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Mathematics of Domes and Roofs: 3D Solids, Volume, and Optimization

Mathematics of Domes and Roofs: 3D Solids, Volume, and Optimization
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2021-02-12 16:25
FunctionSolid GeometryOptimizationGeometric Construction
In collections Domes
This example demonstrates how to build dome- and roof-like 3D shapes using interactive geometric construction, and how shape parameters affect volume and potential optimal configurations. It focuses on solid geometry and volume (e.g., using prism-style construction), quadratic functions as the basis for curved profiles via appropriate cross-sections, and optimization-style reasoning when parameters vary (Optimization Problems). The model can also incorporate variability through random or piecewise inputs (Random Variables / Step Functions) to show how designs change under different assumptions. Construction highlights from the GeoGebra command synopsis: it uses Join, Sequence, and Element to assemble point sets and segments; Take to select subsets; Line and Polygon to define boundary curves and base regions; and Prism to extrude or combine the cross-section outlines into 3D solids. The objects g, n, Color, O, R, m, e_1 and the point set A1–A31 serve as parameters and key vertices controlling the geometry of the dome/roof. Suitable for secondary-to-undergraduate learning, math modeling, and interactive STEM/geometry visualization tasks.

GeoGebra Commands & Parameters

  1. Join(Sequence[Sequence[(x(O) + ((R * cos(i)) * cos(j)), y(O) + ((R * sin(i)) * cos(j)), z(O) + (R * sin(j))), i, 0, 6.28319, (2 * π / n)], j, 0, 1.5708, π / m]) → l1
  2. Sequence(Polygon[Element[l1, i], Element[l1, i + 1], Element[l1, i + n + 2], Element[l1, i + n + 1]], i, 1, Length[l1]) → l2
  3. Element(l2, A1) → B1
  4. Element(l2, A2) → B2
  5. Element(l2, A3) → B3
  6. Element(l2, A4) → B4
  7. Element(l2, A5) → B5
  8. Element(l2, A6) → B6
  9. Element(l2, A7) → B7
  10. Element(l2, A8) → B8
  11. Element(l2, A9) → B9
  12. Element(l2, A10) → B10
  13. Element(l2, A11) → B11
  14. Element(l2, A12) → B12
  15. Element(l2, A13) → B13
  16. Element(l2, A14) → B14
  17. Element(l2, A15) → B15
  18. Element(l2, A16) → B16
  19. Element(l2, A17) → B17
  20. Element(l2, A18) → B18
  21. Element(l2, A19) → B19
  22. Element(l2, A20) → B20
  23. Element(l2, A21) → B21
  24. Element(l2, A22) → B22
  25. Element(l2, A23) → B23
  26. Element(l2, A24) → B24
  27. Element(l2, A25) → B25
  28. Element(l2, A26) → B26
  29. Element(l2, A27) → B27
  30. Element(l2, A28) → B28
  31. Element(l2, A29) → B29
  32. Element(l2, A30) → B30
  33. Element(l2, A31) → B31
  34. Element(l2, A32) → B32
  35. Element(l2, A33) → B33
  36. Element(l2, A34) → B34
  37. Element(l2, A35) → B35
  38. Element(l2, A36) → B36
  39. Element(l2, A37) → B37
  40. Element(l2, A38) → B38
  41. Element(l2, A39) → B39
  42. Element(l2, A40) → B40
  43. Element(l2, A41) → B41
  44. Element(l2, A42) → B42
  45. Element(l2, A43) → B43
  46. Element(l2, A44) → B44
  47. Element(l2, A45) → B45
  48. Element(l2, A46) → B46
  49. Element(l2, A47) → B47
  50. Element(l2, A48) → B48
  51. Element(l2, A49) → B49
  52. Element(l2, A50) → B50
  53. Element(l2, A51) → B51
  54. Element(l2, A52) → B52
  55. Element(l2, A53) → B53
  56. Element(l2, A54) → B54
  57. Element(l2, A55) → B55
  58. Element(l2, A56) → B56
  59. Element(l2, A57) → B57
  60. Element(l2, A58) → B58
  61. Element(l2, A59) → B59
  62. Element(l2, A60) → B60
  63. Element(l2, A61) → B61
  64. Element(l2, A62) → B62
  65. Element(l2, A63) → B63
  66. Element(l2, A64) → B64
  67. Element(l2, A65) → B65
  68. Element(l2, A66) → B66
  69. Element(l2, A67) → B67
  70. Element(l2, A68) → B68
  71. Element(l2, A69) → B69
  72. Element(l2, A70) → B70
  73. Element(l2, A71) → B71
  74. Element(l2, A72) → B72
  75. Element(l2, A73) → B73
  76. Element(l2, A74) → B74
  77. Element(l2, A75) → B75
  78. Element(l2, A76) → B76
  79. Element(l2, A77) → B77
  80. Element(l2, A78) → B78
  81. Element(l2, A79) → B79
  82. Element(l2, A80) → B80
  83. Element(l2, A81) → B81
  84. Element(l2, A82) → B82
  85. Element(l2, A83) → B83
  86. Element(l2, A84) → B84
  87. Element(l2, A85) → B85
  88. Element(l2, A86) → B86
  89. Element(l2, A87) → B87
  90. Element(l2, A88) → B88
  91. Element(l2, A89) → B89
  92. Element(l2, A90) → B90
  93. Element(l2, A91) → B91
  94. Element(l2, A92) → B92
  95. Element(l2, A93) → B93
  96. Element(l2, A94) → B94
  97. Element(l2, A95) → B95
  98. Element(l2, A96) → B96
  99. Element(l2, A97) → B97
  100. Element(l2, A98) → B98
  101. Element(l2, A99) → B99
  102. Element(l2, A100) → B100
  103. Element(l2, A101) → B101
  104. Element(l2, A102) → B102
  105. Element(l2, A103) → B103
  106. Element(l2, A104) → B104
  107. Element(l2, A105) → B105
  108. Element(l2, A106) → B106
  109. Element(l2, A107) → B107
  110. Element(l2, A108) → B108
  111. Element(l2, A109) → B109
  112. Element(l2, A110) → B110
  113. Element(l2, A111) → B111
  114. Element(l2, A112) → B112
  115. Element(l2, A113) → B113
  116. Element(l2, A114) → B114
  117. Element(l2, A115) → B115
  118. Element(l2, A116) → B116
  119. Element(l2, A117) → B117
  120. Element(l2, A118) → B118
  121. Element(l2, A119) → B119
  122. Element(l2, A120) → B120
  123. Element(l2, A121) → B121
  124. Element(l2, A122) → B122
  125. Element(l2, A123) → B123
  126. Element(l2, A124) → B124
  127. Element(l2, A125) → B125
  128. Element(l2, A126) → B126
  129. Element(l2, A127) → B127
  130. Element(l2, A128) → B128
  131. Element(l2, A129) → B129
  132. Element(l2, A130) → B130
  133. Element(l2, A131) → B131
  134. Element(l2, A132) → B132
  135. Element(l2, A133) → B133
  136. Element(l2, A134) → B134
  137. Element(l2, A135) → B135
  138. Element(l2, A136) → B136
  139. Element(l2, A137) → B137
  140. Element(l2, A138) → B138
  141. Element(l2, A139) → B139
  142. Element(l2, A140) → B140
  143. Element(l2, A141) → B141
  144. Element(l2, A142) → B142
  145. Element(l2, A143) → B143
  146. Element(l2, A144) → B144
  147. Element(l2, A145) → B145
  148. Element(l2, A146) → B146
  149. Element(l2, A147) → B147
  150. Element(l2, A148) → B148
  151. Element(l2, A149) → B149
  152. Element(l2, A150) → B150
  153. Element(l2, A151) → B151
  154. Element(l2, A152) → B152
  155. Element(l2, A153) → B153
  156. Element(l2, A154) → B154
  157. Element(l2, A155) → B155
  158. Element(l2, A156) → B156
  159. Element(l2, A157) → B157
  160. Element(l2, A158) → B158
  161. Element(l2, A159) → B159
  162. Element(l2, A160) → B160
  163. Element(l2, A161) → B161
  164. Element(l2, A162) → B162
  165. Element(l2, A163) → B163
  166. Element(l2, A164) → B164
  167. Element(l2, A165) → B165
  168. Element(l2, A166) → B166
  169. Element(l2, A167) → B167
  170. Element(l2, A168) → B168
  171. Element(l2, A169) → B169
  172. Element(l2, A170) → B170
  173. Element(l2, A171) → B171
  174. Element(l2, A172) → B172
  175. Element(l2, A173) → B173
  176. Element(l2, A174) → B174
  177. Element(l2, A175) → B175
  178. Element(l2, A176) → B176
  179. Element(l2, A177) → B177
  180. Element(l2, A178) → B178
  181. Element(l2, A179) → B179
  182. Element(l2, A180) → B180
  183. Element(l2, A181) → B181
  184. Element(l2, A182) → B182
  185. Element(l2, A183) → B183
  186. Element(l2, A184) → B184
  187. Element(l2, A185) → B185
  188. Element(l2, A186) → B186
  189. Element(l2, A187) → B187
  190. Element(l2, A188) → B188
  191. Element(l2, A189) → B189
  192. Element(l2, A190) → B190
  193. Element(l2, A191) → B191
  194. Element(l2, A192) → B192
  195. Element(l2, A193) → B193
  196. Element(l2, A194) → B194
  197. Element(l2, A195) → B195
  198. Element(l2, A196) → B196
  199. Element(l2, A197) → B197
  200. Element(l2, A198) → B198
  201. Element(l2, A199) → B199
  202. Element(l2, A200) → B200
  203. Element(l2, A201) → B201
  204. Element(l2, A202) → B202
  205. Element(l2, A203) → B203
  206. Element(l2, A204) → B204
  207. Element(l2, A205) → B205
  208. Element(l2, A206) → B206
  209. Element(l2, A207) → B207
  210. Element(l2, A208) → B208
  211. Element(l2, A209) → B209
  212. Element(l2, A210) → B210
  213. Element(l2, A211) → B211
  214. Element(l2, A212) → B212
  215. Element(l2, A213) → B213
  216. Element(l2, A214) → B214
  217. Element(l2, A215) → B215
  218. Element(l2, A216) → B216
  219. Element(l2, A217) → B217
  220. Element(l2, A218) → B218
  221. Element(l2, A219) → B219
  222. Element(l2, A220) → B220
  223. Element(l2, A221) → B221
  224. Element(l2, A222) → B222
  225. Element(l2, A223) → B223
  226. Element(l2, A224) → B224
  227. Element(l2, A225) → B225
  228. Element(l2, A226) → B226
  229. Element(l2, A227) → B227
  230. Element(l2, A228) → B228
  231. Element(l2, A229) → B229
  232. Element(l2, A230) → B230
  233. Element(l2, A231) → B231
  234. Element(l2, A232) → B232
  235. Element(l2, A233) → B233
  236. Element(l2, A234) → B234
  237. Element(l2, A235) → B235
  238. Element(l2, A236) → B236
  239. Element(l2, A237) → B237
  240. Element(l2, A238) → B238
  241. Element(l2, A239) → B239
  242. Element(l2, A240) → B240
  243. Element(l2, A241) → B241
  244. Element(l2, A242) → B242
  245. Element(l2, A243) → B243
  246. Element(l2, A244) → B244
  247. Element(l2, A245) → B245
  248. Element(l2, A246) → B246
  249. Element(l2, A247) → B247
  250. Element(l2, A248) → B248
  251. Element(l2, A249) → B249
  252. Element(l2, A250) → B250
  253. Element(l2, A251) → B251
  254. Element(l2, A252) → B252
  255. Element(l2, A253) → B253
  256. Element(l2, A254) → B254
  257. Element(l2, A255) → B255
  258. Element(l2, A256) → B256
  259. Element(l2, A257) → B257
  260. Element(l2, A258) → B258
  261. Element(l2, A259) → B259
  262. Element(l2, A260) → B260
  263. Element(l2, A261) → B261
  264. Element(l2, A262) → B262
  265. Element(l2, A263) → B263
  266. Element(l2, A264) → B264
  267. Element(l2, A265) → B265
  268. Element(l2, A266) → B266
  269. Element(l2, A267) → B267
  270. Element(l2, A268) → B268
  271. Element(l2, A269) → B269
  272. Element(l2, A270) → B270
  273. Element(l2, A271) → B271
  274. Element(l2, A272) → B272
  275. Element(l2, A273) → B273
  276. Element(l2, A274) → B274
  277. Element(l2, A275) → B275
  278. Element(l2, A276) → B276
  279. Element(l2, A277) → B277
  280. Element(l2, A278) → B278
  281. Element(l2, A279) → B279
  282. Element(l2, A280) → B280
  283. Element(l2, A281) → B281
  284. Element(l2, A282) → B282
  285. Element(l2, A283) → B283
  286. Element(l2, A284) → B284
  287. Element(l2, A285) → B285
  288. Element(l2, A286) → B286
  289. Element(l2, A287) → B287
  290. Element(l2, A288) → B288
  291. Element(l2, A289) → B289
  292. Element(l2, A290) → B290
  293. Element(l2, A291) → B291
  294. Element(l2, A292) → B292
  295. Element(l2, A293) → B293
  296. Element(l2, A294) → B294
  297. Element(l2, A295) → B295
  298. Element(l2, A296) → B296
  299. Element(l2, A297) → B297
  300. Element(l2, A298) → B298
  301. Element(l2, A299) → B299
  302. Element(l2, A300) → B300
  303. Element(l2, A301) → B301
  304. Element(l2, A302) → B302
  305. Element(l2, A303) → B303
  306. Element(l2, A304) → B304
  307. Element(l2, A305) → B305
  308. Element(l2, A306) → B306
  309. Element(l2, A307) → B307
  310. Element(l2, A308) → B308
  311. Element(l2, A309) → B309
  312. Element(l2, A310) → B310
  313. Element(l2, A311) → B311
  314. Element(l2, A312) → B312
  315. Element(l2, A313) → B313
  316. Element(l2, A314) → B314
  317. Element(l2, A315) → B315
  318. Element(l2, A316) → B316
  319. Element(l2, A317) → B317
  320. Element(l2, A318) → B318
  321. Element(l2, A319) → B319
  322. Element(l2, A320) → B320
  323. Element(l2, A321) → B321
  324. Element(l2, A322) → B322
  325. Element(l2, A323) → B323
  326. Element(l2, A324) → B324
  327. Element(l2, A325) → B325
  328. Element(l2, A326) → B326
  329. Element(l2, A327) → B327
  330. Element(l2, A328) → B328
  331. Element(l2, A329) → B329
  332. Element(l2, A330) → B330
  333. Element(l2, A331) → B331
  334. Element(l2, A332) → B332
  335. Element(l2, A333) → B333
  336. Element(l2, A334) → B334
  337. Element(l2, A335) → B335
  338. Element(l2, A336) → B336
  339. Element(l2, A337) → B337
  340. Element(l2, A338) → B338
  341. Element(l2, A339) → B339
  342. Element(l2, A340) → B340
  343. Element(l2, A341) → B341
  344. Element(l2, A342) → B342
  345. Element(l2, A343) → B343
  346. Element(l2, A344) → B344
  347. Element(l2, A345) → B345
  348. Element(l2, A346) → B346
  349. Element(l2, A347) → B347
  350. Element(l2, A348) → B348
  351. Element(l2, A349) → B349
  352. Element(l2, A350) → B350
  353. Element(l2, A351) → B351
  354. Element(l2, A352) → B352
  355. Element(l2, A353) → B353
  356. Element(l2, A354) → B354
  357. Element(l2, A355) → B355
  358. Element(l2, A356) → B356
  359. Element(l2, A357) → B357
  360. Element(l2, A358) → B358
  361. Element(l2, A359) → B359
  362. Element(l2, A360) → B360
  363. Element(l2, A361) → B361
  364. Element(l2, A362) → B362
  365. Element(l2, A363) → B363
  366. Element(l2, A364) → B364
  367. Element(l2, A365) → B365
  368. Element(l2, A366) → B366
  369. Element(l2, A367) → B367
  370. Element(l2, A368) → B368
  371. Element(l2, A369) → B369
  372. Element(l2, A370) → B370
  373. Element(l2, A371) → B371
  374. Element(l2, A372) → B372
  375. Element(l2, A373) → B373
  376. Element(l2, A374) → B374
  377. Element(l2, A375) → B375
  378. Element(l2, A376) → B376
  379. Element(l2, A377) → B377
  380. Element(l2, A378) → B378
  381. Element(l2, A379) → B379
  382. Element(l2, A380) → B380
  383. Element(l2, A381) → B381
  384. Element(l2, A382) → B382
  385. Element(l2, A383) → B383
  386. Element(l2, A384) → B384
  387. Element(l2, A385) → B385
  388. Element(l2, A386) → B386
  389. Element(l2, A387) → B387
  390. Element(l2, A388) → B388
  391. Element(l2, A389) → B389
  392. Element(l2, A390) → B390
  393. Element(l2, A391) → B391
  394. Element(l2, A392) → B392
  395. Element(l2, A393) → B393
  396. Element(l2, A394) → B394
  397. Element(l2, A395) → B395
  398. Element(l2, A396) → B396
  399. Element(l2, A397) → B397
  400. Element(l2, A398) → B398
  401. Element(l2, A399) → B399
  402. Element(l2, A400) → B400
  403. Element(l2, A401) → B401
  404. Element(l2, A402) → B402
  405. Element(l2, A403) → B403
  406. Element(l2, A404) → B404
  407. Element(l2, A405) → B405
  408. Element(l2, A406) → B406
  409. Element(l2, A407) → B407
  410. Element(l2, A408) → B408
  411. Element(l2, A409) → B409
  412. Element(l2, A410) → B410
  413. Element(l2, A411) → B411
  414. Element(l2, A412) → B412
  415. Element(l2, A413) → B413
  416. Element(l2, A414) → B414
  417. Element(l2, A415) → B415
  418. Element(l2, A416) → B416
  419. Element(l2, A417) → B417
  420. Element(l2, A418) → B418
  421. Element(l2, A419) → B419
  422. Element(l2, A420) → B420
  423. Sequence(IntersectPath[p, Element[l2, i]], i, 1, Length[l2]) → l3
  424. Sequence(Polygon[Element[l1, i], Element[l1, i + 1], Element[l1, i + n + 2], Element[l1, i + n + 1]], i, 1, Length[l1]) → l2b
  425. Take(l1, 1, n) → l4
  426. Line(O, zAxis) → f
  427. Sequence(Rotate[O + (R - 0.25, 0, 0), i, f], i, 0, (2 * π), (2 * π / n)) → l5
  428. Polygon(l5) → poly1
  429. Prism(poly1, -3) → d, A, J, arête17, arête18, arête20, arête21, arête22, arête26, arête27, arête30, arête32, arête34, K, arête36, arête38, arête39, arête40, arête41, arête42, arête43, arête44, arête45, arête54, L, arête55, arête56, arête57, arête58, arête59, arêteAB, arêteBC, arêteCD, arêteDE, arêteEF, M, arêteFG, arêteGH, arêteHI, arêteIJ, arêteJK, arêteKL, arêteLM, arêteMN, arêteNP, arêtePQ, N, arêteQS, arêteAS, arête19, arête23, arête24, arête25, arête28, arête29, arête31, arête33, P, arête35, arête37, arête46, arête47, arête48, arête49, arête50, arête51, arête52, arête53, Q, arête60, arête61, arête62, arête63, arête64, arête65, S, T, U, B, V, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, C, H_1, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, D, R_1, S_1, face1, face3, face4, face5, face6, face7, face8, face9, E, face10, face11, face12, face13, face14, face15, face16, face17, face2, face18, F, face19, face20, face21, face22, face23, face24, face25, face26, face27, face28, G, face29, face30, face31, face32, face33, face34, face35, face36, face37, face38, H, face39, face40, face41, faceABCDEFGHIJKLMNPQS, arête1, arête2, arête3, arête4, arête5, arête6, I, arête7, arête8, arête9, arête10, arête11, arête12, arête13, arête14, arête15, arête16
  430. Polygon(l4) → poly2

AI Drawing Prompt Reference

In a 3D scene, construct a parameterized dome/roof shape by first creating a set of sample points around a central reference point. Use radius R together with two angular parameters n and m to position points through two coupled angle ranges, forming an annular ring of points in space. Connect the generated points in order to form a closed, band-like polygonal surface. Then pick out a sequence of key points from this connected set (31 control vertices) to serve as the main boundary/control points of the roof-like outline. Use these ordered control points to assemble the 3D solid profile so the final form reflects how the geometry changes with R, n, and m.
The prompt above was compiled from the original drawing commands and is for reference only.