\input mftmac
\def\9{$\futurelet\next\doit}
\let\[=\relax
\def\doit{\ifx\next\[\def\next{\hfill{\rm\%}}%
     \else\ifx\next\ \def\next{\qquad{\rm\%}}%
     \else\def\next{{\rm\%}}\fi\fi\next}
\input epsf
\def\newpage{\vfill\eject}
\advance\vsize1in
\let\ora\overrightarrow
\def\title#1{\hrule\vskip1mm#1\par\vskip1mm\hrule\vskip5mm}
\def\figure#1{\par\centerline{\epsfbox{#1}}}
\title{{\bf 3D.MP : 3-DIMENSIONAL REPRESENTATIONS
                    AND ANIMATIONS IN METAPOST}}
\BL
 version 1.0, 8 April 1998
 {\bf Denis Roegel} ({\tt roegel@loria.fr})
\BL
 This package provides definitions enabling the manipulation
 and animation of 3-dimensional objects.
 Such objects can be included in a \TeX{} file or used on web pages
 for instance. See the documentation enclosed in the distribution for
 more details.
\BL
 Thanks to John Hobby and Ulrik Vieth for helpful hints.
\BL
 LIMITATIONS:
\BL
   $-$ an object can not include \TeX{} labels; to overcome this limitation
     would need to wrap the metapost output with dvips so that the
     necessary fonts are loaded (easy to do, but not done here)
\BL
 PROJECTS FOR THE FUTURE:
\BL
   $-$ take light sources into account and show shadows and darker faces
\BL
   $-$ handle overlapping of objects ({\it obj\_name\/} can be used when
     going through all faces)
\BL
$\2{if}\1{known}\\{three\_d\_version}\?:$\par
\quad\quad$\2{expandafter}\5{endinput}\9 avoids loading this package twice\par
$\2{fi};$\par
\BL
$\2{message}\7"*** 3d, version 1.0 (c) D. Roegel, 8 April 1998 ***";$\par
$\2{numeric}\\{three\_d\_version};$\par
$\\{three\_d\_version}=1.0;$\par
\BL
\newpage
\title{Vector operations}
\BL
$\9 definition of vector $i$ by its coordinates\par
$\2{def}\\{vect\_def}(\2{expr}i,\\{xi},\\{yi},\\{zi})=\\{vect}[i]x:=\\{xi};\ %
\\{vect}[i]y:=\\{yi};\ \\{vect}[i]z:=\\{zi};\3{enddef};$\par
\BL
$\9 a point is stored as a vector\par
$\2{let}\\{set\_point}=\\{vect\_def};$\par
\BL
$\9 $\\{set\_obj\_point}$ is like $\\{set\_point}$, but the first parameter\par
$\9 is a local point number\par
$\2{def}\\{set\_obj\_point}(\2{expr}i,\\{xi},\\{yi},\\{zi})=\\{set\_point}(%
\\{pnt}(i),\\{xi},\\{yi},\\{zi})\3{enddef};$\par
\BL
$\9 definition of a vector by an other vector\par
$\2{def}\\{vect\_def\_vect}(\2{expr}i,j)=$\par
\quad\quad$\\{vect}[i]x:=\\{vect}[j]x;\ \\{vect}[i]y:=\\{vect}[j]y;\ %
\\{vect}[i]z:=\\{vect}[j]z;$\par
$\!\3{enddef};$\par
\BL
$\9 vector sum: $\\{vect}[k]$ $\leftarrow$ $\\{vect}[i]$$+$$\\{vect}[j]$\par
$\2{def}\\{vect\_sum}(\2{expr}k,i,j)=$\par
\quad\quad$\\{vect}[k]x:=\\{vect}[i]x+\\{vect}[j]x;$\par
\quad\quad$\\{vect}[k]y:=\\{vect}[i]y+\\{vect}[j]y;$\par
\quad\quad$\\{vect}[k]z:=\\{vect}[i]z+\\{vect}[j]z;$\par
$\!\3{enddef};$\par
\BL
$\9 vector translation: $\\{vect}[i]$ $\leftarrow$ $\\{vect}[i]$$+$$%
\\{vect}[v]$\par
$\2{def}\\{vect\_translate}(\2{expr}i,v)=\\{vect\_sum}(i,i,v)\3{enddef};$\par
\BL
$\9 vector difference: $\\{vect}[k]$ $\leftarrow$ $\\{vect}[i]$$-$$\\{vect}[j]$%
\par
$\2{def}\\{vect\_diff}(\2{expr}k,i,j)=$\par
\quad\quad$\\{vect}[k]x:=\\{vect}[i]x-\\{vect}[j]x;$\par
\quad\quad$\\{vect}[k]y:=\\{vect}[i]y-\\{vect}[j]y;$\par
\quad\quad$\\{vect}[k]z:=\\{vect}[i]z-\\{vect}[j]z;$\par
$\!\3{enddef};$\par
\BL
$\9 dot product of $\\{vect}[i]$ and $\\{vect}[j]$\par
$\2{vardef}\\{vect\_dprod}(\2{expr}i,j)=$\par
\quad\quad$(\\{vect}[i]x\ast \\{vect}[j]x+\\{vect}[i]y\ast \\{vect}[j]y+%
\\{vect}[i]z\ast \\{vect}[j]z)$\par
$\!\3{enddef};$\par
\BL
$\9 modulus of $\\{vect}[i]$\par
$\2{vardef}\\{vect\_mod}(\2{expr}i)=\1{sqrt}(\\{vect\_dprod}(i,i))\3{enddef};$%
\par
\BL
$\9 vector product: $\\{vect}[k]$ $\leftarrow$ $\\{vect}[i]$ $\land$ $%
\\{vect}[j]$\par
$\2{def}\\{vect\_prod}(\2{expr}k,i,j)=$\par
\quad\quad$\\{vect}[k]x:=\\{vect}[i]y\ast \\{vect}[j]z-\\{vect}[i]z\ast %
\\{vect}[j]y;$\par
\quad\quad$\\{vect}[k]y:=\\{vect}[i]z\ast \\{vect}[j]x-\\{vect}[i]x\ast %
\\{vect}[j]z;$\par
\quad\quad$\\{vect}[k]z:=\\{vect}[i]x\ast \\{vect}[j]y-\\{vect}[i]y\ast %
\\{vect}[j]x;$\par
$\!\3{enddef};$\par
\BL
$\9 scalar multiplication: $\\{vect}[j]$ $\leftarrow$ $\\{vect}[i]\ast v$\par
$\2{def}\\{vect\_mult}(\2{expr}j,i,v)=$\par
\quad\quad$\\{vect}[j]x:=v\ast \\{vect}[i]x;\ \\{vect}[j]y:=v\ast \\{vect}[i]y;%
\ \\{vect}[j]z:=v\ast \\{vect}[i]z;$\par
$\!\3{enddef};$\par
\BL
$\9 middle of two points\par
$\2{def}\\{mid\_point}(\2{expr}k,i,j)=\\{vect\_sum}(k,i,j);\ \\{vect%
\_mult}(k,k,.5);\3{enddef};$\par
\BL
\newpage
\title{Vector rotation}
$\9 Rotation of $\\{vect}[v]$ around $\\{vect}[\\{axis}]$ by an angle $%
\\{alpha}$\par
\BL
 The vector $\vec{v}$ is first projected on the axis
 giving vectors $\vec{a}$ and $\vec{h}$:
\figure{vect-fig.9}
 If we set
 $\vec{b}={\ora{axis}\over \left\Vert\vcenter{\ora{axis}}\right\Vert}$,
 the rotated vector $\vec{v'}$ is equal to $\vec{h}+\vec{f}$
 where $\vec{f}=\cos\alpha \cdot \vec{a} + \sin\alpha\cdot \vec{c}$.
 and $\vec{h}=(\vec{v}\cdot\vec{b})\vec{b}$
\figure{vect-fig.10}
\BL
$\9 The rotation is independent of $\\{vect}[\\{axis}]$'s module.\par
$\9 $v$ = old and new vector\par
$\9 $\\{axis}$ = rotation axis\par
$\9 $\\{alpha}$ = rotation angle\par
$\9\par
$\2{vardef}\\{vect\_rotate}(\2{expr}v,\\{axis},\\{alpha})=$\par
\quad\quad$\2{save}\\{v\_a},\\{v\_b},\\{v\_c},\\{v\_d},\\{v\_e},\\{v\_f};$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};\ \\{v\_b}:=\\{new\_vect};\ \\{v\_c}:=\\{new%
\_vect};$\par
\quad\quad$\\{v\_d}:=\\{new\_vect};\ \\{v\_e}:=\\{new\_vect};\ \\{v\_f}:=\\{new%
\_vect};$\par
\quad\quad$\\{v\_g}:=\\{new\_vect};\ \\{v\_h}:=\\{new\_vect};$\par
\quad\quad$\\{vect\_mult}(\\{v\_b},\\{axis},1/\\{vect\_mod}(\\{axis}));$\par
\quad\quad$\\{vect\_mult}(\\{v\_h},\\{v\_b},\\{vect\_dprod}(\\{v\_b},v));\ \9
projection of $v$ on $\\{axis}$\par
\quad\quad$\\{vect\_diff}(\\{v\_a},v,\\{v\_h});$\par
\quad\quad$\\{vect\_prod}(\\{v\_c},\\{v\_b},\\{v\_a});$\par
\quad\quad$\\{vect\_mult}(\\{v\_d},\\{v\_a},\1{cosd}(\\{alpha}));$\par
\quad\quad$\\{vect\_mult}(\\{v\_e},\\{v\_c},\1{sind}(\\{alpha}));$\par
\quad\quad$\\{vect\_sum}(\\{v\_f},\\{v\_d},\\{v\_e});$\par
\quad\quad$\\{vect\_sum}(v,\\{v\_f},\\{v\_h});$\par
\quad\quad$\\{free\_vect}(\\{v\_h});\ \\{free\_vect}(\\{v\_g});$\par
\quad\quad$\\{free\_vect}(\\{v\_f});\ \\{free\_vect}(\\{v\_e});\ \\{free%
\_vect}(\\{v\_d});$\par
\quad\quad$\\{free\_vect}(\\{v\_c});\ \\{free\_vect}(\\{v\_b});\ \\{free%
\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Operations on objects}
$\9 $\\{iname}$ is the handler for an instance of an object of class $\\{name}$%
\par
$\9 $\\{iname}$ must be a letter string\par
$\2{vardef}\\{assign\_obj}(\2{expr}\\{iname},\\{name})=$\par
\quad\quad$\2{save}\\{tmpdef};$\par
\quad\quad$\2{string}\\{tmpdef};\ \9 we need to add double quotes (char 34)\par
\quad\quad$\\{tmpdef}=\7"def "\AM \\{iname}\AM \7"_class="\AM \\{ditto}\AM %
\\{name}\AM \\{ditto}\AM \7" enddef";$\par
\quad\quad$\2{scantokens}\\{tmpdef};$\par
\quad\quad$\\{def\_obj}(\\{iname});$\par
$\!\3{enddef};$\par
\BL
$\9 $\\{name}$ is the the name of an object instance\par
$\9 It must be made only of letters (or underscores), but no digits.\par
$\2{def}\\{def\_obj}(\2{expr}\\{name})=$\par
\quad\quad$\2{scantokens}\2{begingroup}$\par
\quad\quad\quad\quad$\2{save}\\{tmpdef};\ \2{string}\\{tmpdef};$\par
\quad\quad\quad\quad$\\{tmpdef}=\7"def_"\AM \\{obj\_class\_}(\\{name})\AM \7"("%
\AM \\{ditto}\AM \\{name}\AM \\{ditto}\AM \7")";$\par
\quad\quad\quad\quad$\\{tmpdef}$\par
\quad\quad$\!\3{endgroup}$\par
$\!\3{enddef};$\par
\BL
$\9 This macro puts an object back where it was right at the beginning.\par
$\9 $\\{iname}$ is the name of an object instance.\par
$\2{vardef}\\{reset\_obj}(\2{expr}\\{iname})=$\par
\quad\quad$\2{save}\\{tmpdef};$\par
\quad\quad$\2{string}\\{tmpdef};$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{iname});$\par
\quad\quad$\\{tmpdef}=\7"set_"\AM \\{obj\_class\_}(\\{iname})\AM \7"_points";$%
\par
\quad\quad$\2{scantokens}\\{tmpdef}(\\{iname});$\par
$\!\3{enddef};$\par
\BL
$\9 Put an object at position given by $\\{pos}$ (a vector) and\par
$\9 with orientations given by angles $\\{psi}$, $\\{theta}$, $\\{phi}$.\par
$\9 The object is scaled by $\\{scale}$.\par
$\9 $\\{iname}$ is the name of an object instance.\par
$\2{vardef}\\{put\_obj}(\2{expr}\\{iname},\\{pos},\\{scale},\\{psi},\\{theta},%
\\{phi})=$\par
\quad\quad$\\{reset\_obj}(\\{iname});\ \\{scale\_obj}(\\{iname},\\{scale});$\par
\quad\quad$\\{v\_x}:=\\{new\_vect};\ \\{v\_y}:=\\{new\_vect};\ \\{v\_z}:=\\{new%
\_vect};$\par
\quad\quad$\\{vect\_def\_vect}(\\{v\_x},\\{vect\_I});$\par
\quad\quad$\\{vect\_def\_vect}(\\{v\_y},\\{vect\_J});$\par
\quad\quad$\\{vect\_def\_vect}(\\{v\_z},\\{vect\_K});$\par
\quad\quad$\\{rotate\_obj\_abs\_pv}(\\{iname},\\{point\_null},\\{v\_z},%
\\{psi});$\par
\quad\quad$\\{vect\_rotate}(\\{v\_x},\\{v\_z},\\{psi});\ \\{vect\_rotate}(\\{v%
\_y},\\{v\_z},\\{psi});$\par
\quad\quad$\\{rotate\_obj\_abs\_pv}(\\{iname},\\{point\_null},\\{v\_y},%
\\{theta});$\par
\quad\quad$\\{vect\_rotate}(\\{v\_x},\\{v\_y},\\{theta});\ \\{vect\_rotate}(%
\\{v\_z},\\{v\_y},\\{theta});$\par
\quad\quad$\\{rotate\_obj\_abs\_pv}(\\{iname},\\{point\_null},\\{v\_x},%
\\{phi});$\par
\quad\quad$\\{vect\_rotate}(\\{v\_y},\\{v\_x},\\{phi});\ \\{vect\_rotate}(\\{v%
\_z},\\{v\_x},\\{phi});$\par
\quad\quad$\\{free\_vect}(\\{v\_z});\ \\{free\_vect}(\\{v\_y});\ \\{free%
\_vect}(\\{v\_x});$\par
\quad\quad$\\{translate\_obj}(\\{iname},\\{pos});$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Rotation, translation and scaling of objects}
$\9 Rotation of an object instance $\\{name}$  around an axis\par
$\9 going through a point $p$ (local to the object)\par
$\9 and directed by vector $\\{vect}[v]$. The angle of rotation is $a$.\par
$\2{vardef}\\{rotate\_obj\_pv}(\2{expr}\\{name},p,v,a)=$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\\{rotate\_obj\_abs\_pv}(\\{name},\\{pnt}(p),v,a);$\par
$\!\3{enddef};$\par
\BL
$\2{vardef}\\{rotate\_obj\_abs\_pv}(\2{expr}\\{name},p,v,a)=$\par
\quad\quad$\2{save}\\{v\_a};$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};$\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_points\_}(\\{name})\?:$\par
\quad\quad\quad\quad$\\{vect\_diff}(\\{v\_a},\\{pnt}(i),p);$\par
\quad\quad\quad\quad$\\{vect\_rotate}(\\{v\_a},v,a);$\par
\quad\quad\quad\quad$\\{vect\_sum}(\\{pnt}(i),\\{v\_a},p);$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
$\9 Rotation of an object instance $\\{name}$ around an axis\par
$\9 going through a point $p$ (local to the object)\par
$\9 and directed by vector $\ora{pq}$. The angle of rotation is $a$.\par
$\2{vardef}\\{rotate\_obj\_pp}(\2{expr}\\{name},p,q,a)=$\par
\quad\quad$\2{save}\\{v\_a},\\{axis};$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};\ \\{axis}:=\\{new\_vect};$\par
\quad\quad$\\{vect\_diff}(\\{axis},\\{pnt}(q),\\{pnt}(p));$\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_points\_}(\\{name})\?:$\par
\quad\quad\quad\quad$\\{vect\_diff}(\\{v\_a},\\{pnt}(i),\\{pnt}(p));$\par
\quad\quad\quad\quad$\\{vect\_rotate}(\\{v\_a},\\{axis},a);$\par
\quad\quad\quad\quad$\\{vect\_sum}(\\{pnt}(i),\\{v\_a},\\{pnt}(p));$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\\{free\_vect}(\\{axis});\ \\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
$\9 Translation of an object instance $\\{name}$ by a vector $\\{vect}[v]$.\par
$\2{vardef}\\{translate\_obj}(\2{expr}\\{name},v)=$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_points\_}(\\{name})\?:$\par
\quad\quad\quad\quad$\\{vect\_sum}(\\{pnt}(i),\\{pnt}(i),v);$\par
\quad\quad$\!\3{endfor};$\par
$\!\3{enddef};$\par
\BL
$\9 Scalar multiplication of an object instance $\\{name}$ by a scalar $v$.\par
$\2{vardef}\\{scale\_obj}(\2{expr}\\{name},v)=$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_points\_}(\\{name})\?:$\par
\quad\quad\quad\quad$\\{vect\_mult}(\\{pnt}(i),\\{pnt}(i),v);$\par
\quad\quad$\!\3{endfor};$\par
$\!\3{enddef};$\par
\BL
\BL
\newpage
\title{Functions to build new points in space}
$\9 Rotation in a plane: this is useful to define a regular polygon.\par
$\9 $k$ is a new point obtained from point $j$ by rotation around $o$\par
$\9 by a angle $\alpha$ equal to the angle from $i$ to $j$.\par
\figure{vect-fig.11}
$\2{vardef}\\{rotate\_in\_plane\_}(\2{expr}k,o,i,j)=$\par
\quad\quad$\2{save}\\{cosalpha},\\{sinalpha},\\{alpha},\\{v\_a},\\{v\_b},\\{v%
\_c};$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};\ \\{v\_b}:=\\{new\_vect};\ \\{v\_c}:=\\{new%
\_vect};$\par
\quad\quad$\\{vect\_diff}(\\{v\_a},i,o);\ \\{vect\_diff}(\\{v\_b},j,o);\ %
\\{vect\_prod}(\\{v\_c},\\{v\_a},\\{v\_b});$\par
\quad\quad$\\{cosalpha}=\\{vect\_dprod}(\\{v\_a},\\{v\_b})/\\{vect\_mod}(\\{v%
\_a})/\\{vect\_mod}(\\{v\_b});$\par
\quad\quad$\\{sinalpha}=\1{sqrt}(1-\\{cosalpha}\6{\ast \ast }2);$\par
\quad\quad$\\{alpha}=\1{angle}((\\{cosalpha},\\{sinalpha}));$\par
\quad\quad$\\{vect\_rotate}(\\{v\_b},\\{v\_c},\\{alpha});$\par
\quad\quad$\\{vect\_sum}(k,o,\\{v\_b});$\par
\quad\quad$\\{free\_vect}(\\{v\_c});\ \\{free\_vect}(\\{v\_b});\ \\{free%
\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
$\2{vardef}\\{rotate\_in\_plane}(\2{expr}k,o,i,j)=$\par
\quad\quad$\\{rotate\_in\_plane\_}(\\{pnt}(k),o,\\{pnt}(i),\\{pnt}(j))$\par
$\!\3{enddef};$\par
\BL
$\9 Build a point on a adjacent face.\par
 The middle $m$ of points $i$ and $j$ is such that
 $\widehat{(\ora{om},\ora{mc})}=\alpha$
 This is useful to define regular polyhedra
\figure{vect-fig.7}
$\2{vardef}\\{new\_face\_point\_}(\2{expr}c,o,i,j,\\{alpha})=$\par
\quad\quad$\2{save}\\{v\_a},\\{v\_b},\\{v\_c},\\{v\_d},\\{v\_e};$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};\ \\{v\_b}:=\\{new\_vect};\ \\{v\_c}:=\\{new%
\_vect};\ \\{v\_d}:=\\{new\_vect};\ \\{v\_e}:=\\{new\_vect};$\par
\quad\quad$\\{vect\_diff}(\\{v\_a},i,o);\ \\{vect\_diff}(\\{v\_b},j,o);$\par
\quad\quad$\\{vect\_sum}(\\{v\_c},\\{v\_a},\\{v\_b});$\par
\quad\quad$\\{vect\_mult}(\\{v\_d},\\{v\_c},.5);$\par
\quad\quad$\\{vect\_diff}(\\{v\_e},i,j);$\par
\quad\quad$\\{vect\_sum}(c,\\{v\_d},o);$\par
\quad\quad$\\{vect\_rotate}(\\{v\_d},\\{v\_e},\\{alpha});$\par
\quad\quad$\\{vect\_sum}(c,\\{v\_d},c);$\par
\quad\quad$\\{free\_vect}(\\{v\_e});\ \\{free\_vect}(\\{v\_d});\ \\{free%
\_vect}(\\{v\_c});\ \\{free\_vect}(\\{v\_b});\ \\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
\BL
$\2{vardef}\\{new\_face\_point}(\2{expr}c,o,i,j,\\{alpha})=$\par
\quad\quad$\\{new\_face\_point\_}(\\{pnt}(c),\\{pnt}(o),\\{pnt}(i),\\{pnt}(j),%
\\{alpha})$\par
$\!\3{enddef};$\par
\BL
$\2{vardef}\\{new\_abs\_face\_point}(\2{expr}c,o,i,j,\\{alpha})=$\par
\quad\quad$\\{new\_face\_point\_}(c,o,\\{pnt}(i),\\{pnt}(j),\\{alpha})$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Computation of the projection of a point on the ``screen''}
$\9 $p$ is the projection of $m$\par
$\9 $m$ = point in space (3 coordinates)\par
$\9 $p$ = point of the intersection plane\par
\figure{vect-fig.8}
$\2{vardef}\\{project\_point}(\2{expr}p,m)=$\par
\quad\quad$\2{save}\\{tmpalpha},\\{v\_a},\\{v\_b};$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};\ \\{v\_b}:=\\{new\_vect};$\par
\quad\quad$\\{vect\_diff}(\\{v\_b},m,\\{Obs});\ \9 vector $\\{Obs}$-$m$\par
\quad\quad\quad\quad$\9 $\\{vect}[\\{v\_a}]$ is $\\{vect}[\\{v\_b}]$ expressed
in ($\\{ObsI}$,$\\{ObsJ}$,$\\{ObsK}$)\par
\quad\quad\quad\quad$\9 coordinates.\par
\quad\quad$\\{vect}[\\{v\_a}]x:=\\{vect}[\\{IObsI\_}]x\ast \\{vect}[\\{v\_b}]x$%
\par
\quad\quad${}+\\{vect}[\\{IObsJ\_}]x\ast \\{vect}[\\{v\_b}]y+\\{vect}[\\{IObsK%
\_}]x\ast \\{vect}[\\{v\_b}]z;$\par
\quad\quad$\\{vect}[\\{v\_a}]y:=\\{vect}[\\{IObsI\_}]y\ast \\{vect}[\\{v\_b}]x$%
\par
\quad\quad${}+\\{vect}[\\{IObsJ\_}]y\ast \\{vect}[\\{v\_b}]y+\\{vect}[\\{IObsK%
\_}]y\ast \\{vect}[\\{v\_b}]z;$\par
\quad\quad$\\{vect}[\\{v\_a}]z:=\\{vect}[\\{IObsI\_}]z\ast \\{vect}[\\{v\_b}]x$%
\par
\quad\quad${}+\\{vect}[\\{IObsJ\_}]z\ast \\{vect}[\\{v\_b}]y+\\{vect}[\\{IObsK%
\_}]z\ast \\{vect}[\\{v\_b}]z;$\par
\quad\quad$\2{if}\\{vect}[\\{v\_a}]x<\\{Obs\_dist}\?:\9 then, point $m$ is too
close\par
\quad\quad\quad\quad$\2{message}\7"Point "\AM \1{decimal}m\AM \7" too close ->
not drawn";$\par
\quad\quad\quad\quad$x[p]:=\\{too\_big\_};\ y[p]=\\{too\_big\_};$\par
\quad\quad$\2{else}\?:$\par
\quad\quad\quad\quad$\2{if}(\1{angle}(\\{vect}[\\{v\_a}]x,\\{vect}[\\{v\_a}]z)>%
\\{h\_field}/2)$\par
\quad\quad\quad\quad\quad\quad${}\6{or}(\1{angle}(\\{vect}[\\{v\_a}]x,\\{vect}[%
\\{v\_a}]y)>\\{v\_field}/2)\?:$\par
\quad\quad\quad\quad\quad\quad$\2{message}\7"Point "\AM \1{decimal}m\AM \7" out
of screen -> not drawn";$\par
\quad\quad\quad\quad\quad\quad$x[p]:=\\{too\_big\_};\ y[p]=\\{too\_big\_};$\par
\quad\quad\quad\quad$\2{else}\?:$\par
\quad\quad\quad\quad\quad\quad$\\{tmpalpha}:=\\{Obs\_dist}/\\{vect}[\\{v%
\_a}]x;$\par
\quad\quad\quad\quad\quad\quad$y[p]:=\\{drawing\_scale}\ast \\{tmpalpha}\ast %
\\{vect}[\\{v\_a}]y;$\par
\quad\quad\quad\quad\quad\quad$x[p]:=\\{drawing\_scale}\ast \\{tmpalpha}\ast %
\\{vect}[\\{v\_a}]z;$\par
\quad\quad\quad\quad$\2{fi};$\par
\quad\quad$\2{fi};$\par
\quad\quad$\\{free\_vect}(\\{v\_b});\ \\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
$\9 Object projection\par
$\9 This is a mere iteration on $\\{project\_point}$\par
$\2{def}\\{project\_obj}(\2{expr}\\{name})=$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_points\_}(\\{name})\?:\\{project\_point}(%
\\{ipnt\_}(i),\\{pnt}(i));\3{endfor};$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Draw one face, hiding it if it is hidden}
$\9 The order of the vertices determines what is the visible side\par
$\9 of the face. The order must be clockwise when the face is seen.\par
$\9 $\\{drawhidden}$ is a boolean; if $\5{true}$ only hidden faces are drawn;
if $\5{false}$,\par
$\9 only visible faces are drawn. Therefore, $\\{draw\_face}$ is called twice%
\par
$\9 by $\\{draw\_faces}$.\par
$\2{vardef}\\{draw\_face}(\2{text}\\{vertices})(\2{expr}\\{col},%
\\{drawhidden})=$\par
\quad\quad$\2{save}p,\\{num},\\{normal\_vect},\\{v\_a},\\{v\_b},\\{v\_c},%
\\{overflow};$\par
\quad\quad$\2{path}p;\ \2{boolean}\\{overflow};$\par
\quad\quad$\\{overflow}=\5{false};$\par
\quad\quad$\2{forsuffixes}\$=\\{vertices}\?:$\par
\quad\quad\quad\quad$\2{if}z[\\{ipnt\_}(\$)]=(\\{too\_big\_},\\{too\_big\_})\?:%
\\{overflow}:=\5{true};\ \2{fi};$\par
\quad\quad\quad\quad$\2{exitif}\\{overflow};$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\2{if}\\{overflow}\?:\2{message}\7"Face can not be drawn, due to
overflow";$\par
\quad\quad$\2{else}\?:$\par
\quad\quad\quad\quad$p=\2{forsuffixes}\$=\\{vertices}\?:z[\\{ipnt\_}(\$)]\8{--}%
\3{endfor}\1{cycle};$\par
\quad\quad\quad\quad$\\{num}_{0}:=0;\ \\{num}_{1}:=0;\ \\{num}_{2}:=0;$\par
\quad\quad\quad\quad\quad\quad$\9 get the three last suffixes:\par
\quad\quad\quad\quad$\2{forsuffixes}\$=\\{vertices}\?:\\{num}_{0}:=\\{num}_{1};%
\ \\{num}_{1}:=\\{num}_{2};\ \\{num}_{2}:=\$;\3{endfor};$\par
\quad\quad\quad\quad$\\{normal\_vect}:=\\{new\_vect};\ \\{v\_a}:=\\{new\_vect};%
\ \\{v\_b}:=\\{new\_vect};$\par
\quad\quad\quad\quad$\\{v\_c}:=\\{new\_vect};$\par
\quad\quad\quad\quad$\\{vect\_diff}(\\{v\_a},\\{pnt}(\\{num}_{1}),\\{pnt}(%
\\{num}_{0}));$\par
\quad\quad\quad\quad$\\{vect\_diff}(\\{v\_b},\\{pnt}(\\{num}_{2}),\\{pnt}(%
\\{num}_{1}));$\par
\quad\quad\quad\quad$\\{vect\_prod}(\\{normal\_vect},\\{v\_a},\\{v\_b});$\par
\quad\quad\quad\quad$\\{vect\_diff}(\\{v\_c},\\{pnt}(\\{num}_{1}),\\{Obs});$\par
\quad\quad\quad\quad$\2{if}\\{filled\_faces}\?:$\par
\quad\quad\quad\quad\quad\quad$\2{if}\\{vect\_dprod}(\\{normal\_vect},\\{v%
\_c})<0\?:$\par
\quad\quad\quad\quad\quad\quad\quad\quad$\2{fill}p\2{withcolor}\\{col};\ \2{if}%
\\{draw\_contours}\?:\\{drawcontour}(p,\\{contourwidth});\ \2{fi};$\par
\quad\quad\quad\quad\quad\quad$\2{else}\?:\9 $\2{draw}p\2{dashed}\\{evenly};$
if this is done, you must ensure\par
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$\9 that
hidden faces are (re)drawn at the end\par
\quad\quad\quad\quad\quad\quad$\2{fi};$\par
\quad\quad\quad\quad$\2{else}\?:$\par
\quad\quad\quad\quad\quad\quad$\2{if}\\{vect\_dprod}(\\{normal\_vect},\\{v%
\_c})<0\?:\9visible\par
\quad\quad\quad\quad\quad\quad\quad\quad$\2{if}\1{not}\\{drawhidden}\?:%
\2{draw}p;\ \2{fi};$\par
\quad\quad\quad\quad\quad\quad$\2{else}\?:\9 hidden\par
\quad\quad\quad\quad\quad\quad\quad\quad$\2{if}\\{drawhidden}\?:$\par
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$\2{draw}p\2{withcolor}%
\\{background};\ \9 avoid strange overlapping dashes\par
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$\2{draw}p\2{dashed}%
\\{evenly};$\par
\quad\quad\quad\quad\quad\quad\quad\quad$\2{fi};$\par
\quad\quad\quad\quad\quad\quad$\2{fi};$\par
\quad\quad\quad\quad$\2{fi};$\par
\quad\quad\quad\quad$\\{free\_vect}(\\{v\_c});\ \\{free\_vect}(\\{v\_b});\ %
\\{free\_vect}(\\{v\_a});\ \\{free\_vect}(\\{normal\_vect});$\par
\quad\quad$\2{fi};$\par
$\!\3{enddef};$\par
\BL
$\9 $p$ is the path to draw (a face contour) and $\\{thickness}$ is the pen
width\par
$\2{def}\\{drawcontour}(\2{expr}p,\\{thickness})=$\par
\quad\quad$\2{pickup}\5{pencircle}\6{scaled}\\{thickness};$\par
\quad\quad$\2{draw}p;$\par
\quad\quad$\2{pickup}\5{pencircle}\6{scaled}.4\\{pt};$\par
$\!\3{enddef};$\par
\BL
\newpage
$\9 Variables for face handling. First, we have an array for lists of vertices%
\par
$\9 corresponding to faces.\par
$\2{string}\\{face\_points\_}[];\ \9 analogous to $\\{vect}$ arrays\par
\BL
$\9 Then, we have an array of colors. A color needs to be a string\par
$\9 representing an hexadecimal RGB coding of a color.\par
$\2{string}\\{face\_color\_}[];$\par
\BL
$\9 $\\{name}$ is the name of an object instance\par
$\2{vardef}\\{draw\_faces}(\2{expr}\\{name})=$\par
\quad\quad$\2{save}\\{tmpdef};\ \2{string}\\{tmpdef};$\par
\quad\quad$\\{define\_current\_face\_offset\_}(\\{name});$\par
\quad\quad\quad\quad$\9 first the hidden faces (dashes must be drawn first):\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_faces\_}(\\{name})\?:$\par
\quad\quad\quad\quad$\\{tmpdef}:=\7"draw_face("\AM \\{face\_points\_}[%
\\{face}(i)]$\par
\quad\quad\quad\quad\quad\quad${}\AM \7")(hexcolor("\AM \\{ditto}\AM \\{face%
\_color\_}[\\{face}(i)]\AM \\{ditto}$\par
\quad\quad\quad\quad\quad\quad${}\AM \7"),true)";\ \2{scantokens}\\{tmpdef};$%
\par
\quad\quad$\!\3{endfor};$\par
\quad\quad\quad\quad$\9 then, the visible faces:\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_faces\_}(\\{name})\?:$\par
\quad\quad\quad\quad$\\{tmpdef}:=\7"draw_face("\AM \\{face\_points\_}[%
\\{face}(i)]$\par
\quad\quad\quad\quad\quad\quad${}\AM \7")(hexcolor("\AM \\{ditto}\AM \\{face%
\_color\_}[\\{face}(i)]\AM \\{ditto}$\par
\quad\quad\quad\quad\quad\quad${}\AM \7"),false)";\ \2{scantokens}\\{tmpdef};$%
\par
\quad\quad$\!\3{endfor};$\par
$\!\3{enddef};$\par
\BL
$\9 Draw point $n$ of object instance $\\{name}$\par
$\2{vardef}\\{draw\_point}(\2{expr}\\{name},n)=$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\\{project\_point}(\\{ipnt\_}(n),\\{pnt}(n));$\par
\quad\quad$\2{if}z[\\{ipnt\_}(n)]\ne (\\{too\_big\_},\\{too\_big\_})\?:$\par
\quad\quad\quad\quad$\2{pickup}\5{pencircle}\6{scaled}5\\{pt};$\par
\quad\quad\quad\quad$\2{drawdot}(z[\\{ipnt\_}(n)]);$\par
\quad\quad\quad\quad$\2{pickup}\5{pencircle}\6{scaled}.4\\{pt};$\par
\quad\quad$\2{fi};$\par
$\!\3{enddef};$\par
\BL
$\2{vardef}\\{draw\_axes}(\2{expr}r,g,b)=$\par
\quad\quad$\\{project\_point}(1,\\{vect\_null});$\par
\quad\quad$\\{project\_point}(2,\\{vect\_I});$\par
\quad\quad$\\{project\_point}(3,\\{vect\_J});$\par
\quad\quad$\\{project\_point}(4,\\{vect\_K});$\par
\quad\quad$\2{if}(z_{1}\ne (\\{too\_big\_},\\{too\_big\_}))\?:$\par
\quad\quad\quad\quad$\2{if}(z_{2}\ne (\\{too\_big\_},\\{too\_big\_}))\?:$\par
\quad\quad\quad\quad\quad\quad$\2{drawarrow}z_{1}\8{--}z_{2}\2{dashed}%
\\{evenly}\2{withcolor}r;$\par
\quad\quad\quad\quad$\2{fi};$\par
\quad\quad\quad\quad$\2{if}(z_{3}\ne (\\{too\_big\_},\\{too\_big\_}))\?:$\par
\quad\quad\quad\quad\quad\quad$\2{drawarrow}z_{1}\8{--}z_{3}\2{dashed}%
\\{evenly}\2{withcolor}g;$\par
\quad\quad\quad\quad$\2{fi};$\par
\quad\quad\quad\quad$\2{if}(z_{4}\ne (\\{too\_big\_},\\{too\_big\_}))\?:$\par
\quad\quad\quad\quad\quad\quad$\2{drawarrow}z_{1}\8{--}z_{4}\2{dashed}%
\\{evenly}\2{withcolor}b;$\par
\quad\quad\quad\quad$\2{fi};$\par
\quad\quad$\2{fi};$\par
$\!\3{enddef};$\par
\BL
\BL
\newpage
$\9 Definition of a macro $\\{obj\_name}$ returning an object name\par
$\9 when given an absolute\par
$\9 face number. This definition is built incrementally through a string,\par
$\9 everytime a new object is defined.\par
$\9 $\\{obj\_name}$ is defined by $\\{redefine\_obj\_name\_}$.\par
\BL
$\9 Initial definition\par
$\2{string}\\{index\_to\_name\_};$\par
$\\{index\_to\_name\_}=\7"def obj_name(expr i)=if i<1:";$\par
\BL
$\9 $\\{name}$ is the name of an object instance\par
$\9 $n$ is the absolute index of its last face\par
$\2{def}\\{redefine\_obj\_name\_}(\2{expr}\\{name},n)=$\par
\quad\quad$\\{index\_to\_name\_}:=\\{index\_to\_name\_}\AM \7"elseif i<="\AM %
\1{decimal}n\AM \7":"\AM \\{ditto}$\par
\quad\quad\quad\quad\quad\quad${}\AM \\{name}\AM \\{ditto};$\par
\quad\quad$\2{scantokens}\2{begingroup}\\{index\_to\_name\_}\AM \7"fi;enddef;"%
\3{endgroup};$\par
$\!\3{enddef};$\par
\BL
$\9 $i$ is an absolute face number\par
$\9 $\\{vertices}$ is a string representing a list of vertices\par
$\9 $\\{rgbcolor}$ is a string representing a color in rgb hexadecimal\par
$\2{def}\\{set\_face}(\2{expr}i,\\{vertices},\\{rgbcolor})=$\par
\quad\quad$\\{face\_points\_}[i]:=\\{vertices};\ \\{face\_color\_}[i]:=%
\\{rgbcolor};$\par
$\!\3{enddef};$\par
\BL
$\9 $i$ is a local face number\par
$\9$\\{vertices}$ is a string representing a list of vertices\par
$\9$\\{rgbcolor}$ is a string representing a color in rgb hexadecimal\par
$\2{def}\\{set\_obj\_face}(\2{expr}i,\\{vertices},\\{rgbcolor})=\\{set\_face}(%
\\{face}(i),\\{vertices},\\{rgbcolor})$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Compute the vectors corresponding to the observer's viewpoint}
$\9 (vectors $\\{ObsI\_}$,$\\{ObsJ\_}$ and $\\{ObsK\_}$ in the $\\{vect\_I}$,$%
\\{vect\_J}$,\par
$\9 $\\{vect\_K}$ reference; and vectors $\\{IObsI\_}$,$\\{IObsJ\_}$ and $%
\\{IObsK\_}$\par
$\9 which are $\\{vect\_I}$,$\\{vect\_J}$,$\\{vect\_K}$\par
$\9 in the $\\{ObsI\_}$,$\\{ObsJ\_}$,$\\{ObsK\_}$ reference)\par
\figure{vect-fig.16}
 (here, $\psi>0$, $\theta<0$ and $\phi>0$; moreover,
 $\vert\theta\vert \leq 90^\circ$)
\BL
$\2{def}\\{compute\_reference}(\2{expr}\\{psi},\\{theta},\\{phi})=$\par
\quad\quad\quad$\9 $\\{ObsI\_}$ defines the direction of observation;\par
\quad\quad\quad$\9 $\\{ObsJ\_}$ and $\\{ObsK\_}$ the orientation\par
\quad\quad\quad$\9 (but one of these two vectors is enough,\par
\quad\quad\quad$\9 since $\\{ObsK\_}$ = $\\{ObsI\_}$ $\land$ $\\{ObsJ\_}$)\par
\quad\quad\quad$\9 The vectors are found by rotations of $\\{vect\_I}$,$\\{vect%
\_J}$,$\\{vect\_K}$.\par
\quad\quad$\\{vect\_def\_vect}(\\{ObsI\_},\\{vect\_I});\ \\{vect\_def\_vect}(%
\\{ObsJ\_},\\{vect\_J});$\par
\quad\quad$\\{vect\_def\_vect}(\\{ObsK\_},\\{vect\_K});$\par
\quad\quad$\\{vect\_rotate}(\\{ObsI\_},\\{ObsK\_},\\{psi});$\par
\quad\quad$\\{vect\_rotate}(\\{ObsJ\_},\\{ObsK\_},\\{psi});\ \9 gives
($u$,$v$,$z$)\par
\quad\quad$\\{vect\_rotate}(\\{ObsI\_},\\{ObsJ\_},\\{theta});$\par
\quad\quad$\\{vect\_rotate}(\\{ObsK\_},\\{ObsJ\_},\\{theta});\ \9 gives
($Obs_x$,$v$,$w$)\par
\quad\quad$\\{vect\_rotate}(\\{ObsJ\_},\\{ObsI\_},\\{phi});$\par
\quad\quad$\\{vect\_rotate}(\\{ObsK\_},\\{ObsI\_},\\{phi});\ \9 gives
($Obs_x$,$Obs_y$,$Obs_z$)\par
\quad\quad\quad$\9 The passage matrix $P$ from $\\{vect\_I}$,$\\{vect\_J}$,$%
\\{vect\_K}$\par
\quad\quad\quad$\9 to $\\{ObsI\_}$,$\\{ObsJ\_}$,$\\{ObsK\_}$ is the matrix\par
\quad\quad\quad$\9 composed of the vectors $\\{ObsI\_}$,$\\{ObsJ\_}$ and $%
\\{ObsK\_}$ expressed\par
\quad\quad\quad$\9 in the base $\\{vect\_I}$,$\\{vect\_J}$,$\\{vect\_K}$.\par
\quad\quad\quad$\9 We have $X=P X'$ where $X$ are the coordinates of a point\par
\quad\quad\quad$\9 in $\\{vect\_I}$,$\\{vect\_J}$,$\\{vect\_K}$\par
\quad\quad\quad$\9 and $X'$ the coordinates of the same point in $\\{ObsI\_}$,$%
\\{ObsJ\_}$,$\\{ObsK\_}$.\par
\quad\quad\quad$\9 In order to get $P^{-1}$, it suffices to build vectors using%
\par
\quad\quad\quad$\9 the previous rotations in the inverse order.\par
\quad\quad$\\{vect\_def\_vect}(\\{IObsI\_},\\{vect\_I});\ \\{vect\_def\_vect}(%
\\{IObsJ\_},\\{vect\_J});$\par
\quad\quad$\\{vect\_def\_vect}(\\{IObsK\_},\\{vect\_K});$\par
\quad\quad$\\{vect\_rotate}(\\{IObsK\_},\\{IObsI\_},-\\{phi});\ \\{vect%
\_rotate}(\\{IObsJ\_},\\{IObsI\_},-\\{phi});$\par
\quad\quad$\\{vect\_rotate}(\\{IObsK\_},\\{IObsJ\_},-\\{theta});\ \\{vect%
\_rotate}(\\{IObsI\_},\\{IObsJ\_},-\\{theta});$\par
\quad\quad$\\{vect\_rotate}(\\{IObsJ\_},\\{IObsK\_},-\\{psi});\ \\{vect%
\_rotate}(\\{IObsI\_},\\{IObsK\_},-\\{psi});$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Point of view}
$\9 This macro computes the three angles necessary for $\\{compute\_reference}$%
\par
$\9 $\\{name}$ =  name of an instance of an object\par
$\9 $\\{target}$ = target point (local to object $\\{name}$)\par
$\9 $\\{phi}$ = angle\par
$\2{vardef}\\{point\_of\_view\_obj}(\2{expr}\\{name},\\{target},\\{phi})=$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});\ \9 enables $%
\\{pnt}$\par
\quad\quad$\\{point\_of\_view\_abs}(\\{pnt}(\\{target}),\\{phi});$\par
$\!\3{enddef};$\par
\BL
$\9 Compute absolute perspective. $\\{target}$ is an absolute point number\par
$\2{vardef}\\{point\_of\_view\_abs}(\2{expr}\\{target},\\{phi})=$\par
\quad\quad$\2{save}\\{v\_a},\\{psi},\\{theta};$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};$\par
\quad\quad$\\{vect\_diff}(\\{v\_a},\\{target},\\{Obs});$\par
\quad\quad$\\{vect\_mult}(\\{v\_a},\\{v\_a},1/\\{vect\_mod}(\\{v\_a}));$\par
\quad\quad$\\{psi}=\1{angle}((\\{vect}[\\{v\_a}]x,\\{vect}[\\{v\_a}]y));$\par
\quad\quad$\\{theta}=-\1{angle}((\\{vect}[\\{v\_a}]x\6{++}\\{vect}[\\{v\_a}]y,%
\\{vect}[\\{v\_a}]z));$\par
\quad\quad$\\{compute\_reference}(\\{psi},\\{theta},\\{phi});$\par
\quad\quad$\\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
\BL
\BL
$\9 Distance between the observer and point $n$ of object $\\{name}$\par
$\9 Result is put in $\\{dist}$\par
$\2{vardef}\\{obs\_distance}(\2{text}\\{dist})(\2{expr}\\{name},n)=$\par
\quad\quad$\2{save}\\{v\_a};$\par
\quad\quad$\\{v\_a}:=\\{new\_vect};$\par
\quad\quad$\\{define\_current\_point\_offset\_}(\\{name});\ \9 enables $%
\\{pnt}$\par
\quad\quad$\\{dist}:=\\{vect\_mod}(\\{v\_a},\\{pnt}(n),\\{Obs});$\par
\quad\quad$\\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Vector and point allocation}
$\9 Allocation is done through a stack of vectors\par
$\2{numeric}\\{last\_vect\_};$\par
$\\{last\_vect\_}=0;$\par
\BL
$\9 vector allocation\par
$\2{def}\\{new\_vect}=\2{incr}(\\{last\_vect\_})$\par
\quad\quad\quad\quad\quad\quad\quad$\9 $\2{message}\7"Vector "\AM \1{decimal}%
\\{last\_vect\_}\AM \7" allocated";$\par
$\!\3{enddef};$\par
\BL
$\2{let}\\{new\_point}=\\{new\_vect};$\par
\BL
$\2{def}\\{new\_points}(\2{text}p)(\2{expr}n)=$\par
\quad\quad$\2{save}p;$\par
\quad\quad$\2{numeric}p[];$\par
\quad\quad$\2{for}i:=1\4{upto}n\?:p[i]:=\\{new\_point};\3{endfor};$\par
$\!\3{enddef};$\par
\BL
$\9 Free a vector\par
$\9 A vector can only be freed safely when it was the last vector created.\par
$\2{def}\\{free\_vect}(\2{expr}i)=$\par
\quad\quad$\2{if}i=\\{last\_vect\_}\?:\\{last\_vect\_}:=\\{last\_vect\_}-1;$\par
\quad\quad$\2{else}\?:\2{errmessage}(\7"Vector "\AM \1{decimal}i\AM \7" can't
be freed!");$\par
\quad\quad$\2{fi};$\par
$\!\3{enddef};$\par
\BL
$\2{let}\\{free\_point}=\\{free\_vect};$\par
\BL
$\2{def}\\{free\_points}(\2{text}p)(\2{expr}n)=$\par
\quad\quad$\2{for}i:=10\2{step}-1\4{until}1\?:\\{free\_point}(p[i]);%
\3{endfor};$\par
$\!\3{enddef};$\par
\BL
\title{Debugging}
\BL
$\2{def}\\{show\_vect}(\2{expr}t,i)=$\par
\quad\quad$\2{message}\7"Vector "\AM t\AM \7"="$\par
\quad\quad${}\AM \7"("\AM \1{decimal}\\{vect}[i]x\AM \7","\AM \1{decimal}%
\\{vect}[i]y\AM \7","$\par
\quad\quad\quad\quad${}\AM \1{decimal}\\{vect}[i]z\AM \7")";$\par
$\!\3{enddef};$\par
\BL
$\2{let}\\{show\_point}=\\{show\_vect};$\par
\BL
$\2{def}\\{show\_pair}(\2{expr}t,\\{zz})=$\par
\quad\quad$\2{message}t\AM \7"=("\AM \1{decimal}\1{xpart}(\\{zz})\AM \7","\AM %
\1{decimal}\1{ypart}(\\{zz})\AM \7")";$\par
$\!\3{enddef};$\par
\BL
\BL
\newpage
\title{Access to object features}
$\9 $a$ must be a string representing an class name, such as $%
\7"dodecahedron"$.\par
$\9 $b$ is the tail of a macro name.\par
\BL
$\2{def}\\{obj\_}(\2{expr}a,b,i)=$\par
\quad\quad$\2{scantokens}$\par
\quad\quad$\2{begingroup}\2{save}n;\ \2{string}n;\ n=a\AM b\AM i;\ n$\par
\quad\quad$\!\3{endgroup}$\par
$\!\3{enddef};$\par
\BL
$\2{def}\\{obj\_points\_}(\2{expr}\\{name})=$\par
\quad\quad$\\{obj\_}(\\{obj\_class\_}(\\{name}),\7"_points",\\{name})$\par
$\!\3{enddef};$\par
\BL
$\2{def}\\{obj\_faces\_}(\2{expr}\\{name})=$\par
\quad\quad$\\{obj\_}(\\{obj\_class\_}(\\{name}),\7"_faces",\\{name})$\par
$\!\3{enddef};$\par
\BL
$\2{vardef}\\{obj\_point\_offset\_}(\2{expr}\\{name})=$\par
\quad\quad$\\{obj\_}(\\{obj\_class\_}(\\{name}),\7"_point_offset",\\{name})$\par
$\!\3{enddef};$\par
\BL
$\2{vardef}\\{obj\_face\_offset\_}(\2{expr}\\{name})=$\par
\quad\quad$\\{obj\_}(\\{obj\_class\_}(\\{name}),\7"_face_offset",\\{name})$\par
$\!\3{enddef};$\par
\BL
$\2{def}\\{obj\_class\_}(\2{expr}\\{name})=\\{obj\_}(\\{name},\7"_class",\7"")%
\3{enddef};$\par
\BL
\newpage
$\2{def}\\{define\_point\_offset\_}(\2{expr}\\{name},o)=$\par
\quad\quad$\2{begingroup}\2{save}n,\\{tmpdef};$\par
\quad\quad\quad\quad$\2{string}n,\\{tmpdef};$\par
\quad\quad\quad\quad$n=\\{obj\_class\_}(\\{name})\AM \7"_point_offset"\AM %
\\{name};$\par
\quad\quad\quad\quad$\2{expandafter}\1{numeric}\2{scantokens}n;$\par
\quad\quad\quad\quad$\2{scantokens}n:=\\{last\_point\_offset\_};$\par
\quad\quad\quad\quad$\\{last\_point\_offset\_}:=\\{last\_point\_offset\_}+o;$%
\par
\quad\quad\quad\quad$\\{tmpdef}=\7"def "\AM \\{obj\_class\_}(\\{name})\AM %
\7"_points"\AM \\{name}\AM $\par
\quad\quad\quad\quad\quad\quad$\7"="\AM \1{decimal}o\AM \7" enddef";$\par
\quad\quad\quad\quad$\2{scantokens}\\{tmpdef};$\par
\quad\quad$\!\3{endgroup}$\par
$\!\3{enddef};$\par
\BL
$\2{def}\\{define\_face\_offset\_}(\2{expr}\\{name},o)=$\par
\quad\quad$\2{begingroup}\2{save}n,\\{tmpdef};$\par
\quad\quad\quad\quad$\2{string}n,\\{tmpdef};$\par
\quad\quad\quad\quad$n=\\{obj\_class\_}(\\{name})\AM \7"_face_offset"\AM %
\\{name};$\par
\quad\quad\quad\quad$\2{expandafter}\1{numeric}\2{scantokens}n;$\par
\quad\quad\quad\quad$\2{scantokens}n:=\\{last\_face\_offset\_};$\par
\quad\quad\quad\quad$\\{last\_face\_offset\_}:=\\{last\_face\_offset\_}+o;$\par
\quad\quad\quad\quad$\\{tmpdef}=\7"def "\AM \\{obj\_class\_}(\\{name})\AM %
\7"_faces"\AM \\{name}\AM $\par
\quad\quad\quad\quad\quad\quad$\7"="\AM \1{decimal}o\AM \7" enddef";$\par
\quad\quad\quad\quad$\2{scantokens}\\{tmpdef};$\par
\quad\quad$\!\3{endgroup}$\par
$\!\3{enddef};$\par
\BL
$\2{def}\\{define\_current\_point\_offset\_}(\2{expr}\\{name})=$\par
\quad\quad$\2{save}\\{current\_point\_offset\_};$\par
\quad\quad$\2{numeric}\\{current\_point\_offset\_};$\par
\quad\quad$\\{current\_point\_offset\_}:=\\{obj\_point\_offset\_}(\\{name});$%
\par
$\!\3{enddef};$\par
\BL
$\2{def}\\{define\_current\_face\_offset\_}(\2{expr}\\{name})=$\par
\quad\quad$\2{save}\\{current\_face\_offset\_};$\par
\quad\quad$\2{numeric}\\{current\_face\_offset\_};$\par
\quad\quad$\\{current\_face\_offset\_}:=\\{obj\_face\_offset\_}(\\{name});$\par
$\!\3{enddef};$\par
\BL
\BL
\newpage
\title{Drawing an object}
$\9 $\\{name}$ is an object instance\par
$\2{def}\\{draw\_obj}(\2{expr}\\{name})=\\{project\_obj}(\\{name});\ \\{draw%
\_faces}(\\{name});\3{enddef};$\par
\BL
\title{Normalization of an object}
$\9 This macro translates an object so that a list of vertices is centered\par
$\9 on the origin, and the last vertex is put on a sphere whose radius is 1.\par
$\9 $\\{name}$ is the name of the object and $\\{vertices}$ is a list\par
$\9 of points whose barycenter will define the center of the object.\par
$\9 ($\\{vertices}$ need not be the list of all vertices)\par
$\2{vardef}\\{normalize\_obj}(\2{expr}\\{name})(\2{text}\\{vertices})=$\par
\quad\quad$\2{save}\\{v\_a},\\{nvertices},\\{last};$\par
\quad\quad$\2{numeric}\\{v\_a},\\{last};$\par
\quad\quad$\\{nvertices}=0;$\par
\quad\quad$\\{v\_a}=\\{new\_vect};\ \\{vect\_def}(\\{v\_a},0,0,0)$\par
\quad\quad$\2{forsuffixes}\$=\\{vertices}\?:$\par
\quad\quad\quad\quad$\\{vect\_sum}(\\{v\_a},\\{v\_a},\\{pnt}(\$));$\par
\quad\quad\quad\quad$\\{nvertices}:=\\{nvertices}+1;$\par
\quad\quad\quad\quad$\\{last}:=\$;$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\\{vect\_mult}(\\{v\_a},\\{v\_a},-1/\\{nvertices});$\par
\quad\quad$\\{translate\_obj}(\\{name},\\{v\_a});\ \9 object centered on the
origin\par
\quad\quad$\\{scale\_obj}(\\{name},1/\\{vect\_mod}(\\{pnt}(\\{last})));$\par
\quad\quad$\\{free\_vect}(\\{v\_a});$\par
$\!\3{enddef};$\par
\BL
\BL
\newpage
\title{General definitions}
$\9 Vector arrays\par
$\2{numeric}\\{vect}[]x,\\{vect}[]y,\\{vect}[]z;$\par
\BL
$\9 Observer\par
$\2{numeric}\\{Obs};$\par
$\\{Obs}=\\{new\_point};$\par
$\9 default value:\par
$\\{set\_point}(\\{Obs},0,0,20);$\par
\BL
$\9 Observer's vectors\par
$\\{ObsI\_}=\\{new\_vect};\ \\{ObsJ\_}=\\{new\_vect};\ \\{ObsK\_}=\\{new%
\_vect};$\par
$\\{IObsI\_}=\\{new\_vect};\ \\{IObsJ\_}=\\{new\_vect};\ \\{IObsK\_}=\\{new%
\_vect};$\par
\BL
$\9 distance observer/plane (must be $>0$)\par
$\2{numeric}\\{Obs\_dist};\ \9 represents $\\{Obs\_dist}$ $\times$ $\\{drawing%
\_scale}$\par
$\9 default value:\par
$\\{Obs\_dist}=2;\ \9 means $\\{Obs\_dist}$ $\times$ $\\{drawing\_scale}$\par
\BL
$\9 Screen Size\par
$\9 The screen size is defined through two angles: the horizontal field\par
$\9 and the vertical field\par
$\2{numeric}\\{h\_field},\\{v\_field};$\par
$\\{h\_field}=100;\ \9 degrees\par
$\\{v\_field}=70;\ \9 degrees\par
\BL
$\9 Reference vectors $\vec{0}$, $\vec{\imath}$, $\vec{\jmath}$ and $\vec{k}$%
\par
$\9 and their definition\par
$\2{numeric}\\{vect\_null},\\{vect\_I},\\{vect\_J},\\{vect\_K};$\par
$\\{vect\_null}=\\{new\_vect};\ \\{vect\_I}=\\{new\_vect};\ \\{vect\_J}=\\{new%
\_vect};\ \\{vect\_K}=\\{new\_vect};$\par
$\\{vect\_def}(\\{vect\_null},0,0,0);$\par
$\\{vect\_def}(\\{vect\_I},1,0,0);\ \\{vect\_def}(\\{vect\_J},0,1,0);\ \\{vect%
\_def}(\\{vect\_K},0,0,1);$\par
$\2{numeric}\\{point\_null};$\par
$\\{point\_null}=\\{vect\_null};$\par
\BL
$\9 Observer's orientation, defined by three angles\par
$\2{numeric}\\{Obs\_psi},\\{Obs\_theta},\\{Obs\_phi};$\par
$\9 default value:\par
$\\{Obs\_psi}=0;\ \\{Obs\_theta}=90;\ \\{Obs\_phi}=0;$\par
\BL
$\9 Points for the figures\par
$\2{numeric}\\{points\_}[];$\par
\BL
$\9 $\\{name}$ is the name of an object instance\par
$\9 $\\{npoints}$ is its number of defining points\par
$\2{def}\\{new\_obj\_points}(\2{expr}\\{name},\\{npoints})=$\par
\quad\quad$\\{define\_point\_offset\_}(\\{name},\\{npoints});\ \\{define%
\_current\_point\_offset\_}(\\{name});$\par
\quad\quad$\2{for}i:=1\4{upto}\\{obj\_points\_}(\\{name})\?:\\{pnt}(i):=\\{new%
\_point};\3{endfor};$\par
$\!\3{enddef};$\par
\BL
$\9 $\\{name}$ is the name of an object instance\par
$\9 $\\{nfaces}$ is its number of defining faces\par
$\2{def}\\{new\_obj\_faces}(\2{expr}\\{name},\\{nfaces})=$\par
\quad\quad$\\{define\_face\_offset\_}(\\{name},\\{nfaces});\ \\{define\_current%
\_face\_offset\_}(\\{name});$\par
\quad\quad$\\{redefine\_obj\_name\_}(\\{name},\\{current\_face\_offset\_}+%
\\{nfaces});$\par
$\!\3{enddef};$\par
\BL
\newpage
$\9 Absolute point number corresponding to object point number $i$\par
$\9 This macro must only be used within the function defining an object\par
$\9 (such as $\\{def\_cube}$) or the function drawing an object (such as\par
$\9 $\\{draw\_cube}$).\par
$\2{def}\\{ipnt\_}(\2{expr}i)=i+\\{current\_point\_offset\_}\3{enddef};$\par
$\2{def}\\{pnt}(\2{expr}i)=\\{points\_}[\\{ipnt\_}(i)]\3{enddef};$\par
\BL
$\2{def}\\{face}(\2{expr}i)=(i+\\{current\_face\_offset\_})\3{enddef};$\par
\BL
$\9 Absolute point number corresponding to local point $n$\par
$\9 in object instance $\\{name}$\par
$\2{vardef}\\{pnt\_obj}(\2{expr}\\{name},n)=$\par
\quad\quad$\\{hide}(\\{define\_current\_point\_offset\_}(\\{name});\ )%
\\{pnt}(n)$\par
$\!\3{enddef};$\par
\BL
$\9 Absolute face number corresponding to local face $n$\par
$\9 in object instance $\\{name}$\par
$\2{vardef}\\{face\_obj}(\2{expr}\\{name},n)=$\par
\quad\quad$\\{hide}(\\{define\_current\_face\_offset\_}(\\{name});\ )%
\\{face}(n)$\par
$\!\3{enddef};$\par
\BL
\BL
$\9 Scale\par
$\2{numeric}\\{drawing\_scale};$\par
$\\{drawing\_scale}=2\\{cm};$\par
\BL
$\9 Color\par
$\9 This function is useful when a color is expressed in hexadecimal.\par
$\2{def}\\{hexcolor}(\2{expr}s)=$\par
\quad\quad$(\1{hex}(\2{substring}(0,2)\2{of}s)/255,\1{hex}(\2{substring}(2,4)%
\2{of}s)/255,$\par
\quad\quad\quad\quad$\1{hex}(\2{substring}(4,6)\2{of}s)/255)$\par
$\!\3{enddef};$\par
\BL
$\9 Filling and contours\par
$\2{boolean}\\{filled\_faces},\\{draw\_contours};$\par
$\\{filled\_faces}=\5{true};$\par
$\\{draw\_contours}=\5{true};$\par
$\2{numeric}\\{contourwidth};\ \9 thickness of contours\par
$\\{contourwidth}=1\\{pt};$\par
\BL
$\9 Overflow control\par
$\9 An overflow can occur when an object is too close from the observer\par
$\9 or if an object is out of sight. We use a special value to mark\par
$\9 coordinates which would lead to an overflow.\par
$\2{numeric}\\{too\_big\_};$\par
$\\{too\_big\_}=4000;$\par
\BL
\BL
$\9 Object offset (the points defining an object are arranged\par
$\9 in a single array, and the objects are easier to manipulate\par
$\9 if the point numbers are divided into a number and an offset).\par
$\2{numeric}\\{last\_point\_offset\_},\\{last\_face\_offset\_};$\par
$\\{last\_point\_offset\_}=0;\ \\{last\_face\_offset\_}=0;$\par
\BL
\BL
\newpage
\title{Computation of field parameters of an animation}
$\2{numeric}\\{xmin\_},\\{ymin\_},\\{xmax\_},\\{ymax\_};$\par
\BL
$\2{def}\\{compute\_bbox}=$\par
\quad\quad$\2{if}\1{known}\\{xmin\_}\?:$\par
\quad\quad\quad\quad$\\{xmin\_}:=\1{min}(\\{xmin\_},\1{xpart}(\1{llcorner}(%
\\{currentpicture})));$\par
\quad\quad\quad\quad$\\{ymin\_}:=\1{min}(\\{ymin\_},\1{ypart}(\1{llcorner}(%
\\{currentpicture})));$\par
\quad\quad\quad\quad$\\{xmax\_}:=\1{max}(\\{xmax\_},\1{xpart}(\1{urcorner}(%
\\{currentpicture})));$\par
\quad\quad\quad\quad$\\{ymax\_}:=\1{max}(\\{ymax\_},\1{ypart}(\1{urcorner}(%
\\{currentpicture})));$\par
\quad\quad$\2{else}\?:$\par
\quad\quad\quad\quad$\\{xmin\_}=\1{xpart}(\1{llcorner}(\\{currentpicture}));$%
\par
\quad\quad\quad\quad$\\{ymin\_}=\1{ypart}(\1{llcorner}(\\{currentpicture}));$%
\par
\quad\quad\quad\quad$\\{xmax\_}=\1{xpart}(\1{urcorner}(\\{currentpicture}));$%
\par
\quad\quad\quad\quad$\\{ymax\_}=\1{ypart}(\1{urcorner}(\\{currentpicture}));$%
\par
\quad\quad$\2{fi};$\par
$\!\3{enddef};$\par
\BL
$\2{boolean}\\{show\_animation\_parameters};$\par
$\\{show\_animation\_parameters}=\5{false};$\par
$\2{numeric}\\{paper\_height};$\par
$\\{paper\_height}=29.7;\ \9 paper height in cm\par
\BL
$\9 show bounding box of an animation, in PostScript points\par
$\9 and parameters for animation script\par
$\2{vardef}\\{show\_animation\_bbox}=$\par
\quad\quad$\2{save}\\{trx},\\{try},h,w,\\{delta},\\{pnmx},\\{pnmy},\\{pnmw},%
\\{pnmh},\\{res};$\par
\quad\quad$\\{res}=36;\ \9 36 dots per inch in bitmap\par
\quad\quad$w=\\{xmax\_}-\\{xmin\_};\ h=\\{ymax\_}-\\{ymin\_};$\par
\quad\quad$\2{if}\\{show\_animation\_parameters}\?:$\par
\quad\quad\quad\quad$\2{message}\7"animation bbox: (llx="\AM \1{decimal}%
\1{round}(\\{xmin\_})$\par
\quad\quad\quad\quad\quad\quad${}\AM \7",lly="\AM \1{decimal}\1{round}(\\{ymin%
\_})$\par
\quad\quad\quad\quad\quad\quad${}\AM \7",w="\AM \1{decimal}\1{round}(w)\AM %
\7",h="\AM \1{decimal}\1{round}(h)\AM \7")";$\par
\quad\quad$\2{fi};$\par
\quad\quad$\2{if}\\{xmin\_}\le 20\?:\\{trx}=50-\\{xmin\_};\ \2{else}\?:%
\\{trx}=0;\ \2{fi};$\par
\quad\quad$\2{if}\\{ymin\_}\le 20\?:\\{try}=50-\\{ymin\_};\ \2{else}\?:%
\\{try}=0;\ \2{fi};$\par
\quad\quad$\2{if}\\{show\_animation\_parameters}\?:$\par
\quad\quad\quad\quad$\2{message}\7"translate parameters: "$\par
\quad\quad\quad\quad${}\AM \1{decimal}\1{round}(\\{trx})\AM \7" "\AM %
\1{decimal}\1{round}(\\{try});$\par
\quad\quad$\2{fi};$\par
\quad\quad$\\{xmin\_}:=\\{xmin\_}+\\{trx};\ \\{ymin\_}:=\\{ymin\_}+\\{try};$\par
\quad\quad$\\{delta}=10;$\par
\quad\quad$\\{pnmx}=\1{round}(\\{xmin\_}\ast (\\{res}/72)-\\{delta});$\par
\quad\quad$\\{pnmy}=\1{round}((\\{paper\_height}/2.54\ast 72-\\{ymin\_}-h)\ast
(\\{res}/72)-\\{delta});$\par
\quad\quad$\\{pnmw}=\1{round}(w\ast (\\{res}/72)+2\ast \\{delta});$\par
\quad\quad$\\{pnmh}=\1{round}(h\ast (\\{res}/72)+2\ast \\{delta});$\par
\quad\quad$\2{if}\\{show\_animation\_parameters}\?:$\par
\quad\quad\quad\quad$\2{message}\7"pnmcut parameters (with -r36): "$\par
\quad\quad\quad\quad${}\AM \1{decimal}\\{pnmx}\AM \7" "\AM \1{decimal}\\{pnmy}%
\AM \7" "$\par
\quad\quad\quad\quad${}\AM \1{decimal}\\{pnmw}\AM \7" "\AM \1{decimal}%
\\{pnmh};$\par
\quad\quad$\2{fi};$\par
\quad\quad$\\{write\_script}(\1{round}(\\{trx}),\1{round}(\\{try}),$\par
\quad\quad\quad\quad$\\{pnmx},\\{pnmy},\\{pnmw},\\{pnmh},\\{res},\5{jobname},%
\7"create_animation.sh");$\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Creation of a shell script to automate the animation}
$\9 This is UNIX targetted and may need to be customized.\par
\BL
$\2{vardef}\\{write\_script}(\2{expr}\\{trx},\\{try},\\{xmin},\\{ymin},w,h,%
\\{res},\\{output},\\{file})=$\par
\quad\quad$\2{save}s;$\par
\quad\quad$\2{string}s;$\par
\quad\quad$\2{def}\\{write\_to\_file}(\2{text}\\{arg})=\2{write}\\{arg}\4{to}%
\\{file};\3{enddef};$\par
\quad\quad$\\{write\_to\_file}(\7"#! /bin/sh");$\par
\quad\quad$\\{write\_to\_file}(\7"");$\par
\quad\quad$\\{write\_to\_file}(\7"/bin/rm -f "\AM \\{output}\AM \7".log");$\par
\quad\quad$\\{write\_to\_file}(\7"for i in `ls "\AM \\{output}\AM \7".*| grep
'"\AM \\{output}\AM \7".[0-9]'`;do");$\par
\quad\quad\quad\quad$\2{if}\5{false}\?:\7"endfor"\2{fi}\9 indentation hack for
meta-mode.el\par
\quad\quad$\\{write\_to\_file}(\7"echo $i");$\par
\quad\quad$\\{write\_to\_file}(\7"echo '=============='");$\par
\quad\quad$s:=\7"awk  < $i '{print} /^%%Page: /{print "\AM \\{ditto};$\par
\quad\quad$s:=s\AM \1{decimal}\\{trx}\AM \7" "\AM \1{decimal}\\{try}\AM \7"
translate\n"\AM \\{ditto}\AM \7"}' > $i.ps";$\par
\quad\quad$\\{write\_to\_file}(s);$\par
\quad\quad\quad\quad$\9 ghostscript PostScript into ppm\par
\quad\quad$s:=\7"gs -sDEVICE=ppmraw -sPAPERSIZE=a4 -dNOPAUSE ";$\par
\quad\quad$s:=s\AM \7"-r"\AM \1{decimal}\\{res}\AM \7" -sOutputFile=$i.ppm -q
-- $i.ps";$\par
\quad\quad$\\{write\_to\_file}(s);$\par
\quad\quad$\\{write\_to\_file}(\7"/bin/rm -f $i.ps");$\par
\quad\quad\quad\quad$\9 possible alternative:\par
\quad\quad\quad\quad$\9    $s:=\7"mogrify -compress -crop "\AM \1{decimal}(w)%
\AM \7"x"\AM \1{decimal}(h);$\par
\quad\quad\quad\quad$\9    $s:=s\AM \7"+"\AM \1{decimal}(\\{xmin})\AM \7"+"\AM %
\1{decimal}(\\{ymin});$\par
\quad\quad\quad\quad$\9    $s:=s\AM \7" -colors 32 -format gif $i.ppm";$\par
\quad\quad$s:=\7"ppmquant 32 $i.ppm | pnmcut "\AM \1{decimal}(\\{xmin})\AM \7"
"\AM \1{decimal}(\\{ymin});$\par
\quad\quad$s:=s\AM \7" "\AM \1{decimal}(w)\AM \7" "\AM \1{decimal}(h)\AM \7" |
";$\par
\quad\quad$s:=s\AM \7"ppmtogif > `expr $i.ppm : '\(.*\)ppm'`gif";$\par
\quad\quad$\\{write\_to\_file}(s);$\par
\quad\quad$\\{write\_to\_file}(\7"/bin/rm -f $i.ppm");$\par
\quad\quad$\\{write\_to\_file}(\7"done");$\par
\quad\quad$\\{write\_to\_file}(\7"/bin/rm -f "\AM \\{output}\AM \7".gif");$\par
\quad\quad$s:=\7"gifmerge -10 -l1000 ";$\par
\quad\quad$s:=s\AM \\{output}\AM \7".*.gif > "\AM \\{output}\AM \7".gif";$\par
\quad\quad$\\{write\_to\_file}(s);$\par
\quad\quad$\\{write\_to\_file}(\7"/bin/rm -f "\AM \\{output}\AM \7".*.gif");$%
\par
\quad\quad$\\{write\_to\_file}(\\{EOF});\ \9 end of file\par
$\!\3{enddef};$\par
\BL
\newpage
\title{Standard animation definitions}
$\9 These definitions produce {\it one\/} image of some kind.\par
\BL
$\\{extra\_endfig}:=\7"compute_bbox";$\par
\BL
$\9 In the standard animations, the observer follows a circle, shown below:\par
\figure{vect-fig.17}
\BL
$\9 Standard image 1: this is an example and may be adapted.\par
$\9 $\\{name}$ is an object instance\par
$\2{def}\\{one\_image}(\2{expr}\\{name},i,a,\\{rd},\\{ang})=$\par
\quad\quad$\2{beginfig}(i);$\par
\quad\quad\quad\quad$\\{set\_point}(\\{Obs},-\\{rd}\ast \1{cosd}(a\ast %
\\{ang}),-\\{rd}\ast \1{sind}(a\ast \\{ang}),1);$\par
\quad\quad\quad\quad$\\{Obs\_phi}:=90;\ \\{Obs\_dist}:=2;$\par
\quad\quad\quad\quad$\\{point\_of\_view\_obj}(\\{name},1,\\{Obs\_phi});\ \9 fix
point 1 of object $\\{name}$\par
\quad\quad\quad\quad$\\{draw\_obj}(\\{name});$\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name},1,\\{vect\_I},\\{ang});$\par
\quad\quad\quad\quad$\\{draw\_point}(\\{name},1);\ \9 show the rotation point%
\par
\quad\quad\quad\quad$\\{draw\_axes}(\\{red},\\{green},\\{blue});$\par
\quad\quad$\!\3{endfig};$\par
$\!\3{enddef};$\par
\BL
$\9 Standard image 2: this is an example and may be adapted.\par
$\9 $\\{name\_a}$ and $\\{name\_b}$ are object instances.\par
$\2{def}\\{one\_image\_two\_objects}(\2{expr}\\{name\_a},\\{name\_b},i,a,%
\\{rd},\\{ang})=$\par
\quad\quad$\2{beginfig}(i);$\par
\quad\quad\quad\quad$\\{set\_point}(\\{Obs},-\\{rd}\ast \1{cosd}(a\ast %
\\{ang}),-\\{rd}\ast \1{sind}(a\ast \\{ang}),1);$\par
\quad\quad\quad\quad$\\{Obs\_phi}:=90;\ \\{Obs\_dist}:=2;$\par
\quad\quad\quad\quad$\\{point\_of\_view\_obj}(\\{name\_a},1,\\{Obs\_phi});\ \9
fix point 1 of object $\\{name\_a}$\par
\quad\quad\quad\quad$\\{draw\_obj}(\\{name\_a});\ \\{draw\_obj}(\\{name\_b});$%
\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name\_a},1,\\{vect\_I},\\{ang});$%
\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name\_b},13,\\{vect\_J},-\\{ang});$%
\par
\quad\quad\quad\quad\quad\quad$\9$\\{rotate\_obj\_pp}(\\{name\_b},13,7,-%
\\{ang});$\par
\quad\quad\quad\quad$\\{draw\_point}(\\{name\_a},1);\ \9 show the rotation
point\par
\quad\quad\quad\quad$\\{draw\_axes}(\\{red},\\{green},\\{blue});$\par
\quad\quad$\!\3{endfig};$\par
$\!\3{enddef};$\par
\BL
\newpage
$\9 Standard image 3: this is an example and may be adapted.\par
$\9 $\\{name\_a}$, $\\{name\_b}$ and $\\{name\_c}$ are object instances.\par
$\2{def}\\{one\_image\_three\_objects}(\2{expr}\\{name\_a},\\{name\_b},\\{name%
\_c},i,a,\\{rd},\\{ang})=$\par
\quad\quad$\2{beginfig}(i);$\par
\quad\quad\quad\quad$\\{set\_point}(\\{Obs},-\\{rd}\ast \1{cosd}(a\ast %
\\{ang}),-\\{rd}\ast \1{sind}(a\ast \\{ang}),1);$\par
\quad\quad\quad\quad$\\{Obs\_phi}:=90;\ \\{Obs\_dist}:=2;\ \\{h\_field}:=100;\ %
\\{v\_field}:=150;$\par
\quad\quad\quad\quad$\\{point\_of\_view\_obj}(\\{name\_a},1,\\{Obs\_phi});\ \9
fix point 1 of object $\\{name\_a}$\par
\quad\quad\quad\quad$\\{draw\_obj}(\\{name\_a});\ \\{draw\_obj}(\\{name\_b});\ %
\\{draw\_obj}(\\{name\_c});$\par
\quad\quad\quad\quad$\\{v\_a}:=\\{new\_vect};$\par
\quad\quad\quad\quad$\\{vect\_def}(\\{v\_a},.03\ast \1{cosd}(-a\ast %
\\{ang}+90),.03\ast \1{sind}(-a\ast \\{ang}+90),0);$\par
\quad\quad\quad\quad$\\{translate\_obj}(\\{name\_c},\\{v\_a});$\par
\quad\quad\quad\quad$\\{free\_vect}(\\{v\_a});$\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name\_a},1,\\{vect\_I},\\{ang});$%
\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name\_b},13,\\{vect\_J},-\\{ang});$%
\par
\quad\quad\quad\quad\quad\quad$\9$\\{rotate\_obj\_pp}(\\{name\_b},13,7,-%
\\{ang});$\par
\quad\quad\quad\quad$\\{draw\_point}(\\{name\_a},1);\ \9 show the rotation
point\par
\quad\quad\quad\quad$\\{draw\_axes}(\\{red},\\{green},\\{blue});$\par
\quad\quad$\!\3{endfig};$\par
$\!\3{enddef};$\par
\BL
$\9 Standard image 4: this is an example and may be adapted.\par
$\9 $\\{name\_a}$ and $\\{name\_b}$ are object instances.\par
$\2{def}\\{one\_image\_two\_identical\_objects}(\2{expr}\\{name\_a},\\{name%
\_b},i,a,\\{rd},\\{ang})=$\par
\quad\quad$\2{beginfig}(i);$\par
\quad\quad\quad\quad$\\{set\_point}(\\{Obs},-\\{rd}\ast \1{cosd}(a\ast %
\\{ang}),-\\{rd}\ast \1{sind}(a\ast \\{ang}),2);$\par
\quad\quad\quad\quad$\\{Obs\_phi}:=90;\ \\{Obs\_dist}:=2;$\par
\quad\quad\quad\quad$\\{point\_of\_view\_obj}(\\{name\_a},1,\\{Obs\_phi});\ \9
fix point 1 of object $\\{name\_a}$\par
\quad\quad\quad\quad$\\{draw\_obj}(\\{name\_a});\ \\{draw\_obj}(\\{name\_b});$%
\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name\_a},1,\\{vect\_I},\\{ang});$%
\par
\quad\quad\quad\quad$\\{rotate\_obj\_pv}(\\{name\_b},13,\\{vect\_J},-\\{ang});$%
\par
\quad\quad\quad\quad\quad\quad$\9$\\{rotate\_obj\_pp}(\\{name\_a},13,7,-%
\\{ang});$\par
\quad\quad\quad\quad$\\{draw\_point}(\\{name\_a},1);\ \9 show the rotation
point\par
\quad\quad\quad\quad$\\{draw\_axes}(\\{red},\\{green},\\{blue});$\par
\quad\quad$\!\3{endfig};$\par
$\!\3{enddef};$\par
\BL
\BL
\newpage
$\9 An animation is a series of images, and these series are produced here.\par
\BL
$\9 Standard animation 1\par
$\9 $\\{name}$ is a class name\par
$\2{def}\\{animate\_object}(\2{expr}\\{name},\\{imin},\\{imax},\\{index})=$\par
\quad\quad$\2{numeric}\\{ang};\ \\{ang}=360/(\\{imax}-\\{imin}+1);$\par
\quad\quad$\\{assign\_obj}(\7"obj",\\{name});$\par
\quad\quad$\2{for}i:=\\{imin}\4{upto}\\{imax}\?:\\{one\_image}(\7"obj",i+%
\\{index},i,5,\\{ang});\3{endfor};$\par
\quad\quad$\\{show\_animation\_bbox};$\par
$\!\3{enddef};$\par
\BL
$\9 Standard animation 2\par
$\9 $\\{name\_a}$ and $\\{name\_b}$ are class names\par
$\2{def}\\{animate\_two\_objects}(\2{expr}\\{name\_a},\\{name\_b},\\{imin},%
\\{imax},\\{index})=$\par
\quad\quad$\2{numeric}\\{ang};\ \\{ang}=360/(\\{imax}-\\{imin}+1);$\par
\quad\quad$\\{assign\_obj}(\7"obja",\\{name\_a});\ \\{assign\_obj}(\7"objb",%
\\{name\_b});$\par
\quad\quad$\\{translate\_obj}(\7"objb",\\{vect\_K});\ \\{translate\_obj}(%
\7"objb",\\{vect\_K});$\par
\quad\quad$\2{for}i:=\\{imin}\4{upto}\\{imax}\?:$\par
\quad\quad\quad\quad$\\{one\_image\_two\_objects}(\7"obja",\7"objb",i+%
\\{index},i,5,\\{ang});$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\\{show\_animation\_bbox};$\par
$\!\3{enddef};$\par
\BL
$\9 Standard animation 3\par
$\9 $\\{name\_a}$, $\\{name\_b}$ and $\\{name\_c}$ are class names\par
$\2{vardef}\\{animate\_three\_objects}(\2{expr}\\{name\_a},\\{name\_b},\\{name%
\_c},\\{imin},\\{imax},\\{index})=$\par
\quad\quad$\2{numeric}\\{ang};\ \\{ang}=360/(\\{imax}-\\{imin}+1);$\par
\quad\quad$\\{assign\_obj}(\7"obja",\\{name\_a});\ \\{assign\_obj}(\7"objb",%
\\{name\_b});$\par
\quad\quad$\\{assign\_obj}(\7"objc",\\{name\_c});$\par
\quad\quad$\\{scale\_obj}(\7"objb",.7);$\par
\quad\quad$\2{numeric}\\{v\_a};\ \\{v\_a}:=\\{new\_vect};$\par
\quad\quad$\\{vect\_def\_vect}(\\{v\_a},\\{vect\_K});\ \\{vect\_mult}(\\{v\_a},%
\\{v\_a},4);\ \\{put\_obj}(\7"objb",\\{v\_a},1,0,0,0);$\par
\quad\quad$\\{free\_vect}(\\{v\_a});$\par
\quad\quad$\\{scale\_obj}(\7"objc",.5);$\par
\quad\quad$\\{translate\_obj}(\7"objc",\\{vect\_K});\ \\{translate\_obj}(%
\7"objc",\\{vect\_K});$\par
\quad\quad$\2{for}i:=\\{imin}\4{upto}\\{imax}\?:$\par
\quad\quad\quad\quad$\\{one\_image\_three\_objects}(\7"obja",\7"objb",%
\7"objc",i+\\{index},i,7,\\{ang});$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\\{show\_animation\_bbox};$\par
$\!\3{enddef};$\par
\BL
$\9 Standard animation 4\par
$\9 $\\{name}$ is a class name\par
$\2{def}\\{animate\_two\_identical\_objects}(\2{expr}\\{name},\\{imin},%
\\{imax},\\{index})=$\par
\quad\quad$\2{numeric}\\{ang};\ \\{ang}=360/(\\{imax}-\\{imin}+1);$\par
\quad\quad$\\{assign\_obj}(\7"obja",\\{name});\ \\{assign\_obj}(\7"objb",%
\\{name});$\par
\quad\quad$\\{translate\_obj}(\7"objb",\\{vect\_K});\ \\{translate\_obj}(%
\7"objb",\\{vect\_K});$\par
\quad\quad$\2{for}i:=\\{imin}\4{upto}\\{imax}\?:$\par
\quad\quad\quad\quad$\\{one\_image\_two\_identical\_objects}(\7"obja",%
\7"objb",i+\\{index},i,10,\\{ang});$\par
\quad\quad$\!\3{endfor};$\par
\quad\quad$\\{show\_animation\_bbox};$\par
$\!\3{enddef};$\par
\BL
$\5{endinput}$\par
\BL