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
|
.PS
# Dini.m4
# This diagram can be produced using, e.g.
# m4 pdf.m4 Dini.m4 | dpic -d > Dini.pdf
# https://tex.stackexchange.com/questions/516734/plot-dinis-surface
threeD_init
NeedDpicTools
scale = 2.0
[
viewazimuth = 0 # View angles in degrees
viewelevation = 20
setview(viewazimuth,viewelevation,0)
a = 1.5
b = 0.25
define(`Dini',
`a*cos(`$1')*sin(`$2'),dnl
a*sin(`$1')*sin(`$2'),dnl
(a*(cos(`$2')+loge(tan((`$2')/2))) + b*(`$1'))')
umin = 0; umax = 4*pi_
vmin = 0.2; vmax = 2.0
ugrid = 80; vgrid = 20
ur = umax-umin
vr = vmax-vmin
`define cu {umin+($1)/ugrid*ur}'
`define cv {vmin+($1)/vgrid*vr}'
# Generate points and sort
n = -1
for iu=0 to ugrid do {
for iv=0 to vgrid do {
n +=1
ciu[n] = cu(iu)
civ[n] = cv(iv)
d[n] = dot3D(View3D,Dini(ciu[n],civ[n]))
ix[n] = n
}
}
dpquicksort(d,0,n,ix)
# Draw the facets
linethick = 0.4
du = ur/ugrid/2
dv = vr/vgrid/2
for i=0 to n do {
NE: Project(Dini(ciu[ix[i]]+du,civ[ix[i]]+dv))
SE: Project(Dini(ciu[ix[i]]+du,civ[ix[i]]-dv))
SW: Project(Dini(ciu[ix[i]]-du,civ[ix[i]]-dv))
NW: Project(Dini(ciu[ix[i]]-du,civ[ix[i]]+dv))
f = (-cos(ciu[ix[i]])+1)/2
g = ( sin(ciu[ix[i]])+1)/2
line from NE to SE then to SW then to NW then to NE \
outlined rgbstring(f,g,0) shaded rgbstring(f,g,0)
# outlined rgbstring(0,0.5,0) shaded rgbstring(f,g,0)
}
]
# Icosahedron.m4
[
#threeD_init
#NeedDpicTools
# Computing facets is complicated, so list them.
# 60 vertices:
define(`m4x',-1)
array2(v,m4inx,0, 0, 1.021)
array2(v,m4inx,0.4035482, 0, 0.9378643)
array2(v,m4inx,-0.2274644, 0.3333333, 0.9378643)
array2(v,m4inx,-0.1471226, -0.375774, 0.9378643)
array2(v,m4inx,0.579632, 0.3333333, 0.7715933)
array2(v,m4inx,0.5058321, -0.375774, 0.8033483)
array2(v,m4inx,-0.6020514, 0.2908927, 0.7715933)
array2(v,m4inx,-0.05138057, 0.6666667, 0.7715933)
array2(v,m4inx,0.1654988, -0.6080151, 0.8033483)
array2(v,m4inx,-0.5217096, -0.4182147, 0.7715933)
array2(v,m4inx,0.8579998, 0.2908927, 0.4708062)
array2(v,m4inx,0.3521676, 0.6666667, 0.6884578)
array2(v,m4inx,0.7841999, -0.4182147, 0.5025612)
array2(v,m4inx,-0.657475, 0.5979962, 0.5025612)
array2(v,m4inx,-0.749174, -0.08488134, 0.6884578)
array2(v,m4inx,-0.3171418, 0.8302373, 0.5025612)
array2(v,m4inx,0.1035333, -0.8826969, 0.5025612)
array2(v,m4inx,-0.5836751, -0.6928964, 0.4708062)
array2(v,m4inx,0.8025761, 0.5979962, 0.2017741)
array2(v,m4inx,0.9602837, -0.08488134, 0.3362902)
array2(v,m4inx,0.4899547, 0.8302373, 0.3362902)
array2(v,m4inx,0.7222343, -0.6928964, 0.2017741)
array2(v,m4inx,-0.8600213, 0.5293258, 0.1503935)
array2(v,m4inx,-0.9517203, -0.1535518, 0.3362902)
array2(v,m4inx,-0.1793548, 0.993808, 0.1503935)
array2(v,m4inx,0.381901, -0.9251375, 0.2017741)
array2(v,m4inx,-0.2710537, -0.9251375, 0.3362902)
array2(v,m4inx,-0.8494363, -0.5293258, 0.2017741)
array2(v,m4inx,0.8494363, 0.5293258, -0.2017741)
array2(v,m4inx,1.007144, -0.1535518, -0.06725804)
array2(v,m4inx,0.2241935, 0.993808, 0.06725804)
array2(v,m4inx,0.8600213, -0.5293258, -0.1503935)
array2(v,m4inx,-0.7222343, 0.6928964, -0.2017741)
array2(v,m4inx,-1.007144, 0.1535518, 0.06725804)
array2(v,m4inx,-0.381901, 0.9251375, -0.2017741)
array2(v,m4inx,0.1793548, -0.993808, -0.1503935)
array2(v,m4inx,-0.2241935, -0.993808, -0.06725804)
array2(v,m4inx,-0.8025761, -0.5979962, -0.2017741)
array2(v,m4inx,0.5836751, 0.6928964, -0.4708062)
array2(v,m4inx,0.9517203, 0.1535518, -0.3362902)
array2(v,m4inx,0.2710537, 0.9251375, -0.3362902)
array2(v,m4inx,0.657475, -0.5979962, -0.5025612)
array2(v,m4inx,-0.7841999, 0.4182147, -0.5025612)
array2(v,m4inx,-0.9602837, 0.08488134, -0.3362902)
array2(v,m4inx,-0.1035333, 0.8826969, -0.5025612)
array2(v,m4inx,0.3171418, -0.8302373, -0.5025612)
array2(v,m4inx,-0.4899547, -0.8302373, -0.3362902)
array2(v,m4inx,-0.8579998, -0.2908927, -0.4708062)
array2(v,m4inx,0.5217096, 0.4182147, -0.7715933)
array2(v,m4inx,0.749174, 0.08488134, -0.6884578)
array2(v,m4inx,0.6020514, -0.2908927, -0.7715933)
array2(v,m4inx,-0.5058321, 0.375774, -0.8033483)
array2(v,m4inx,-0.1654988, 0.6080151, -0.8033483)
array2(v,m4inx,0.05138057, -0.6666667, -0.7715933)
array2(v,m4inx,-0.3521676, -0.6666667, -0.6884578)
array2(v,m4inx,-0.579632, -0.3333333, -0.7715933)
array2(v,m4inx,0.1471226, 0.375774, -0.9378643)
array2(v,m4inx,0.2274644, -0.3333333, -0.9378643)
array2(v,m4inx,-0.4035482, 0, -0.9378643)
array2(v,m4inx,0, 0, -1.021)
# 12 pentagons
undefine(`m4x')
array2(p,m4inx,0,3,8,5,1)
array2(p,m4inx,2,7,15,13,6)
array2(p,m4inx,4,10,18,20,11)
array2(p,m4inx,9,14,23,27,17)
array2(p,m4inx,12,21,31,29,19)
array2(p,m4inx,16,26,36,35,25)
array2(p,m4inx,22,32,42,43,33)
array2(p,m4inx,24,30,40,44,34)
array2(p,m4inx,28,39,49,48,38)
array2(p,m4inx,37,47,55,54,46)
array2(p,m4inx,41,45,53,57,50)
array2(p,m4inx,51,52,56,59,58)
# 20 hexagons
undefine(`m4x')
array2(h,m4inx,0,1,4,11,7,2)
array2(h,m4inx,0,2,6,14,9,3)
array2(h,m4inx,1,5,12,19,10,4)
array2(h,m4inx,3,9,17,26,16,8)
array2(h,m4inx,5,8,16,25,21,12)
array2(h,m4inx,6,13,22,33,23,14)
array2(h,m4inx,7,11,20,30,24,15)
array2(h,m4inx,10,19,29,39,28,18)
array2(h,m4inx,13,15,24,34,32,22)
array2(h,m4inx,17,27,37,46,36,26)
array2(h,m4inx,18,28,38,40,30,20)
array2(h,m4inx,21,25,35,45,41,31)
array2(h,m4inx,23,33,43,47,37,27)
array2(h,m4inx,29,31,41,50,49,39)
array2(h,m4inx,32,34,44,52,51,42)
array2(h,m4inx,35,36,46,54,53,45)
array2(h,m4inx,38,48,56,52,44,40)
array2(h,m4inx,42,51,58,55,47,43)
array2(h,m4inx,48,49,50,57,59,56)
array2(h,m4inx,53,54,55,58,59,57)
viewazimuth = -10 # View angles in degrees
viewelevation = 40
setview(viewazimuth,viewelevation,0)
n = 0 # Find and draw the visible hexagons
for i=1 to 20 do { # The normal is the line to the center
for_(1,3,1,`a[m4x] = (v[h[i,1],m4x] + v[h[i,4],m4x])/2')
if dot3D(View3D,a[1],a[2],a[3]) >= 0 then { n +=1; ih[n] = i }
}
for i=1 to n do {
for j=1 to 6 do { x[j]=h[ih[i],j] }
line from Project((v[x[1],1]),(v[x[1],2]),(v[x[1],3]))\
for_(2,6,1,`to Project((v[x[m4x],1]),(v[x[m4x],2]),(v[x[m4x],3]))\')\
to Project((v[x[1],1]),(v[x[1],2]),(v[x[1],3])) thick 0.2 fill_(0.95)
}
m = 0
for i=1 to 12 do { # Visible pentagons
for_(1,3,1,`a[m4x] = 0; for j=1 to 5 do { a[m4x] += v[p[i,j],m4x] }')
if dot3D(View3D,a[1]/5,a[2]/5,a[3]/5) >= 0 then { m +=1; ip[m] = i }
}
for i=1 to m do {
for j=1 to 5 do { x[j]=p[ip[i],j] }
line from Project((v[x[1],1]),(v[x[1],2]),(v[x[1],3])) \
for_(2,5,1,`to Project((v[x[m4x],1]),(v[x[m4x],2]),(v[x[m4x],3]))\')\
to Project((v[x[1],1]),(v[x[1],2]),(v[x[1],3])) thick 0.2 fill_(0)
}
] scaled 2.5 with .sw at last [].se+(0.5,0)
[
#.PE
#.PS
## BlueBall.m4
## https://latex.org/forum/viewtopic.php?f=5&t=30639
#threeD_init
#NeedDpicTools
ballrad = 2.5
cylrad = ballrad*0.87
hht =sqrt(ballrad^2-cylrad^2)
azimuth = 0
elevation = atan2(hht,cylrad)*rtod_ # disks touch at C
setview(azimuth,elevation)
sinelev = sind(elevation)
linethick_(1)
C: shadedball(ballrad,,,,(0,1,1)) at (0,0)
Op: C+Project(0,0,hht)
O: C+Project(0,0,-hht)
line dashed from Op+( cylrad,0) down Op.y-O.y
line dashed from Op+(-cylrad,0) down Op.y-O.y
define(`Normal',`cosd(`$1')*cylrad,sind(`$1')*cylrad,-hht')
define Cylvis { $`'2 = dot3D(Normal($`'1),View3D) }
findroot(Cylvis,5,170,1e-8,rA)
A: Project(Normal(rA))
B: O+(-(A.x-O.x),A.y-O.y)
aA = atan2(A.y-O.y,(A.x-O.x)*sinelev)
ellipsearc(cylrad*2,cylrad*2*sinelev,aA,-pi_-aA,0,cw) with .C at O
ellipsearc(cylrad*2,cylrad*2*sinelev,-pi_-aA-4*dtor_,aA+4*dtor_,,cw,dashed) \
with .C at O
ellipsearc(cylrad*2,cylrad*2*sinelev,-aA,pi_+aA,0,cw) with .C at Op
ellipsearc(cylrad*2,cylrad*2*sinelev,pi_+aA+4*dtor_,-aA-4*dtor_,,cw,dashed) \
with .C at Op
dot(at C); # "$T$" at last [].w rjust below
dot(at O); # "$O$" at last [].s below
dot(at A); # "$A$" at last [].s ljust below
dot(at B); # "$B$" at last [].s rjust below
dot(at Op); # "$O'$" at last [].n above
Ap: dot(at Op+(A-O)); # "$A'$" at Ap.n ljust above
Bp: dot(at Op+(B-O)); # "$B'$" at Bp.n rjust above
linethick_(0.4)
line dashed from Ap to A then to O then to Op
line dashed from A to C
line dashed from Bp to B
V: Between_(O,A,0.15)
U: C+Project(0,0,-hht+0.15)
line from U to U+(V-O) then to V
] with .w at last [].e+(0.5,0)
.PE
|
