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几何与概率模拟

几何与概率模拟
几何画板及绘图内容加载中，请稍等……
2014-08-09 20:09
向量三角函数解析几何概率统计几何作图
所属专题 Statistics Simulations Statistics Simulations
①本范例演示了通过几何图形构造与概率统计相结合的现象；②涉及的核心数学概念包括概率分布、几何作图、随机过程；③作图要点——从下方『构造摘要』提炼用到了 Polygon（多边形）、Segment（线段）、Intersect（交点）、If（条件判断）、Sum（求和）、DotPlot（点图）、Mean（均值）、Vector（向量）等 GeoGebra 命令/对象；④适合高中及以上学生或数学爱好者用于理解几何与概率的交叉应用。

GeoGebra 命令及参数

  1. Polygon((0, 0), (1, 0), 4) → poly1
  2. Segment(B, (0, 0)) → a
  3. Segment(A, (1, 0)) → b
  4. Intersect(b, a) → C
  5. Segment((0, 0), (random(), random())) → A1
  6. Segment((1, 0), (random(), random())) → B1
  7. Intersect(A1, B1) → C1
  8. If((x(C1) * y(C1)) > 0, 1, 0) → D1
  9. Segment((0, 0), (random(), random())) → A2
  10. Segment((1, 0), (random(), random())) → B2
  11. Intersect(A2, B2) → C2
  12. If((x(C2) * y(C2)) > 0, 1, 0) → D2
  13. Segment((0, 0), (random(), random())) → A3
  14. Segment((1, 0), (random(), random())) → B3
  15. Intersect(A3, B3) → C3
  16. If((x(C3) * y(C3)) > 0, 1, 0) → D3
  17. Segment((0, 0), (random(), random())) → A4
  18. Segment((1, 0), (random(), random())) → B4
  19. Intersect(A4, B4) → C4
  20. If((x(C4) * y(C4)) > 0, 1, 0) → D4
  21. Segment((0, 0), (random(), random())) → A5
  22. Segment((1, 0), (random(), random())) → B5
  23. Intersect(A5, B5) → C5
  24. If((x(C5) * y(C5)) > 0, 1, 0) → D5
  25. Segment((0, 0), (random(), random())) → A6
  26. Segment((1, 0), (random(), random())) → B6
  27. Intersect(A6, B6) → C6
  28. If((x(C6) * y(C6)) > 0, 1, 0) → D6
  29. Segment((0, 0), (random(), random())) → A7
  30. Segment((1, 0), (random(), random())) → B7
  31. Intersect(A7, B7) → C7
  32. If((x(C7) * y(C7)) > 0, 1, 0) → D7
  33. Segment((0, 0), (random(), random())) → A8
  34. Segment((1, 0), (random(), random())) → B8
  35. Intersect(A8, B8) → C8
  36. If((x(C8) * y(C8)) > 0, 1, 0) → D8
  37. Segment((0, 0), (random(), random())) → A9
  38. Segment((1, 0), (random(), random())) → B9
  39. Intersect(A9, B9) → C9
  40. If((x(C9) * y(C9)) > 0, 1, 0) → D9
  41. Segment((0, 0), (random(), random())) → A10
  42. Segment((1, 0), (random(), random())) → B10
  43. Intersect(A10, B10) → C10
  44. If((x(C10) * y(C10)) > 0, 1, 0) → D10
  45. Segment((0, 0), (random(), random())) → A11
  46. Segment((1, 0), (random(), random())) → B11
  47. Intersect(A11, B11) → C11
  48. If((x(C11) * y(C11)) > 0, 1, 0) → D11
  49. Segment((0, 0), (random(), random())) → A12
  50. Segment((1, 0), (random(), random())) → B12
  51. Intersect(A12, B12) → C12
  52. If((x(C12) * y(C12)) > 0, 1, 0) → D12
  53. Segment((0, 0), (random(), random())) → A13
  54. Segment((1, 0), (random(), random())) → B13
  55. Intersect(A13, B13) → C13
  56. If((x(C13) * y(C13)) > 0, 1, 0) → D13
  57. Segment((0, 0), (random(), random())) → A14
  58. Segment((1, 0), (random(), random())) → B14
  59. Intersect(A14, B14) → C14
  60. If((x(C14) * y(C14)) > 0, 1, 0) → D14
  61. Segment((0, 0), (random(), random())) → A15
  62. Segment((1, 0), (random(), random())) → B15
  63. Intersect(A15, B15) → C15
  64. If((x(C15) * y(C15)) > 0, 1, 0) → D15
  65. Segment((0, 0), (random(), random())) → A16
  66. Segment((1, 0), (random(), random())) → B16
  67. Intersect(A16, B16) → C16
  68. If((x(C16) * y(C16)) > 0, 1, 0) → D16
  69. Segment((0, 0), (random(), random())) → A17
  70. Segment((1, 0), (random(), random())) → B17
  71. Intersect(A17, B17) → C17
  72. If((x(C17) * y(C17)) > 0, 1, 0) → D17
  73. Segment((0, 0), (random(), random())) → A18
  74. Segment((1, 0), (random(), random())) → B18
  75. Intersect(A18, B18) → C18
  76. If((x(C18) * y(C18)) > 0, 1, 0) → D18
  77. Segment((0, 0), (random(), random())) → A19
  78. Segment((1, 0), (random(), random())) → B19
  79. Intersect(A19, B19) → C19
  80. If((x(C19) * y(C19)) > 0, 1, 0) → D19
  81. Segment((0, 0), (random(), random())) → A20
  82. Segment((1, 0), (random(), random())) → B20
  83. Intersect(A20, B20) → C20
  84. If((x(C20) * y(C20)) > 0, 1, 0) → D20
  85. Segment((0, 0), (random(), random())) → A21
  86. Segment((1, 0), (random(), random())) → B21
  87. Intersect(A21, B21) → C21
  88. If((x(C21) * y(C21)) > 0, 1, 0) → D21
  89. Segment((0, 0), (random(), random())) → A22
  90. Segment((1, 0), (random(), random())) → B22
  91. Intersect(A22, B22) → C22
  92. If((x(C22) * y(C22)) > 0, 1, 0) → D22
  93. Segment((0, 0), (random(), random())) → A23
  94. Segment((1, 0), (random(), random())) → B23
  95. Intersect(A23, B23) → C23
  96. If((x(C23) * y(C23)) > 0, 1, 0) → D23
  97. Segment((0, 0), (random(), random())) → A24
  98. Segment((1, 0), (random(), random())) → B24
  99. Intersect(A24, B24) → C24
  100. If((x(C24) * y(C24)) > 0, 1, 0) → D24
  101. Segment((0, 0), (random(), random())) → A25
  102. Segment((1, 0), (random(), random())) → B25
  103. Intersect(A25, B25) → C25
  104. If((x(C25) * y(C25)) > 0, 1, 0) → D25
  105. Segment((0, 0), (random(), random())) → A26
  106. Segment((1, 0), (random(), random())) → B26
  107. Intersect(A26, B26) → C26
  108. If((x(C26) * y(C26)) > 0, 1, 0) → D26
  109. Segment((0, 0), (random(), random())) → A27
  110. Segment((1, 0), (random(), random())) → B27
  111. Intersect(A27, B27) → C27
  112. If((x(C27) * y(C27)) > 0, 1, 0) → D27
  113. Segment((0, 0), (random(), random())) → A28
  114. Segment((1, 0), (random(), random())) → B28
  115. Intersect(A28, B28) → C28
  116. If((x(C28) * y(C28)) > 0, 1, 0) → D28
  117. Segment((0, 0), (random(), random())) → A29
  118. Segment((1, 0), (random(), random())) → B29
  119. Intersect(A29, B29) → C29
  120. If((x(C29) * y(C29)) > 0, 1, 0) → D29
  121. Segment((0, 0), (random(), random())) → A30
  122. Segment((1, 0), (random(), random())) → B30
  123. Intersect(A30, B30) → C30
  124. If((x(C30) * y(C30)) > 0, 1, 0) → D30
  125. Segment((0, 0), (random(), random())) → A31
  126. Segment((1, 0), (random(), random())) → B31
  127. Intersect(A31, B31) → C31
  128. If((x(C31) * y(C31)) > 0, 1, 0) → D31
  129. Segment((0, 0), (random(), random())) → A32
  130. Segment((1, 0), (random(), random())) → B32
  131. Intersect(A32, B32) → C32
  132. If((x(C32) * y(C32)) > 0, 1, 0) → D32
  133. Segment((0, 0), (random(), random())) → A33
  134. Segment((1, 0), (random(), random())) → B33
  135. Intersect(A33, B33) → C33
  136. If((x(C33) * y(C33)) > 0, 1, 0) → D33
  137. Segment((0, 0), (random(), random())) → A34
  138. Segment((1, 0), (random(), random())) → B34
  139. Intersect(A34, B34) → C34
  140. If((x(C34) * y(C34)) > 0, 1, 0) → D34
  141. Segment((0, 0), (random(), random())) → A35
  142. Segment((1, 0), (random(), random())) → B35
  143. Intersect(A35, B35) → C35
  144. If((x(C35) * y(C35)) > 0, 1, 0) → D35
  145. Segment((0, 0), (random(), random())) → A36
  146. Segment((1, 0), (random(), random())) → B36
  147. Intersect(A36, B36) → C36
  148. If((x(C36) * y(C36)) > 0, 1, 0) → D36
  149. Segment((0, 0), (random(), random())) → A37
  150. Segment((1, 0), (random(), random())) → B37
  151. Intersect(A37, B37) → C37
  152. If((x(C37) * y(C37)) > 0, 1, 0) → D37
  153. Segment((0, 0), (random(), random())) → A38
  154. Segment((1, 0), (random(), random())) → B38
  155. Intersect(A38, B38) → C38
  156. If((x(C38) * y(C38)) > 0, 1, 0) → D38
  157. Segment((0, 0), (random(), random())) → A39
  158. Segment((1, 0), (random(), random())) → B39
  159. Intersect(A39, B39) → C39
  160. If((x(C39) * y(C39)) > 0, 1, 0) → D39
  161. Segment((0, 0), (random(), random())) → A40
  162. Segment((1, 0), (random(), random())) → B40
  163. Intersect(A40, B40) → C40
  164. If((x(C40) * y(C40)) > 0, 1, 0) → D40
  165. Segment((0, 0), (random(), random())) → A41
  166. Segment((1, 0), (random(), random())) → B41
  167. Intersect(A41, B41) → C41
  168. If((x(C41) * y(C41)) > 0, 1, 0) → D41
  169. Segment((0, 0), (random(), random())) → A42
  170. Segment((1, 0), (random(), random())) → B42
  171. Intersect(A42, B42) → C42
  172. If((x(C42) * y(C42)) > 0, 1, 0) → D42
  173. Segment((0, 0), (random(), random())) → A43
  174. Segment((1, 0), (random(), random())) → B43
  175. Intersect(A43, B43) → C43
  176. If((x(C43) * y(C43)) > 0, 1, 0) → D43
  177. Segment((0, 0), (random(), random())) → A44
  178. Segment((1, 0), (random(), random())) → B44
  179. Intersect(A44, B44) → C44
  180. If((x(C44) * y(C44)) > 0, 1, 0) → D44
  181. Segment((0, 0), (random(), random())) → A45
  182. Segment((1, 0), (random(), random())) → B45
  183. Intersect(A45, B45) → C45
  184. If((x(C45) * y(C45)) > 0, 1, 0) → D45
  185. Segment((0, 0), (random(), random())) → A46
  186. Segment((1, 0), (random(), random())) → B46
  187. Intersect(A46, B46) → C46
  188. If((x(C46) * y(C46)) > 0, 1, 0) → D46
  189. Segment((0, 0), (random(), random())) → A47
  190. Segment((1, 0), (random(), random())) → B47
  191. Intersect(A47, B47) → C47
  192. If((x(C47) * y(C47)) > 0, 1, 0) → D47
  193. Segment((0, 0), (random(), random())) → A48
  194. Segment((1, 0), (random(), random())) → B48
  195. Intersect(A48, B48) → C48
  196. If((x(C48) * y(C48)) > 0, 1, 0) → D48
  197. Segment((0, 0), (random(), random())) → A49
  198. Segment((1, 0), (random(), random())) → B49
  199. Intersect(A49, B49) → C49
  200. If((x(C49) * y(C49)) > 0, 1, 0) → D49
  201. Segment((0, 0), (random(), random())) → A50
  202. Segment((1, 0), (random(), random())) → B50
  203. Intersect(A50, B50) → C50
  204. If((x(C50) * y(C50)) > 0, 1, 0) → D50
  205. Segment((0, 0), (random(), random())) → A51
  206. Segment((1, 0), (random(), random())) → B51
  207. Intersect(A51, B51) → C51
  208. If((x(C51) * y(C51)) > 0, 1, 0) → D51
  209. Segment((0, 0), (random(), random())) → A52
  210. Segment((1, 0), (random(), random())) → B52
  211. Intersect(A52, B52) → C52
  212. If((x(C52) * y(C52)) > 0, 1, 0) → D52
  213. Segment((0, 0), (random(), random())) → A53
  214. Segment((1, 0), (random(), random())) → B53
  215. Intersect(A53, B53) → C53
  216. If((x(C53) * y(C53)) > 0, 1, 0) → D53
  217. Segment((0, 0), (random(), random())) → A54
  218. Segment((1, 0), (random(), random())) → B54
  219. Intersect(A54, B54) → C54
  220. If((x(C54) * y(C54)) > 0, 1, 0) → D54
  221. Segment((0, 0), (random(), random())) → A55
  222. Segment((1, 0), (random(), random())) → B55
  223. Intersect(A55, B55) → C55
  224. If((x(C55) * y(C55)) > 0, 1, 0) → D55
  225. Segment((0, 0), (random(), random())) → A56
  226. Segment((1, 0), (random(), random())) → B56
  227. Intersect(A56, B56) → C56
  228. If((x(C56) * y(C56)) > 0, 1, 0) → D56
  229. Segment((0, 0), (random(), random())) → A57
  230. Segment((1, 0), (random(), random())) → B57
  231. Intersect(A57, B57) → C57
  232. If((x(C57) * y(C57)) > 0, 1, 0) → D57
  233. Segment((0, 0), (random(), random())) → A58
  234. Segment((1, 0), (random(), random())) → B58
  235. Intersect(A58, B58) → C58
  236. If((x(C58) * y(C58)) > 0, 1, 0) → D58
  237. Segment((0, 0), (random(), random())) → A59
  238. Segment((1, 0), (random(), random())) → B59
  239. Intersect(A59, B59) → C59
  240. If((x(C59) * y(C59)) > 0, 1, 0) → D59
  241. Segment((0, 0), (random(), random())) → A60
  242. Segment((1, 0), (random(), random())) → B60
  243. Intersect(A60, B60) → C60
  244. If((x(C60) * y(C60)) > 0, 1, 0) → D60
  245. Segment((0, 0), (random(), random())) → A61
  246. Segment((1, 0), (random(), random())) → B61
  247. Intersect(A61, B61) → C61
  248. If((x(C61) * y(C61)) > 0, 1, 0) → D61
  249. Segment((0, 0), (random(), random())) → A62
  250. Segment((1, 0), (random(), random())) → B62
  251. Intersect(A62, B62) → C62
  252. If((x(C62) * y(C62)) > 0, 1, 0) → D62
  253. Segment((0, 0), (random(), random())) → A63
  254. Segment((1, 0), (random(), random())) → B63
  255. Intersect(A63, B63) → C63
  256. If((x(C63) * y(C63)) > 0, 1, 0) → D63
  257. Segment((0, 0), (random(), random())) → A64
  258. Segment((1, 0), (random(), random())) → B64
  259. Intersect(A64, B64) → C64
  260. If((x(C64) * y(C64)) > 0, 1, 0) → D64
  261. Segment((0, 0), (random(), random())) → A65
  262. Segment((1, 0), (random(), random())) → B65
  263. Intersect(A65, B65) → C65
  264. If((x(C65) * y(C65)) > 0, 1, 0) → D65
  265. Segment((0, 0), (random(), random())) → A66
  266. Segment((1, 0), (random(), random())) → B66
  267. Intersect(A66, B66) → C66
  268. If((x(C66) * y(C66)) > 0, 1, 0) → D66
  269. Segment((0, 0), (random(), random())) → A67
  270. Segment((1, 0), (random(), random())) → B67
  271. Intersect(A67, B67) → C67
  272. If((x(C67) * y(C67)) > 0, 1, 0) → D67
  273. Segment((0, 0), (random(), random())) → A68
  274. Segment((1, 0), (random(), random())) → B68
  275. Intersect(A68, B68) → C68
  276. If((x(C68) * y(C68)) > 0, 1, 0) → D68
  277. Segment((0, 0), (random(), random())) → A69
  278. Segment((1, 0), (random(), random())) → B69
  279. Intersect(A69, B69) → C69
  280. If((x(C69) * y(C69)) > 0, 1, 0) → D69
  281. Segment((0, 0), (random(), random())) → A70
  282. Segment((1, 0), (random(), random())) → B70
  283. Intersect(A70, B70) → C70
  284. If((x(C70) * y(C70)) > 0, 1, 0) → D70
  285. Segment((0, 0), (random(), random())) → A71
  286. Segment((1, 0), (random(), random())) → B71
  287. Intersect(A71, B71) → C71
  288. If((x(C71) * y(C71)) > 0, 1, 0) → D71
  289. Segment((0, 0), (random(), random())) → A72
  290. Segment((1, 0), (random(), random())) → B72
  291. Intersect(A72, B72) → C72
  292. If((x(C72) * y(C72)) > 0, 1, 0) → D72
  293. Segment((0, 0), (random(), random())) → A73
  294. Segment((1, 0), (random(), random())) → B73
  295. Intersect(A73, B73) → C73
  296. If((x(C73) * y(C73)) > 0, 1, 0) → D73
  297. Segment((0, 0), (random(), random())) → A74
  298. Segment((1, 0), (random(), random())) → B74
  299. Intersect(A74, B74) → C74
  300. If((x(C74) * y(C74)) > 0, 1, 0) → D74
  301. Segment((0, 0), (random(), random())) → A75
  302. Segment((1, 0), (random(), random())) → B75
  303. Intersect(A75, B75) → C75
  304. If((x(C75) * y(C75)) > 0, 1, 0) → D75
  305. Segment((0, 0), (random(), random())) → A76
  306. Segment((1, 0), (random(), random())) → B76
  307. Intersect(A76, B76) → C76
  308. If((x(C76) * y(C76)) > 0, 1, 0) → D76
  309. Segment((0, 0), (random(), random())) → A77
  310. Segment((1, 0), (random(), random())) → B77
  311. Intersect(A77, B77) → C77
  312. If((x(C77) * y(C77)) > 0, 1, 0) → D77
  313. Segment((0, 0), (random(), random())) → A78
  314. Segment((1, 0), (random(), random())) → B78
  315. Intersect(A78, B78) → C78
  316. If((x(C78) * y(C78)) > 0, 1, 0) → D78
  317. Segment((0, 0), (random(), random())) → A79
  318. Segment((1, 0), (random(), random())) → B79
  319. Intersect(A79, B79) → C79
  320. If((x(C79) * y(C79)) > 0, 1, 0) → D79
  321. Segment((0, 0), (random(), random())) → A80
  322. Segment((1, 0), (random(), random())) → B80
  323. Intersect(A80, B80) → C80
  324. If((x(C80) * y(C80)) > 0, 1, 0) → D80
  325. Segment((0, 0), (random(), random())) → A81
  326. Segment((1, 0), (random(), random())) → B81
  327. Intersect(A81, B81) → C81
  328. If((x(C81) * y(C81)) > 0, 1, 0) → D81
  329. Segment((0, 0), (random(), random())) → A82
  330. Segment((1, 0), (random(), random())) → B82
  331. Intersect(A82, B82) → C82
  332. If((x(C82) * y(C82)) > 0, 1, 0) → D82
  333. Segment((0, 0), (random(), random())) → A83
  334. Segment((1, 0), (random(), random())) → B83
  335. Intersect(A83, B83) → C83
  336. If((x(C83) * y(C83)) > 0, 1, 0) → D83
  337. Segment((0, 0), (random(), random())) → A84
  338. Segment((1, 0), (random(), random())) → B84
  339. Intersect(A84, B84) → C84
  340. If((x(C84) * y(C84)) > 0, 1, 0) → D84
  341. Segment((0, 0), (random(), random())) → A85
  342. Segment((1, 0), (random(), random())) → B85
  343. Intersect(A85, B85) → C85
  344. If((x(C85) * y(C85)) > 0, 1, 0) → D85
  345. Segment((0, 0), (random(), random())) → A86
  346. Segment((1, 0), (random(), random())) → B86
  347. Intersect(A86, B86) → C86
  348. If((x(C86) * y(C86)) > 0, 1, 0) → D86
  349. Segment((0, 0), (random(), random())) → A87
  350. Segment((1, 0), (random(), random())) → B87
  351. Intersect(A87, B87) → C87
  352. If((x(C87) * y(C87)) > 0, 1, 0) → D87
  353. Segment((0, 0), (random(), random())) → A88
  354. Segment((1, 0), (random(), random())) → B88
  355. Intersect(A88, B88) → C88
  356. If((x(C88) * y(C88)) > 0, 1, 0) → D88
  357. Segment((0, 0), (random(), random())) → A89
  358. Segment((1, 0), (random(), random())) → B89
  359. Intersect(A89, B89) → C89
  360. If((x(C89) * y(C89)) > 0, 1, 0) → D89
  361. Segment((0, 0), (random(), random())) → A90
  362. Segment((1, 0), (random(), random())) → B90
  363. Intersect(A90, B90) → C90
  364. If((x(C90) * y(C90)) > 0, 1, 0) → D90
  365. Segment((0, 0), (random(), random())) → A91
  366. Segment((1, 0), (random(), random())) → B91
  367. Intersect(A91, B91) → C91
  368. If((x(C91) * y(C91)) > 0, 1, 0) → D91
  369. Segment((0, 0), (random(), random())) → A92
  370. Segment((1, 0), (random(), random())) → B92
  371. Intersect(A92, B92) → C92
  372. If((x(C92) * y(C92)) > 0, 1, 0) → D92
  373. Segment((0, 0), (random(), random())) → A93
  374. Segment((1, 0), (random(), random())) → B93
  375. Intersect(A93, B93) → C93
  376. If((x(C93) * y(C93)) > 0, 1, 0) → D93
  377. Segment((0, 0), (random(), random())) → A94
  378. Segment((1, 0), (random(), random())) → B94
  379. Intersect(A94, B94) → C94
  380. If((x(C94) * y(C94)) > 0, 1, 0) → D94
  381. Segment((0, 0), (random(), random())) → A95
  382. Segment((1, 0), (random(), random())) → B95
  383. Intersect(A95, B95) → C95
  384. If((x(C95) * y(C95)) > 0, 1, 0) → D95
  385. Segment((0, 0), (random(), random())) → A96
  386. Segment((1, 0), (random(), random())) → B96
  387. Intersect(A96, B96) → C96
  388. If((x(C96) * y(C96)) > 0, 1, 0) → D96
  389. Segment((0, 0), (random(), random())) → A97
  390. Segment((1, 0), (random(), random())) → B97
  391. Intersect(A97, B97) → C97
  392. If((x(C97) * y(C97)) > 0, 1, 0) → D97
  393. Segment((0, 0), (random(), random())) → A98
  394. Segment((1, 0), (random(), random())) → B98
  395. Intersect(A98, B98) → C98
  396. If((x(C98) * y(C98)) > 0, 1, 0) → D98
  397. Segment((0, 0), (random(), random())) → A99
  398. Segment((1, 0), (random(), random())) → B99
  399. Intersect(A99, B99) → C99
  400. If((x(C99) * y(C99)) > 0, 1, 0) → D99
  401. Segment((0, 0), (random(), random())) → A100
  402. Segment((1, 0), (random(), random())) → B100
  403. Intersect(A100, B100) → C100
  404. If((x(C100) * y(C100)) > 0, 1, 0) → D100
  405. Sum(D1:D100) → E1
  406. DotPlot(data) → c
  407. Mean(data) → d
  408. Vector(D, E) → u
  409. Vector(F, G) → v

AI 绘图提示语参考

画一个正方形，从原点出发绘制两条边，再从正方形的两个顶点分别向内随机画线段，交点用圆圈表示，根据交点坐标的乘积正负用不同颜色填充，最后用直方图展示这些随机点的分布情况，整体体现几何与概率的结合。
以上提示语为根据原始绘图命令整理而来，仅供参考。