vv];
x = r*Cos[vv]
z = r*Sin[vv]
y = dd*(1 \[Minus] s) + s*bb*Sin[u]
thePlot =
 ParametricPlot3D[{x, y, z}, {u, 0, 2 Pi}, {v, 0, 2 Pi},
  NormalsFunction -> None, Boxed -> False, Axes -> False]

Export["Snailshell.ply", thePlot, VertexNormals -> Automatic]