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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
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