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双曲线中的重复反射

双曲线中的重复反射
几何画板及绘图内容加载中，请稍等……
2018-02-25 03:18
函数解析几何变换几何作图
①本范例演示了在双曲线内部进行重复反射的现象；②涉及的核心数学概念包括双曲线、反射、解析几何、几何作图；③作图要点——从下方『构造摘要』提炼用到了 Point（点）、Mirror（镜像）、Hyperbola（双曲线）、Line（直线）、Tangent（切线）、Intersect（交点）等 GeoGebra 命令/对象；④适合高中及以上学生学习和理解几何变换与双曲线性质。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Mirror(A, yAxis) → A'
  3. Point(xAxis) → B
  4. Hyperbola(A, A', B) → c
  5. Point(c) → C
  6. Point(c) → D
  7. Line(C, D) → f
  8. Tangent(D, c) → g,
  9. Mirror(f, g) → f'
  10. Intersect(c, f', 2) → E
  11. Tangent(E, c) → h, n
  12. Mirror(f', h) → f''
  13. Intersect(c, f'', 2) → F
  14. Tangent(F, c) → i, p
  15. Mirror(f'', i) → f'''
  16. Intersect(c, f''', 2) → G
  17. Tangent(G, c) → j, q
  18. Mirror(f''', j) → k
  19. Intersect(c, k, 2) → H
  20. Tangent(H, c) → l, r
  21. Mirror(k, l) → k'
  22. Intersect(c, k', 2) → I
  23. Tangent(I, c) → m, s
  24. Mirror(k', m) → k''
  25. Intersect(c, A1) → B1, C1
  26. Tangent(C1, c) → D1, t
  27. Mirror(A1, D1) → A2
  28. Intersect(c, A2) → B2, C2
  29. Mirror(B1, D1) → B1'
  30. Tangent(C2, c) → D2, a
  31. Mirror(A2, D2) → A3
  32. Intersect(c, A3) → B3, C3
  33. Tangent(C3, c) → D3, b
  34. Mirror(A3, D3) → A4
  35. Intersect(c, A4) → B4, C4
  36. Tangent(C4, c) → D4, d
  37. Mirror(A4, D4) → A5
  38. Intersect(c, A5) → B5, C5
  39. Tangent(C5, c) → D5, e
  40. Mirror(A5, D5) → A6
  41. Intersect(c, A6) → B6, C6
  42. Tangent(C6, c) → D6, f_1
  43. Mirror(A6, D6) → A7
  44. Intersect(c, A7) → B7, C7
  45. Tangent(C7, c) → D7, g_1
  46. Mirror(A7, D7) → A8
  47. Intersect(c, A8) → B8, C8
  48. Tangent(C8, c) → D8, h_1
  49. Mirror(A8, D8) → A9
  50. Intersect(c, A9) → B9, C9
  51. Tangent(C9, c) → D9, i_1
  52. Mirror(A9, D9) → A10
  53. Intersect(c, A10) → B10, C10
  54. Tangent(C10, c) → D10, j_1
  55. Mirror(A10, D10) → A11
  56. Intersect(c, A11) → B11, C11
  57. Tangent(C11, c) → D11, k_1
  58. Mirror(A11, D11) → A12
  59. Intersect(c, A12) → B12, C12
  60. Tangent(C12, c) → D12, l_1
  61. Mirror(A12, D12) → A13
  62. Intersect(c, A13) → B13, C13
  63. Tangent(C13, c) → D13, m_1
  64. Mirror(A13, D13) → A14
  65. Intersect(c, A14) → B14, C14
  66. Tangent(C14, c) → D14, n_1
  67. Mirror(A14, D14) → A15
  68. Intersect(c, A15) → B15, C15
  69. Tangent(C15, c) → D15, p_1
  70. Mirror(A15, D15) → A16
  71. Intersect(c, A16) → B16, C16
  72. Tangent(C16, c) → D16, q_1
  73. Mirror(A16, D16) → A17
  74. Intersect(c, A17) → B17, C17
  75. Tangent(C17, c) → D17, r_1
  76. Mirror(A17, D17) → A18
  77. Intersect(c, A18) → B18, C18
  78. Tangent(C18, c) → D18, s_1
  79. Mirror(A18, D18) → A19
  80. Intersect(c, A19) → B19, C19
  81. Tangent(C19, c) → D19, t_1
  82. Mirror(A19, D19) → A20
  83. Intersect(c, A20) → B20, C20
  84. Tangent(C20, c) → D20, a_1
  85. Mirror(A20, D20) → A21
  86. Intersect(c, A21) → B21, C21
  87. Tangent(C21, c) → D21, b_1
  88. Mirror(A21, D21) → A22
  89. Intersect(c, A22) → B22, C22
  90. Tangent(C22, c) → D22, c_1
  91. Mirror(A22, D22) → A23
  92. Intersect(c, A23) → B23, C23
  93. Tangent(C23, c) → D23, d_1
  94. Mirror(A23, D23) → A24
  95. Intersect(c, A24) → B24, C24
  96. Tangent(C24, c) → D24, e_1
  97. Mirror(A24, D24) → A25
  98. Intersect(c, A25) → B25, C25
  99. Tangent(C25, c) → D25, f_2
  100. Mirror(A25, D25) → A26
  101. Intersect(c, A26) → B26, C26
  102. Tangent(C26, c) → D26, g_2
  103. Mirror(A26, D26) → A27
  104. Intersect(c, A27) → B27, C27
  105. Tangent(C27, c) → D27, h_2
  106. Mirror(A27, D27) → A28
  107. Intersect(c, A28) → B28, C28
  108. Tangent(C28, c) → D28, i_2
  109. Mirror(A28, D28) → A29
  110. Intersect(c, A29) → B29, C29
  111. Tangent(C29, c) → D29, j_2
  112. Mirror(A29, D29) → A30
  113. Intersect(c, A30) → B30, C30
  114. Tangent(C30, c) → D30, k_2
  115. Mirror(A30, D30) → A31
  116. Intersect(c, A31) → B31, C31
  117. Tangent(C31, c) → D31, l_2
  118. Mirror(A31, D31) → A32
  119. Intersect(c, A32) → B32, C32
  120. Tangent(C32, c) → D32, m_2
  121. Mirror(A32, D32) → A33
  122. Intersect(c, A33) → B33, C33
  123. Tangent(C33, c) → D33, n_2
  124. Mirror(A33, D33) → A34
  125. Intersect(c, A34) → B34, C34
  126. Tangent(C34, c) → D34, p_2
  127. Mirror(A34, D34) → A35
  128. Intersect(c, A35) → B35, C35
  129. Tangent(C35, c) → D35, q_2
  130. Mirror(A35, D35) → A36
  131. Intersect(c, A36) → B36, C36
  132. Tangent(C36, c) → D36, r_2
  133. Mirror(A36, D36) → A37
  134. Intersect(c, A37) → B37, C37
  135. Tangent(C37, c) → D37, s_2
  136. Mirror(A37, D37) → A38
  137. Intersect(c, A38) → B38, C38
  138. Tangent(C38, c) → D38, t_2
  139. Mirror(A38, D38) → A39
  140. Intersect(c, A39) → B39, C39
  141. Tangent(C39, c) → D39, a_2
  142. Mirror(A39, D39) → A40
  143. Intersect(c, A40) → B40, C40
  144. Tangent(C40, c) → D40, b_2
  145. Mirror(A40, D40) → A41
  146. Intersect(c, A41) → B41, C41
  147. Tangent(C41, c) → D41, c_2
  148. Mirror(A41, D41) → A42
  149. Intersect(c, A42) → B42, C42
  150. Tangent(C42, c) → D42, d_2
  151. Mirror(A42, D42) → A43
  152. Intersect(c, A43) → B43, C43
  153. Tangent(C43, c) → D43, e_2
  154. Mirror(A43, D43) → A44
  155. Intersect(c, A44) → B44, C44
  156. Tangent(C44, c) → D44, f_3
  157. Mirror(A44, D44) → A45
  158. Intersect(c, A45) → B45, C45
  159. Tangent(C45, c) → D45, g_3
  160. Mirror(A45, D45) → A46
  161. Intersect(c, A46) → B46, C46
  162. Tangent(C46, c) → D46, h_3
  163. Mirror(A46, D46) → A47
  164. Intersect(c, A47) → B47, C47
  165. Tangent(C47, c) → D47, i_3
  166. Mirror(A47, D47) → A48
  167. Intersect(c, A48) → B48, C48
  168. Tangent(C48, c) → D48, j_3
  169. Mirror(A48, D48) → A49
  170. Intersect(c, A49) → B49, C49
  171. Tangent(C49, c) → D49, k_3
  172. Mirror(A49, D49) → A50
  173. Intersect(c, A50) → B50, C50
  174. Tangent(C50, c) → D50, l_3
  175. Mirror(A50, D50) → A51
  176. Intersect(c, A51) → B51, C51
  177. Tangent(C51, c) → D51, m_3
  178. Mirror(A51, D51) → A52
  179. Intersect(c, A52) → B52, C52
  180. Tangent(C52, c) → D52, n_3
  181. Mirror(A52, D52) → A53
  182. Intersect(c, A53) → B53, C53
  183. Tangent(C53, c) → D53, p_3
  184. Mirror(A53, D53) → A54
  185. Intersect(c, A54) → B54, C54
  186. Tangent(C54, c) → D54, q_3
  187. Mirror(A54, D54) → A55
  188. Intersect(c, A55) → B55, C55
  189. Tangent(C55, c) → D55, r_3
  190. Mirror(A55, D55) → A56
  191. Intersect(c, A56) → B56, C56
  192. Tangent(C56, c) → D56, s_3
  193. Mirror(A56, D56) → A57
  194. Intersect(c, A57) → B57, C57
  195. Tangent(C57, c) → D57, t_3
  196. Mirror(A57, D57) → A58
  197. Intersect(c, A58) → B58, C58
  198. Tangent(C58, c) → D58, a_3
  199. Mirror(A58, D58) → A59
  200. Intersect(c, A59) → B59, C59
  201. Tangent(C59, c) → D59, b_3
  202. Mirror(A59, D59) → A60
  203. Intersect(c, A60) → B60, C60
  204. Tangent(C60, c) → D60, c_3
  205. Mirror(A60, D60) → A61
  206. Intersect(c, A61) → B61, C61
  207. Tangent(C61, c) → D61, d_3
  208. Mirror(A61, D61) → A62
  209. Intersect(c, A62) → B62, C62
  210. Tangent(C62, c) → D62, e_3
  211. Mirror(A62, D62) → A63
  212. Intersect(c, A63) → B63, C63
  213. Tangent(C63, c) → D63, f_4
  214. Mirror(A63, D63) → A64
  215. Intersect(c, A64) → B64, C64
  216. Tangent(C64, c) → D64, g_4
  217. Mirror(A64, D64) → A65
  218. Intersect(c, A65) → B65, C65
  219. Tangent(C65, c) → D65, h_4
  220. Mirror(A65, D65) → A66
  221. Intersect(c, A66) → B66, C66
  222. Tangent(C66, c) → D66, i_4
  223. Mirror(A66, D66) → A67
  224. Intersect(c, A67) → B67, C67
  225. Tangent(C67, c) → D67, j_4
  226. Mirror(A67, D67) → A68
  227. Intersect(c, A68) → B68, C68
  228. Tangent(C68, c) → D68, k_4
  229. Mirror(A68, D68) → A69
  230. Intersect(c, A69) → B69, C69
  231. Tangent(C69, c) → D69, l_4
  232. Mirror(A69, D69) → A70
  233. Intersect(c, A70) → B70, C70
  234. Tangent(C70, c) → D70, m_4
  235. Mirror(A70, D70) → A71
  236. Intersect(c, A71) → B71, C71
  237. Tangent(C71, c) → D71, n_4
  238. Mirror(A71, D71) → A72
  239. Intersect(c, A72) → B72, C72
  240. Tangent(C72, c) → D72, p_4
  241. Mirror(A72, D72) → A73
  242. Intersect(c, A73) → B73, C73
  243. Tangent(C73, c) → D73, q_4
  244. Mirror(A73, D73) → A74
  245. Intersect(c, A74) → B74, C74
  246. Tangent(C74, c) → D74, r_4
  247. Mirror(A74, D74) → A75
  248. Intersect(c, A75) → B75, C75
  249. Tangent(C75, c) → D75, s_4
  250. Mirror(A75, D75) → A76
  251. Intersect(c, A76) → B76, C76
  252. Tangent(C76, c) → D76, t_4
  253. Mirror(A76, D76) → A77
  254. Intersect(c, A77) → B77, C77
  255. Tangent(C77, c) → D77, a_4
  256. Mirror(A77, D77) → A78
  257. Intersect(c, A78) → B78, C78
  258. Tangent(C78, c) → D78, b_4
  259. Mirror(A78, D78) → A79
  260. Intersect(c, A79) → B79, C79
  261. Tangent(C79, c) → D79, c_4
  262. Mirror(A79, D79) → A80
  263. Intersect(c, A80) → B80, C80
  264. Tangent(C80, c) → D80, d_4
  265. Mirror(A80, D80) → A81
  266. Intersect(c, A81) → B81, C81
  267. Tangent(C81, c) → D81, e_4
  268. Mirror(A81, D81) → A82
  269. Intersect(c, A82) → B82, C82
  270. Tangent(C82, c) → D82, f_5
  271. Mirror(A82, D82) → A83
  272. Intersect(c, A83) → B83, C83
  273. Tangent(C83, c) → D83, g_5
  274. Mirror(A83, D83) → A84
  275. Intersect(c, A84) → B84, C84
  276. Tangent(C84, c) → D84, h_5
  277. Mirror(A84, D84) → A85
  278. Intersect(c, A85) → B85, C85
  279. Tangent(C85, c) → D85, i_5
  280. Mirror(A85, D85) → A86
  281. Intersect(c, A86) → B86, C86
  282. Tangent(C86, c) → D86, j_5
  283. Mirror(A86, D86) → A87
  284. Intersect(c, A87) → B87, C87
  285. Tangent(C87, c) → D87, k_5
  286. Mirror(A87, D87) → A88
  287. Intersect(c, A88) → B88, C88
  288. Tangent(C88, c) → D88, l_5
  289. Mirror(A88, D88) → A89
  290. Intersect(c, A89) → B89, C89
  291. Tangent(C89, c) → D89, m_5
  292. Mirror(A89, D89) → A90
  293. Intersect(c, A90) → B90, C90
  294. Tangent(C90, c) → D90, n_5
  295. Mirror(A90, D90) → A91
  296. Intersect(c, A91) → B91, C91
  297. Tangent(C91, c) → D91, p_5
  298. Mirror(A91, D91) → A92
  299. Intersect(c, A92) → B92, C92
  300. Tangent(C92, c) → D92, q_5
  301. Mirror(A92, D92) → A93
  302. Intersect(c, A93) → B93, C93
  303. Tangent(C93, c) → D93, r_5
  304. Mirror(A93, D93) → A94
  305. Intersect(c, A94) → B94, C94
  306. Tangent(C94, c) → D94, s_5
  307. Mirror(A94, D94) → A95
  308. Intersect(c, A95) → B95, C95
  309. Tangent(C95, c) → D95, t_5
  310. Mirror(A95, D95) → A96
  311. Intersect(c, A96) → B96, C96
  312. Tangent(C96, c) → D96, a_5
  313. Mirror(A96, D96) → A97
  314. Intersect(c, A97) → B97, C97
  315. Tangent(C97, c) → D97, b_5
  316. Mirror(A97, D97) → A98
  317. Intersect(c, A98) → B98, C98
  318. Tangent(C98, c) → D98, c_5
  319. Mirror(A98, D98) → A99
  320. Intersect(c, A99) → B99, C99
  321. Tangent(C99, c) → D99, d_5
  322. Mirror(A99, D99) → A100
  323. Intersect(c, A100) → B100, C100
  324. Tangent(C100, c) → D100, e_5
  325. Mirror(A100, D100) → A101
  326. Intersect(c, A101) → B101, C101
  327. Tangent(C101, c) → D101,
  328. Mirror(A101, D101) → A102
  329. Intersect(c, A102) → B102, C102
  330. Tangent(C102, c) → D102,
  331. Mirror(A102, D102) → A103
  332. Intersect(c, A103) → B103, C103
  333. Tangent(C103, c) → D103,
  334. Mirror(A103, D103) → A104
  335. Intersect(c, A104) → B104, C104
  336. Tangent(C104, c) → D104,
  337. Mirror(A104, D104) → A105
  338. Intersect(c, A105) → B105, C105
  339. Tangent(C105, c) → D105,
  340. Mirror(A105, D105) → A106
  341. Intersect(c, A106) → B106, C106
  342. Tangent(C106, c) → D106,
  343. Mirror(A106, D106) → A107
  344. Intersect(c, A107) → B107, C107
  345. Tangent(C107, c) → D107,
  346. Mirror(A107, D107) → A108
  347. Intersect(c, A108) → B108, C108
  348. Tangent(C108, c) → D108,
  349. Mirror(A108, D108) → A109
  350. Intersect(c, A109) → B109, C109
  351. Tangent(C109, c) → D109,
  352. Mirror(A109, D109) → A110
  353. Intersect(c, A110) → B110, C110
  354. Tangent(C110, c) → D110,
  355. Mirror(A110, D110) → A111
  356. Intersect(c, A111) → B111, C111
  357. Tangent(C111, c) → D111,
  358. Mirror(A111, D111) → A112
  359. Intersect(c, A112) → B112, C112
  360. Tangent(C112, c) → D112,
  361. Mirror(A112, D112) → A113
  362. Intersect(c, A113) → B113, C113
  363. Tangent(C113, c) → D113,
  364. Mirror(A113, D113) → A114
  365. Intersect(c, A114) → B114, C114
  366. Tangent(C114, c) → D114,
  367. Mirror(A114, D114) → A115
  368. Intersect(c, A115) → B115, C115
  369. Tangent(C115, c) → D115,
  370. Mirror(A115, D115) → A116
  371. Intersect(c, A116) → B116, C116
  372. Tangent(C116, c) → D116,
  373. Mirror(A116, D116) → A117
  374. Intersect(c, A117) → B117, C117
  375. Tangent(C117, c) → D117,
  376. Mirror(A117, D117) → A118
  377. Intersect(c, A118) → B118, C118
  378. Tangent(C118, c) → D118,
  379. Mirror(A118, D118) → A119
  380. Intersect(c, A119) → B119, C119
  381. Tangent(C119, c) → D119,
  382. Mirror(A119, D119) → A120
  383. Intersect(c, A120) → B120, C120
  384. Tangent(C120, c) → D120,
  385. Mirror(A120, D120) → A121
  386. Intersect(c, A121) → B121, C121
  387. Tangent(C121, c) → D121,
  388. Mirror(A121, D121) → A122
  389. Intersect(c, A122) → B122, C122
  390. Tangent(C122, c) → D122,
  391. Mirror(A122, D122) → A123
  392. Intersect(c, A123) → B123, C123
  393. Tangent(C123, c) → D123,
  394. Mirror(A123, D123) → A124
  395. Intersect(c, A124) → B124, C124
  396. Tangent(C124, c) → D124,
  397. Mirror(A124, D124) → A125
  398. Intersect(c, A125) → B125, C125
  399. Tangent(C125, c) → D125,
  400. Mirror(A125, D125) → A126
  401. Intersect(c, A126) → B126, C126
  402. Tangent(C126, c) → D126,
  403. Mirror(A126, D126) → A127
  404. Intersect(c, A127) → B127, C127
  405. Tangent(C127, c) → D127,
  406. Mirror(A127, D127) → A128
  407. Intersect(c, A128) → B128, C128
  408. Tangent(C128, c) → D128,
  409. Mirror(A128, D128) → A129
  410. Intersect(c, A129) → B129, C129
  411. Tangent(C129, c) → D129,
  412. Mirror(A129, D129) → A130
  413. Intersect(c, A130) → B130, C130
  414. Tangent(C130, c) → D130,
  415. Mirror(A130, D130) → A131
  416. Intersect(c, A131) → B131, C131
  417. Tangent(C131, c) → D131,
  418. Mirror(A131, D131) → A132
  419. Intersect(c, A132) → B132, C132
  420. Tangent(C132, c) → D132,
  421. Mirror(A132, D132) → A133
  422. Intersect(c, A133) → B133, C133
  423. Tangent(C133, c) → D133,
  424. Mirror(A133, D133) → A134
  425. Intersect(c, A134) → B134, C134
  426. Tangent(C134, c) → D134,
  427. Mirror(A134, D134) → A135
  428. Intersect(c, A135) → B135, C135
  429. Tangent(C135, c) → D135,
  430. Mirror(A135, D135) → A136
  431. Intersect(c, A136) → B136, C136
  432. Tangent(C136, c) → D136,
  433. Mirror(A136, D136) → A137
  434. Intersect(c, A137) → B137, C137
  435. Tangent(C137, c) → D137,
  436. Mirror(A137, D137) → A138
  437. Intersect(c, A138) → B138, C138
  438. Tangent(C138, c) → D138,
  439. Mirror(A138, D138) → A139
  440. Intersect(c, A139) → B139, C139
  441. Tangent(C139, c) → D139,
  442. Mirror(A139, D139) → A140
  443. Intersect(c, A140) → B140, C140
  444. Tangent(C140, c) → D140,
  445. Mirror(A140, D140) → A141
  446. Intersect(c, A141) → B141, C141
  447. Tangent(C141, c) → D141,
  448. Mirror(A141, D141) → A142
  449. Intersect(c, A142) → B142, C142
  450. Tangent(C142, c) → D142,
  451. Mirror(A142, D142) → A143
  452. Intersect(c, A143) → B143, C143
  453. Tangent(C143, c) → D143,
  454. Mirror(A143, D143) → A144
  455. Intersect(c, A144) → B144, C144
  456. Tangent(C144, c) → D144,
  457. Mirror(A144, D144) → A145
  458. Intersect(c, A145) → B145, C145
  459. Tangent(C145, c) → D145,
  460. Mirror(A145, D145) → A146
  461. Intersect(c, A146) → B146, C146
  462. Tangent(C146, c) → D146,
  463. Mirror(A146, D146) → A147
  464. Intersect(c, A147) → B147, C147
  465. Tangent(C147, c) → D147,
  466. Mirror(A147, D147) → A148
  467. Intersect(c, A148) → B148, C148
  468. Tangent(C148, c) → D148,
  469. Mirror(A148, D148) → A149
  470. Intersect(c, A149) → B149, C149
  471. Tangent(C149, c) → D149,
  472. Mirror(A149, D149) → A150
  473. Intersect(c, A150) → B150, C150
  474. Tangent(C150, c) → D150,

AI 绘图提示语参考

画一个双曲线，并在其内部进行重复反射操作。首先，在x轴上取两点，以这两点为焦点和一点在x轴上的点构造双曲线。接着，从双曲线上的一点出发，沿x轴画一条线，然后找到这条线与双曲线的切线，并以此切线为轴，将线进行反射，得到新的线。重复此过程，将新线继续进行反射，并在双曲线上标记反射后的点。最终展示双曲线内部由重复反射形成的图案。
以上提示语为根据原始绘图命令整理而来，仅供参考。