Re: How to put text on a curved surface?

*To*: mathgroup at smc.vnet.net*Subject*: [mg87471] Re: How to put text on a curved surface?*From*: Narasimham <mathma18 at hotmail.com>*Date*: Fri, 11 Apr 2008 01:45:13 -0400 (EDT)*References*: <fsvels$sto$1@smc.vnet.net> <ft2aok$pbg$1@smc.vnet.net>

On Apr 5, 2:23 pm, Narasimham <mathm... at hotmail.com> wrote: > On Apr 3, 3:15 pm, Narasimham <mathm... at hotmail.com> wrote: > > > On Apr 2, 1:04 pm, P_ter <peter_van_summe... at yahoo.co.uk> wrote: > > > > I would like to put "Mathematica" on a curved surface, e.g. a torus. > > > Can anyone help here? > > > with friendly greetings, > > > P_ter > > > You mean the projected/mapped letters should be curved ? Try > > projecting stencil slots onto the torus from a suitable bright point > > as light source, which can be even at infinity.. as intersection with > > the conical rays. > > It should be mentioned that the mapping can never be faithful ( per > isometric mappings between flat to flat or curvrd to curved surfaces ) > as there is bound to be alteration due to: > > 1) magnification/reduction ( strain of tension or compression), > > 2) distortion ( angle changes in shear). > > It is in other words so stated by Gauss Egregium theorem. After > writing or printing the text on a flat label when you try to stick it > on a torus of suitable size, a tendency to tear on areas outside the > crown (Gauss Curvature > 0) and some folds/frills on the inside ( G.C > < 0) is inevitable. The most faithful reproduction takes place at the > crown ( G.C ~ 0). Continued... Simple example below is for a faithful map from a flat label to a cone in 3D plot 'drape' : t=0; umin=1;umax=16;vmin=3;vmax=5; (* before draping flatstrip on cone ; can be rasterized as suggested by Fred Klingener *) Strp={u Cos[t]-v Sin[t],u Sin[t]+v Cos[t],0}; aa=ParametricPlot3D[Strp,{u,umin,umax}, {v,vmin,vmax},PlotPoints=AE{17,4}]; th[u_,v_,gt_]:=ArcTan[u Cos[gt]-v Sin[gt],u Sin[gt]+v Cos[gt]]; gt=Pi/4; bb=ParametricPlot3D[Sqrt[u^2+v^2] {Cos[th[u,v,gt]],Sin[th[u,v,gt]],0}, {u,umin,umax},{v,vmin,vmax},PlotPoints=AE{17,4}] Show[aa,bb] (* PlotPoints=AE4 is partly does not come through, bb is also not Shown in full*) al=Pi/12; cone=ParametricPlot3D[RHO {Sin[al] Cos[t],Sin[al] Sin[t],Cos[al]} +{0,0,.03},{RHO,2,17},{t,0,2 Pi},PlotPoints=AE{41,55}] (*after rotating gt in the plane,Drape/Bend/Roll the flat strip into a cone of semi vertex angle al*) t=1.4; rho=Sqrt[u^2+v^2]; STRP=rho*{Sin[al] Cos[th[u,v,t]/Sin[al]],Sin[al] Sin[th[u,v,t]/ Sin[al]],Cos[al]} ; drape=ParametricPlot3D[STRP,{u,umin,umax}, {v,vmin,vmax},PlotPoints=AE{65,11}] Show[cone,drape] Narasimham