       Projection of curley 3D curves to represent a cylindrical surface

• To: mathgroup at smc.vnet.net
• Subject: [mg99102] Projection of curley 3D curves to represent a cylindrical surface
• From: Bill <WDWNORWALK at aol.com>
• Date: Mon, 27 Apr 2009 01:15:42 -0400 (EDT)

```Hi:

I have the following plot in 3D:

Clear[T,pnt,M,g1,g2];
T[t_]:={t^2,t,1};
pnt={{{0,0,0},{1,0,0},{2,0,0}},{{0,1,0},{1,1,1},
{2,1,-0.5}},{{0,2,0},{1,2,0},{2,2,0}}};
M={{2,-4,2},{-3,4,-1},{1,0,0}};
pts=Graphics3D[{AbsolutePointSize,
Table[Point[pnt[[i,j]]],{i,1,3},{j,1,3}]}];
comb[i_]:=(T[u].M.pnt)[[i]] (Transpose[M].T[w])[[i]];
surf=ParametricPlot3D[comb+comb+comb,{u,0,1},{w,0,1},
Mesh->False,BoundaryStyle->Directive[Magenta,Thickness[0.005]],
PlotPoints->100,RegionFunction->(1/16+1/32 Sin[10
ArcTan[#4-1/2,#5-1/2]]
<(#4-1/2)^2+(#5-1/2)^2<1/8+1/32 Sin[10 ArcTan[#4-1/2,
#5-1/2]]&),
Axes->True]//Quiet;
pic1=Show[surf,ViewPoint->{0,0,2},BaseStyle->{10,FontFamily->"cmr10"},
AxesLabel->{"x","y","z"},ImageSize->500]

I'd like to project the magenta colored curves from their z coordinate positions down to the x-y=0 plane, creating cylindrical walls that match the curves. Perhaps the plot and code that I've given above can be converted to 3D Graphics,and the function "Scale" could be used to do this. I've tried to do this but haven't had any success. However, I'm not sure that this is the right approach.

As an end result, I'd like to show the above plot with the cylindrical surfaces in one 3D plot.

Suggestions and Mathematica 6.0.1 code would be greatly appreciated.

Thanks,

Bill

```

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