%%%% sphrmd03.txt
%%%% Created by Laurence D. Finston (LDF) Fri Jul 16 18:26:21 CEST 2010


%% * (1) Copyright and License.

%%%% This file is part of GNU 3DLDF, a package for three-dimensional drawing. 
%%%% Copyright (C) 2010, 2011, 2012, 2013 The Free Software Foundation
   
%%%% GNU 3DLDF is free software; you can redistribute it and/or modify 
%%%% it under the terms of the GNU General Public License as published by 
%%%% the Free Software Foundation; either version 3 of the License, or 
%%%% (at your option) any later version. 

%%%% GNU 3DLDF is distributed in the hope that it will be useful, 
%%%% but WITHOUT ANY WARRANTY; without even the implied warranty of 
%%%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the 
%%%% GNU General Public License for more details. 

%%%% You should have received a copy of the GNU General Public License 
%%%% along with GNU 3DLDF; if not, write to the Free Software 
%%%% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA

%%%% GNU 3DLDF is a GNU package.  
%%%% It is part of the GNU Project of the  
%%%% Free Software Foundation 
%%%% and is published under the GNU General Public License. 
%%%% See the website http://www.gnu.org 
%%%% for more information.   
%%%% GNU 3DLDF is available for downloading from 
%%%% http://www.gnu.org/software/3dldf/LDF.html. 

%%%% Please send bug reports to Laurence.Finston@gmx.de.
%%%% The mailing list help-3dldf@gnu.org is available for people to 
%%%% ask other users for help.  
%%%% The mailing list info-3dldf@gnu.org is for the maintainer of 
%%%% GNU 3DLDF to send announcements to users. 
%%%% To subscribe to these mailing lists, send an 
%%%% email with ``subscribe <email-address>'' as the subject.  

%%%% The author can be contacted at: 

%%%% Laurence D. Finston 
%%%% c/o Free Software Foundation, Inc. 
%%%% 51 Franklin St, Fifth Floor 
%%%% Boston, MA  02110-1301  
%%%% USA

%%%% Laurence.Finston@gmx.de



%% * (1)

\special{papersize=297mm, 420mm} %% DIN A3 Portrait
\vsize=420mm
\hsize=297mm

%% \special{papersize=420mm, 297mm} %% DIN A3 Landscape
%% \vsize=297mm
%% \hsize=420mm

\input epsf 
\nopagenumbers

\pageno=1

\parindent=0pt

\def\epsfsize#1#2{#1}

\font\small=cmr8
\font\smalltt=cmtt8
\font\medium=cmr10
\font\largemed=cmr11
\font\large=cmr12
\font\largebx=cmbx12
\font\Large=cmr17
\font\huge=cmr17 scaled \magstep1

%% *** (3) 

\headline={\hfil}

\advance\voffset by -1in
\advance\voffset by 2cm

\advance\hoffset by -1in
\advance\hoffset by 1cm

%% *** (3) Title page

\pageno=1

\begingroup
\hsize=26cm
%\pageno=0
%\vskip2cm
\centerline{Sphere Model 3}
\vskip\baselineskip
\centerline{Laurence D. Finston}
\vskip\baselineskip
\centerline{Created:  July 16, 2010}
\vskip\baselineskip
\centerline{\hskip 1cm Last updated:  August 13, 2010}
\vskip0cm
%
{\small
\hsize=.75\hsize
%\hskip1cm
\vbox{\vskip2\baselineskip
This document is part of GNU 3DLDF, a package for three-dimensional
drawing.
\vskip\baselineskip

Copyright (C) 2010, 2011, 2012, 2013 The Free Software Foundation
\vskip\baselineskip

GNU 3DLDF is free software; you can redistribute it and/or modify 
it under the terms of the GNU General Public License as published by 
the Free Software Foundation; either version 3 of the License, or 
(at your option) any later version. 
\vskip\baselineskip

GNU 3DLDF is distributed in the hope that it will be useful, 
but WITHOUT ANY WARRANTY; without even the implied warranty of 
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the 
GNU General Public License for more details. 
\vskip\baselineskip

You should have received a copy of the GNU General Public License 
along with GNU 3DLDF; if not, write to the Free Software 
Foundation, Inc.,\hfil\break
51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
\vskip\baselineskip

See the GNU Free Documentation License for the copying conditions 
that apply to this document.
\vskip\baselineskip

You should have received a copy of the GNU Free Documentation License 
along with GNU 3DLDF; if not, write to the Free Software 
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301  USA
\vskip\baselineskip

The mailing list {\smalltt info-3dldf@gnu.org} is for sending 
announcements to users. To subscribe to this mailing list, send an 
email with ``subscribe $\langle$email-address$\rangle$'' as the subject.  
\vskip\baselineskip

The webpages for GNU 3DLDF are here:
{\smalltt http://www.gnu.org/software/3dldf/LDF.html}
\vskip\baselineskip

The author can be contacted at:\vskip\baselineskip 

Laurence D. Finston\hfil\break
c/o Free Software Foundation, Inc.\hfil\break
51 Franklin St, Fifth Floor \hfil\break
Boston, MA  02110-1301 \hfil\break 
USA
\vskip\baselineskip

{\smalltt Laurence.Finston@gmx.de}}}
\par

\vskip2cm

\def\epsfsize#1#2{1.5#1}

\centerline{\epsffile{sphrmd03.2}}

%\centerline{\special{psfile=sphrmd03.2 angle=20}} %% Rotation of image

\def\epsfsize#1#2{#1}

\vfil\eject

%% *** (3) Headline

\pageno=2

\headline={\hskip 2cm Sphere Model 3\quad %
           Copyright {\copyright} 2010, 2011, 2012, 2013 The Free Software Foundation\hfil %
           \folio\hfil Author:  Laurence D. Finston\hfil}

%% *** (3) Instructions

\begingroup

\parskip=\baselineskip

\vbox{}
\vskip\baselineskip

\centerline{{\largebx Instructions}}
\vskip 1.5\baselineskip

{\bf PLEASE NOTE!}  The author has tried to ensure that the following
plans are correct, but as of August 6, 2010, he has not tested them
yet himself.  As mentioned above, this material is distributed {\bf without
a warranty}.  I recommend that users check it themselves before
investing a lot of time and effort into cutting out the paper or cardboard 
model.

Any corrections or other feedback will be gratefully received by the author.
Contact information can be found above.

These plans represent a sphere divided into unlike triangles.
The source code can be found in the file {\tt sphrmd03.ldf} which 
is available from the GNU 3DLDF Sphere and Dome Models webpage:\hfil\break
{\tt http://www.gnu.org/software/3dldf/sphrdmmd.html\char`\#Sphere\char`\_Models}

The GNU 3DLDF program that generates the plans and other images in this
document, i.e., the code in {\tt sphrmd03.ldf}, uses parameters for various
values;  in particular, the radius of the sphere and the arc angle that
separates the points on the sphere that are used to find the points on the
triangles.  By changing the values of the parameters, variations of these
plans can be generated.  However, one must be careful, since some values may
cause unforeseen problems or unsatisfactory results, or require changes to
other parts of the program. 

The center of the sphere is located at the origin.
The radius of the sphere used here is $5.375\rm{cm}$.  It was chosen in order
to have (approx.) the largest possible sphere for which the plans would fit
onto a single sheet of DIN A3 paper.

The sphere is divided into 5 ``bands'' plus two pyramids with dodecagonal
bases for the poles.  The bands, in turn, each consist of 12 sets of 4
triangles.  When the model is folded together, each set of 4 triangles becomes 
a pyramid with a trapezoidal base, whereby the the base of the pyramids
belonging to the middle band are rectangular.

The corner points of the trapezoidal or rectangular bases represent line
segments that join points on $30^\circ$ arcs of circles that lie on the 
surface of the sphere and are parallel to one of the major planes 
(and perpendicular to the others).  Other plans may be created by changing the 
angle used for the arcs, but this would certainly require modifying other
parts of the program.  However, it shouldn't be too difficult.

To use these plans, tape, tack or otherwise attach them to a sheet of
paper or cardboard.  I generally prefer tacking to taping, where possible.
Cardboard could be used for this model, since all of the pieces are flat, but
heavy paper would probably also result in an adequately stable model.

This model uses ``double tabs'' for attaching pieces to each other.  The
triangles on the plans on page 4 have traces of the tabs drawn on them.  The
tabs themselves are on page 5.  The tabs can be used for sewing the model
together, either in combination with gluing or alone.  The model can be
assembled by gluing alone, in which case no holes need to be pricked on the
tabs or the triangles.  Nor is it necessary in this case for the tabs to
match up with the traces when assembling the model.

Sewing, however, will generally lead to better results, especially with regard
to the alignment of the pieces.  In this case, the crosshairs in black mark
the places where holes should be pricked, on both the pieces and the tabs.  It
may also be advisable to transfer the lines of the traces to the cardboard or
paper using graphite paper or in some other way.

Felt mats are available at hobby supply stores.  I find they work well for
pricking out the holes.

The solid (outer) lines should be cut out, while the dashed (inner) lines
should be scored only.  I have tried to be careful about drawing dashed and
solid lines correctly, but due to the complexity of this model, I can't be
sure that I haven't made any mistakes until I've tried it myself.  (Even then,
mistakes like this can slip past one easily.)  Please exercise care when
cutting out the model and don't cut through the paper or cardboard if you have
any doubts about a solid line being correct!  However, in an emergency, an
additional tab should fix the problem, if a line is cut through by mistake,
and the difference in the finished model will be negligible.

The knife must be sharp as heavy paper or cardboard will dull the blade quickly.
I have been using knives with disposable blades.  I've been meaning to
try sharpening them but haven't done so yet.  I therefore can't say
whether this will work.  It seems a shame to waste so many blades,
which is why I have a jar full of them.  They must be good for
something.  I don't think scissors would work well for this model, and I don't
recommend scoring with scissors, because this will dull its blade
unnecessarily.  However, this isn't a hard-and-fast rule, if someone prefers
scissors.

It will be necessary to reattach the plans parts of them are cut out.

Make sure that the plan is attached smoothly or you will introduce
inaccuracies.  {\it Do not detach it or let it slip until you are
done!\/}  You will never get it back where it's supposed to go. 

If you use tape, please use the removable kind.  Ordinary masking tape will
damage the paper when it is removed.  Be aware that ``removable tape'' isn't
completely reliable, especially if left too long on the drawing.  Sometimes
it's possible to reuse pieces of it, which avoids wasting large amounts of
it. 

Pages 6--7 contain patterns for the various triangles that appear in the
plans with dashed lines to make it easier to cut them out.  These patterns can
used to cut triangles out of better quality paper, such as watercolor paper,
which can then be attached to the assembled model.  This may improve the
appearance of the model, especially if it is to be painted.  One might also
cut the triangles out of good quality colored paper.
\par
\endgroup
\endgroup


\headline={\hskip 2cm Sphere Model 3\quad %
           Copyright {\copyright} 2010, 2011, 2012, 2013 The Free Software Foundation\hfil %
           \folio\hfil Author:  Laurence D. Finston\hfil}


\vfil\eject

%% *** (3) Page 3

\advance\hoffset by -1in
\advance\hoffset by 1.5cm

\pageno=3

%% *** (3) Page 3

\setbox0=\hbox{\epsffile{sphrmd03.1}}
\setbox1=\hbox to \wd0{\hss Parallel Projection onto the x-z plane\hss}

\setbox2=\vbox{\box0
  \vskip 1cm
  \box1}

\setbox0=\hbox{\epsffile{sphrmd03.2}}
\setbox1=\hbox to \wd0{\hss Parallel Projection onto the x-y plane\hss}

\setbox3=\vbox{\box0
  \vskip 1cm
  \box1}

\setbox0=\hbox{\epsffile{sphrmd03.3}}
\setbox1=\hbox to \wd0{\hss Parallel Projection onto the z-y plane\hss}

\setbox4=\vbox{\box0
  \vskip 1cm
  \box1}

\setbox0=\hbox{\epsffile{sphrmd03.4}}
\setbox1=\hbox to \wd0{\hss\vtop{\hbox{Perspective Projection}%
               \hbox{Position of focus: $(3\rm{cm}, 10\rm{cm}, -20\rm{cm})$}%
               \hbox{Direction of view: $(3\rm{cm}, 10\rm{cm}, 10\rm{cm})$}
               \hbox{Distance from plane of projection: $20\rm{cm}$}}\hss}

\setbox5=\vbox{\vskip1cm\box0
  \vskip 1cm
  \box1}

\vbox to \vsize{%
\vskip1cm
\centerline{\vbox{\hbox{Radius of sphere:  $5.375\rm{cm}$}%
                  \hbox{Center at origin}}}
\vskip2cm
\line{\hskip2cm \box2\hskip2cm\box3}
\vskip1cm
\line{\hskip2cm \box4\hskip2cm\box5\hss}
\vss}

%% *** (3) Page 4

\vbox to \vsize{%
\vskip1cm
\line{\hskip .75cm\epsffile{sphrmd03.5}\hss}
\vskip 2cm
\centerline{Model Plans}
\vss}

\vbox to \vsize{%
%\vskip1cm
\centerline{{\large Tabs}}
\vskip 0cm
\line{\hskip 2cm\epsffile{sphrmd03.6}\hss}
\vskip .5cm
\centerline{{\large Extra Tabs}}
%\vskip .5cm
\line{\hskip 2cm\epsffile{sphrmd03.6}\hss}
\vss}

%% Figure 8.  Triangles for cutting out and gluing to model, first page
%% LDF 2010.08.11.

\vbox to \vsize{%
\vskip1cm
\centerline{{\large Triangles for Cutting-Out 1}}
\vskip1cm
\line{\hskip.5cm\epsffile{sphrmd03.8}\hss}
\vss}

%% Figure 9.  Triangles for cutting out and gluing to model, second page
%% LDF 2010.08.13.

\vbox to \vsize{%
\vskip1cm
\centerline{{\large Triangles for Cutting-Out 2}}
\vskip1cm
\line{\hskip.25cm\epsffile{sphrmd03.9}\hss}
\vss}


%% %% Figure 7.  For use on website.

%% \vbox to \vsize{%
%% \vskip1cm
%% \line{\hskip 2cm\epsffile{sphrmd03.7}\hss}
%% \vss}





%% *** (3) End here

\bye

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