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Repeated Reflections in an Ellipse

Repeated Reflections in an Ellipse
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2018-02-24 20:51
Coordinate GeometryTransformationGeometric ConstructionLocus
This example demonstrates the complex trajectories and patterns formed by repeated reflections of a point or light ray inside an ellipse. The core mathematical concepts include the optical property of an ellipse (a ray from one focus reflects to the other focus), reflection transformations, similarity transformations, and geometric mean. The construction uses GeoGebra commands such as Point, Mirror, Ellipse, Line, Tangent, and Intersect to create the ellipse, incident rays, reflected rays, and a series of reflection points (e.g., A, A', B, C, etc.), applying reflection operations iteratively to generate multi-level reflected figures. This example is suitable for high school and above students studying analytic geometry, geometric transformations, and ellipse properties, and can be used for mathematical exploration or dynamic demonstrations to understand the iterative process of reflection and geometric patterns.

GeoGebra Commands & Parameters

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