summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/metapost/mp3d/3dpoly.mp
blob: 5d3b8f5f2fa73cc9bbaf47d88150cc31416739fc (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
%%\input epsf
%%\def\newpage{\vfill\eject}
%%\def\vc#1{$\vcenter{#1}$}
%%\def\title#1{\hrule\vskip1mm#1\par\vskip1mm\hrule\vskip5mm}
%%\def\figure#1{\par\centerline{\epsfbox{#1}}}
%%\title{{\bf 3DPOLY.MP: DEFINITION OF REGULAR CONVEX POLYHEDRA IN METAPOST}}

%% version 1.34, 17 August 2003
%% {\bf Denis Roegel} ({\tt roegel@loria.fr})
%%
%%This package defines the five regular convex polyhedra classes and
%%must be used with the {\bf 3d} package.

if known three_d_poly_version: 
  expandafter endinput % avoids loading this package twice
fi;

message "*** 3dpoly,      v1.34 (c) D. Roegel, 17 August 2003 ***";
numeric three_d_poly_version;
three_d_poly_version=1.34;

%%Each object definition is parameterized by
%%an instance identification.

%%All polyhedra are normalized (centered on the origin and inscriptible
%%in a sphere of radius 1). 

%%The polyhedra are defined with a certain number of points and faces.
%%The points comprise both the vertices and the center of the object.

%%It might be useful to remind the Euler formula linking 
%%the number of faces $f$,
%%vertices $v$ and edges $e$ in polyhedra with no hole in dimension 3:
%%$f+v-e=2$. This can be verified in the following table:

%%\centerline{\vbox{\halign{\quad#\hfil\quad&&\quad\hfil#\quad\cr
%% \omit \hfil type\hfil&\omit\hfil faces\hfil
%%                      &\omit\hfil vertices\hfil&\omit\hfil edges\hfil\cr
%% tetrahedron&4&4&6\cr
%% cube&6&8&12\cr
%% octahedron&8&6&12\cr
%% dodecahedron&12&20&30\cr
%% icosahedron&20&12&30\cr}}}

%%For more details, see the documentation enclosed in the distribution.

%%\newpage\title{TETRAHEDRON (4 faces)\kern1cm\epsfbox{tetra.ps}}

%% Construction of a tetrahedron:

%% One face is \vc{\epsfbox{vect-fig.1}} with $h=c\sqrt{3}/2$

%% The angle $\alpha$ between two faces is given by 
%% $\sin(\alpha/2)={\displaystyle c/2\over \displaystyle c\sqrt{3}/2}=1/\sqrt3$

%% \figure{vect-fig.2}

def set_tetrahedron_points(expr inst)=
  set_point(1)(0,0,0);set_point(2)(1,0,0);
  set_point(3)(cosd(60),sind(60),0);% $60=360/6$
  sinan=1/sqrt(3);cosan=sqrt(1-sinan**2);
  an=180-2*angle((cosan,sinan));
  new_face_point(4,1,2,3,an);
  normalize_obj(inst)(1,2,3,4);
  set_point(5)(0,0,0);% center of tetrahedron
enddef;

vardef def_tetrahedron(expr inst)=
  new_obj_points(inst,5);% 4 vertices and the center
  new_obj_faces(inst,4);% 4 faces in the definition
  set_tetrahedron_points(inst);
  set_obj_face(1,"1,2,4","b4fefe");
  set_obj_face(2,"2,3,4","b49bc0");
  set_obj_face(3,"1,4,3","b4c8fe");
  set_obj_face(4,"1,3,2","b4fe40");
enddef;

%%\newpage\title{CUBE (HEXAHEDRON) (6 faces)\kern1cm\epsfbox{cube.ps}}

%%The definition of a cube is straightforward.

def set_cube_points(expr inst)=
  set_point(1)(0,0,0);
  set_point(2)(1,0,0);
  set_point(3)(1,1,0);
  set_point(4)(0,1,0);
  set_point(5)(0,0,1);
  set_point(6)(1,0,1);
  set_point(7)(1,1,1);
  set_point(8)(0,1,1);
  normalize_obj(inst)(1,7);% 1 and 7 are opposite vertices
  set_point(9)(0,0,0);% center of cube
enddef;

vardef def_cube(expr inst)=
  new_obj_points(inst,9);% 8 vertices and the center
  new_obj_faces(inst,6);% 6 faces in the definition
  set_cube_points(inst);
  set_obj_face(1,"1,2,6,5","b4fe40");
  set_obj_face(2,"2,3,7,6","45d040");
  set_obj_face(3,"4,8,7,3","45a114");
  set_obj_face(4,"1,5,8,4","45a1d4");
  set_obj_face(5,"5,6,7,8","4569d4");
  set_obj_face(6,"4,3,2,1","112da1");
enddef;


%%\newpage\title{OCTAHEDRON (8 faces)\kern1cm\epsfbox{octa.ps}}

%% A section of a pyramid is: \vc{\epsfbox{vect-fig.3}}

%% The height of this pyramid is given 
%% by $H^2={(c\sqrt3/2)}^2-{(c/2)}^2=c^2/2$, hence $H=c/\sqrt2$

def set_octahedron_points(expr inst)=
  set_point(1)(0,0,0);
  set_point(2)(1,0,0);
  set_point(3)(1,1,0);
  set_point(4)(0,1,0);
  set_point(5)(.5,.5,1/sqrt(2));
  set_point(6)(.5,.5,-1/sqrt(2));
  normalize_obj(inst)(5,6);% 5 and 6 are opposite vertices
  set_point(7)(0,0,0);% center of octahedron
enddef;

vardef def_octahedron(expr inst)=
  new_obj_points(inst,7);% 6 vertices and the center
  new_obj_faces(inst,8);% 8 faces in the definition
  set_octahedron_points(inst);
  set_obj_face(1,"1,2,5","b4fefe");
  set_obj_face(2,"2,3,5","45d040");
  set_obj_face(3,"3,4,5","4569d4");
  set_obj_face(4,"4,1,5","b49bc0");
  set_obj_face(5,"6,1,4","45a1d4");
  set_obj_face(6,"6,2,1","b4c8fe");
  set_obj_face(7,"6,3,2","b49b49");
  set_obj_face(8,"6,4,3","112da1");
enddef;


%%\newpage\title{DODECAHEDRON (12 faces)\kern1cm\epsfbox{dodeca.ps}}

%% Two adjacent faces (pentagons) of the dodecahedron are as follows:
%%\figure{vect-fig.13}
%% The angle between the faces is the angle between $\overrightarrow{OA}$
%% and $\overrightarrow{OB}$.
%% $OA=OB=d_3$ as defined in \figure{vect-fig.14}
%% $d_1=2r\sin(\pi/5)$, $d_2=2r\sin(2\pi/5)$ 
%% and $d_3=d_1\sin(2\pi/5)=2r\sin(\pi/5)\sin(2\pi/5)$
%%
%% The angle $\alpha$ is defined by the following conditions:
%% \figure{vect-fig.15}
%% Thus $d_2^2=d_3^2+d_3^2-2d_3^2\cos\alpha$, which leads to
%% $\cos\alpha=1-{\displaystyle d_2^2\over\displaystyle2d_3^2}
%%      =-{\displaystyle\cos(2\pi/5)\over\displaystyle2\sin^2(\pi/5)}$
%%\newpage

def set_dodecahedron_points(expr inst)=
  new_points(fc)(10);% face centers
  set_point_(fc1,0,0,0);set_point(1)(1,0,0);
  set_point(2)(cosd(72),sind(72),0);% 72=360/5
  rotate_in_plane(3,fc1,1,2);
  rotate_in_plane(4,fc1,2,3);
  rotate_in_plane(5,fc1,3,4);
  cosan=-cosd(72)/(2*sind(36)*sind(36));sinan=sqrt(1-cosan**2);
  an=180-angle((cosan,sinan));
  new_abs_face_point(fc2,fc1,1,2,an);
  new_abs_face_point(fc3,fc1,2,3,an);
  new_abs_face_point(fc4,fc1,3,4,an);
  new_abs_face_point(fc5,fc1,4,5,an);
  new_abs_face_point(fc6,fc1,5,1,an);
  rotate_in_plane(6,fc2,2,1);
  rotate_in_plane(7,fc2,1,6);
  rotate_in_plane(8,fc2,6,7);
  rotate_in_plane(9,fc3,2,8);
  rotate_in_plane(10,fc3,8,9);
  rotate_in_plane(11,fc4,3,10);
  rotate_in_plane(12,fc4,10,11);
  rotate_in_plane(13,fc5,4,12);
  rotate_in_plane(14,fc5,12,13);
  rotate_in_plane(15,fc6,5,14);
  new_abs_face_point(fc7,fc2,6,7,an);
  new_abs_face_point(fc8,fc3,8,9,an);
  new_abs_face_point(fc9,fc4,10,11,an);
  new_abs_face_point(fc10,fc5,12,13,an);
  rotate_in_plane(16,fc7,6,15);
  rotate_in_plane(17,fc7,15,16);
  rotate_in_plane(18,fc8,7,17);
  rotate_in_plane(19,fc9,9,18);
  rotate_in_plane(20,fc10,11,19);   
  normalize_obj(inst)(2,20);% opposite vertices
  set_point(21)(0,0,0);% center of dodecahedron
  free_points(fc)(10);
enddef;
%%\newpage
vardef def_dodecahedron(expr inst)=
  save cosan,sinan,an;
  new_obj_points(inst,21);% 21 points in the definition
  new_obj_faces(inst,12);% 12 faces in the definition
  set_dodecahedron_points(inst);
  set_obj_face(1,"5,4,3,2,1","ff0fa1");
  set_obj_face(2,"8,7,6,1,2","b40000");
  set_obj_face(3,"10,9,8,2,3","b49b49");
  set_obj_face(4,"12,11,10,3,4","b49bc0");
  set_obj_face(5,"14,13,12,4,5","b4c8fe");
  set_obj_face(6,"6,15,14,5,1","b4fefe");
  set_obj_face(7,"6,7,17,16,15","b4fe40");
  set_obj_face(8,"8,9,18,17,7","45d040");
  set_obj_face(9,"10,11,19,18,9","45a114");
  set_obj_face(10,"11,12,13,20,19","45a1d4");
  set_obj_face(11,"14,15,16,20,13","4569d4");
  set_obj_face(12,"16,17,18,19,20","112da1");
enddef;

%%\newpage\title{ICOSAHEDRON (20 faces)\kern1cm\epsfbox{icosa.ps}}

%%Two faces of an icosahedron are linked in the following way:

%% \vc{\epsfbox{vect-fig.4}} where $h=c\sqrt3/2$

%% $d$ is also a diagonal in a pentagon: \vc{\epsfbox{vect-fig.5}}

%% $c=2r\sin(\pi/5)$,
%% $d=2r\sin(2\pi/5)=2c\cos(\pi/5)$

%% The angle between two faces is computed as follows: 
%% \vc{\epsfbox{vect-fig.6}}

%% We have ${(2\cos(\pi/5))}^2=2{(\sqrt3/2)}^2-2{(\sqrt3/2)}^2\cos\alpha$

%% Thus $cos\alpha=1-{8\over3}\cos^2(\pi/5)$
%%\newpage

def set_icosahedron_points(expr inst)=
  set_point(1)(0,0,0);set_point(2)(1,0,0);
  set_point(3)(cosd(60),sind(60),0);% 60=360/6
  cosan=1-8/3*cosd(36)*cosd(36);sinan=sqrt(1-cosan**2);
  an=180-angle((cosan,sinan));
  new_face_point(4,1,2,3,an);
  new_face_point(5,2,3,1,an);
  new_face_point(6,3,1,2,an);
  new_face_point(7,2,4,3,an);
  new_face_point(8,3,5,1,an);
  new_face_point(9,1,6,2,an);
  new_face_point(10,3,4,7,an);
  new_face_point(11,3,7,5,an);
  new_face_point(12,1,8,6,an);
  normalize_obj(inst)(1,10);% opposite vertices
  set_point(13)(0,0,0);% center of icosahedron
enddef;

vardef def_icosahedron(expr inst)=
  save cosan,sinan,an;
  new_obj_points(inst,13);% 12 vertices and the center
  new_obj_faces(inst,20);% 20 faces in the definition
  set_icosahedron_points(inst);
  set_obj_face(1,"3,2,1","b40000");
  set_obj_face(2,"2,3,4","ff0fa1");
  set_obj_face(3,"3,7,4","b49b49");
  set_obj_face(4,"3,5,7","b49bc0");
  set_obj_face(5,"3,1,5","b4c8fe");
  set_obj_face(6,"1,8,5","b4fefe");
  set_obj_face(7,"1,6,8","b4fe40");
  set_obj_face(8,"1,2,6","45d040");
  set_obj_face(9,"2,9,6","45a114");
  set_obj_face(10,"2,4,9","45a1d4");
  set_obj_face(11,"9,4,10","4569d4");
  set_obj_face(12,"4,7,10","112da1");
  set_obj_face(13,"7,5,11","b4fefe");
  set_obj_face(14,"5,8,11","b49bc0");
  set_obj_face(15,"8,6,12","45a114");
  set_obj_face(16,"6,9,12","b49b49");
  set_obj_face(17,"8,12,11","b40000");
  set_obj_face(18,"7,11,10","45a1d4");
  set_obj_face(19,"12,10,11","b4c8fe");
  set_obj_face(20,"9,10,12","ff0fa1");
enddef;

%%\newpage\title{General draw functions}

def draw_polyhedron(expr name)=
  draw_faces(name);
enddef;

let draw_tetrahedron=draw_polyhedron;
let draw_cube=draw_polyhedron;
let draw_octahedron=draw_polyhedron;
let draw_dodecahedron=draw_polyhedron;
let draw_icosahedron=draw_polyhedron;

let new_poly=assign_obj;

endinput