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Repeated Reflections in a Hyperbola

Repeated Reflections in a Hyperbola
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2018-02-25 03:18
FunctionCoordinate GeometryTransformationGeometric Construction
This example demonstrates the phenomenon of repeated reflections inside a hyperbola. It involves core mathematical concepts such as hyperbola, reflection, coordinate geometry, and geometric construction. The construction utilizes GeoGebra commands/objects like Point, Mirror, Hyperbola, Line, Tangent, and Intersect. It is suitable for high school and above students to learn about geometric transformations and properties of hyperbolas.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a hyperbola and perform repeated reflections inside it. Start by taking two points on the x-axis, then construct the hyperbola using these two points as foci and a third point on the x-axis. Next, draw a line from a point on the hyperbola along the x-axis, find its tangent line to the hyperbola, and reflect the line across this tangent. Repeat this reflection process, each time using the new line and its corresponding tangent, marking the reflected points on the hyperbola. The final image should show the pattern formed by these repeated reflections inside the hyperbola.
The prompt above was compiled from the original drawing commands and is for reference only.