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Patterns from Tracing Two Ellipses

Patterns from Tracing Two Ellipses
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2021-01-07 15:43
VectorTransformationGeometric ConstructionLocus
This example demonstrates how visually intricate, symmetric patterns emerge when you trace two ellipses together—using relative motion (via parameter steps and vector-driven updates) to accumulate a locus-like drawing. The central ideas are analytic geometry (ellipse curves), vector-based construction, and the way transformations/relative rotation create structured geometric effects. In GeoGebra, the construction uses Vector to define vector quantities (e.g., direction/offset components), Sequence to build parameter or sampling chains (such as n and multiple step parameters), Take and Element to extract particular sequence items during the iterative trace, and CurveCartesian to generate the Cartesian coordinate curves that represent the traced paths. Intermediate objects (like Phase, points A/B/C/D/E/F and vectors u/v) coordinate the two ellipses and the resulting combined pattern. Suitable for secondary school through early university use in analytic geometry, geometric construction, and dynamic visualization.

GeoGebra Commands & Parameters

  1. Vector(A, C) → v
  2. Vector(A, B) → u
  3. Vector(D, F) → a
  4. Vector(D, E) → w
  5. Sequence(A + (u * cos(k + Phase)) + (v * sin(k + Phase)), k, 0, (2 * π), (2 * π / n)) → l1
  6. Sequence(D + (w * cos(k)) + (a * sin(k)), k, 0, (2 * π), (2 * π / n)) → l2
  7. Sequence(Segment[Element[l1, i], Element[l2, i]], i, 1, Length[l1]) → l3
  8. Take(l3, 1, b) → l4
  9. CurveCartesian(x(A + (u * cos(k)) + (v * sin(k))), y(A + (u * cos(k)) + (v * sin(k))), k, 0, (2 * π)) → c
  10. CurveCartesian(x(D + (w * cos(k)) + (a * sin(k))), y(D + (w * cos(k)) + (a * sin(k))), k, 0, (2 * π)) → d
  11. Element(l4, A1) → B1
  12. Element(l4, A2) → B2
  13. Element(l4, A3) → B3
  14. Element(l4, A4) → B4
  15. Element(l4, A5) → B5
  16. Element(l4, A6) → B6
  17. Element(l4, A7) → B7
  18. Element(l4, A8) → B8
  19. Element(l4, A9) → B9
  20. Element(l4, A10) → B10
  21. Element(l4, A11) → B11
  22. Element(l4, A12) → B12
  23. Element(l4, A13) → B13
  24. Element(l4, A14) → B14
  25. Element(l4, A15) → B15
  26. Element(l4, A16) → B16
  27. Element(l4, A17) → B17
  28. Element(l4, A18) → B18
  29. Element(l4, A19) → B19
  30. Element(l4, A20) → B20
  31. Element(l4, A21) → B21
  32. Element(l4, A22) → B22
  33. Element(l4, A23) → B23
  34. Element(l4, A24) → B24
  35. Element(l4, A25) → B25
  36. Element(l4, A26) → B26
  37. Element(l4, A27) → B27
  38. Element(l4, A28) → B28
  39. Element(l4, A29) → B29
  40. Element(l4, A30) → B30
  41. Element(l4, A31) → B31
  42. Element(l4, A32) → B32
  43. Element(l4, A33) → B33
  44. Element(l4, A34) → B34
  45. Element(l4, A35) → B35
  46. Element(l4, A36) → B36
  47. Element(l4, A37) → B37
  48. Element(l4, A38) → B38
  49. Element(l4, A39) → B39
  50. Element(l4, A40) → B40
  51. Element(l4, A41) → B41
  52. Element(l4, A42) → B42
  53. Element(l4, A43) → B43
  54. Element(l4, A44) → B44
  55. Element(l4, A45) → B45
  56. Element(l4, A46) → B46
  57. Element(l4, A47) → B47
  58. Element(l4, A48) → B48
  59. Element(l4, A49) → B49
  60. Element(l4, A50) → B50
  61. Element(l4, A51) → B51
  62. Element(l4, A52) → B52
  63. Element(l4, A53) → B53
  64. Element(l4, A54) → B54
  65. Element(l4, A55) → B55
  66. Element(l4, A56) → B56
  67. Element(l4, A57) → B57
  68. Element(l4, A58) → B58
  69. Element(l4, A59) → B59
  70. Element(l4, A60) → B60
  71. Element(l4, A61) → B61
  72. Element(l4, A62) → B62
  73. Element(l4, A63) → B63
  74. Element(l4, A64) → B64
  75. Element(l4, A65) → B65
  76. Element(l4, A66) → B66
  77. Element(l4, A67) → B67
  78. Element(l4, A68) → B68
  79. Element(l4, A69) → B69
  80. Element(l4, A70) → B70
  81. Element(l4, A71) → B71
  82. Element(l4, A72) → B72
  83. Element(l4, A73) → B73
  84. Element(l4, A74) → B74
  85. Element(l4, A75) → B75
  86. Element(l4, A76) → B76
  87. Element(l4, A77) → B77
  88. Element(l4, A78) → B78
  89. Element(l4, A79) → B79
  90. Element(l4, A80) → B80
  91. Element(l4, A81) → B81
  92. Element(l4, A82) → B82
  93. Element(l4, A83) → B83
  94. Element(l4, A84) → B84
  95. Element(l4, A85) → B85
  96. Element(l4, A86) → B86
  97. Element(l4, A87) → B87
  98. Element(l4, A88) → B88
  99. Element(l4, A89) → B89
  100. Element(l4, A90) → B90
  101. Element(l4, A91) → B91
  102. Element(l4, A92) → B92
  103. Element(l4, A93) → B93
  104. Element(l4, A94) → B94
  105. Element(l4, A95) → B95
  106. Element(l4, A96) → B96
  107. Element(l4, A97) → B97
  108. Element(l4, A98) → B98
  109. Element(l4, A99) → B99
  110. Element(l4, A100) → B100
  111. Element(l4, A101) → B101
  112. Element(l4, A102) → B102
  113. Element(l4, A103) → B103
  114. Element(l4, A104) → B104
  115. Element(l4, A105) → B105
  116. Element(l4, A106) → B106
  117. Element(l4, A107) → B107
  118. Element(l4, A108) → B108
  119. Element(l4, A109) → B109
  120. Element(l4, A110) → B110
  121. Element(l4, A111) → B111
  122. Element(l4, A112) → B112
  123. Element(l4, A113) → B113
  124. Element(l4, A114) → B114
  125. Element(l4, A115) → B115
  126. Element(l4, A116) → B116
  127. Element(l4, A117) → B117
  128. Element(l4, A118) → B118
  129. Element(l4, A119) → B119
  130. Element(l4, A120) → B120
  131. Element(l4, A121) → B121
  132. Element(l4, A122) → B122
  133. Element(l4, A123) → B123
  134. Element(l4, A124) → B124
  135. Element(l4, A125) → B125
  136. Element(l4, A126) → B126
  137. Element(l4, A127) → B127
  138. Element(l4, A128) → B128
  139. Element(l4, A129) → B129
  140. Element(l4, A130) → B130
  141. Element(l4, A131) → B131
  142. Element(l4, A132) → B132
  143. Element(l4, A133) → B133
  144. Element(l4, A134) → B134
  145. Element(l4, A135) → B135
  146. Element(l4, A136) → B136
  147. Element(l4, A137) → B137
  148. Element(l4, A138) → B138
  149. Element(l4, A139) → B139
  150. Element(l4, A140) → B140
  151. Element(l4, A141) → B141
  152. Element(l4, A142) → B142
  153. Element(l4, A143) → B143
  154. Element(l4, A144) → B144
  155. Element(l4, A145) → B145
  156. Element(l4, A146) → B146
  157. Element(l4, A147) → B147
  158. Element(l4, A148) → B148
  159. Element(l4, A149) → B149
  160. Element(l4, A150) → B150
  161. Element(l4, A151) → B151
  162. Element(l4, A152) → B152
  163. Element(l4, A153) → B153
  164. Element(l4, A154) → B154
  165. Element(l4, A155) → B155
  166. Element(l4, A156) → B156
  167. Element(l4, A157) → B157
  168. Element(l4, A158) → B158
  169. Element(l4, A159) → B159
  170. Element(l4, A160) → B160
  171. Element(l4, A161) → B161
  172. Element(l4, A162) → B162
  173. Element(l4, A163) → B163
  174. Element(l4, A164) → B164
  175. Element(l4, A165) → B165
  176. Element(l4, A166) → B166
  177. Element(l4, A167) → B167
  178. Element(l4, A168) → B168
  179. Element(l4, A169) → B169
  180. Element(l4, A170) → B170
  181. Element(l4, A171) → B171
  182. Element(l4, A172) → B172
  183. Element(l4, A173) → B173
  184. Element(l4, A174) → B174
  185. Element(l4, A175) → B175
  186. Element(l4, A176) → B176
  187. Element(l4, A177) → B177
  188. Element(l4, A178) → B178
  189. Element(l4, A179) → B179
  190. Element(l4, A180) → B180
  191. Element(l4, A181) → B181
  192. Element(l4, A182) → B182
  193. Element(l4, A183) → B183
  194. Element(l4, A184) → B184
  195. Element(l4, A185) → B185
  196. Element(l4, A186) → B186
  197. Element(l4, A187) → B187
  198. Element(l4, A188) → B188
  199. Element(l4, A189) → B189
  200. Element(l4, A190) → B190
  201. Element(l4, A191) → B191
  202. Element(l4, A192) → B192
  203. Element(l4, A193) → B193
  204. Element(l4, A194) → B194
  205. Element(l4, A195) → B195
  206. Element(l4, A196) → B196
  207. Element(l4, A197) → B197
  208. Element(l4, A198) → B198
  209. Element(l4, A199) → B199
  210. Element(l4, A200) → B200
  211. Element(l4, A201) → B201
  212. Element(l4, A202) → B202
  213. Element(l4, A203) → B203
  214. Element(l4, A204) → B204
  215. Element(l4, A205) → B205
  216. Element(l4, A206) → B206
  217. Element(l4, A207) → B207
  218. Element(l4, A208) → B208
  219. Element(l4, A209) → B209
  220. Element(l4, A210) → B210
  221. Element(l4, A211) → B211
  222. Element(l4, A212) → B212
  223. Element(l4, A213) → B213
  224. Element(l4, A214) → B214
  225. Element(l4, A215) → B215
  226. Element(l4, A216) → B216
  227. Element(l4, A217) → B217
  228. Element(l4, A218) → B218
  229. Element(l4, A219) → B219
  230. Element(l4, A220) → B220
  231. Element(l4, A221) → B221
  232. Element(l4, A222) → B222
  233. Element(l4, A223) → B223
  234. Element(l4, A224) → B224
  235. Element(l4, A225) → B225
  236. Element(l4, A226) → B226
  237. Element(l4, A227) → B227
  238. Element(l4, A228) → B228
  239. Element(l4, A229) → B229
  240. Element(l4, A230) → B230
  241. Element(l4, A231) → B231
  242. Element(l4, A232) → B232
  243. Element(l4, A233) → B233
  244. Element(l4, A234) → B234
  245. Element(l4, A235) → B235
  246. Element(l4, A236) → B236
  247. Element(l4, A237) → B237
  248. Element(l4, A238) → B238
  249. Element(l4, A239) → B239
  250. Element(l4, A240) → B240
  251. Element(l4, A241) → B241
  252. Element(l4, A242) → B242
  253. Element(l4, A243) → B243
  254. Element(l4, A244) → B244
  255. Element(l4, A245) → B245
  256. Element(l4, A246) → B246
  257. Element(l4, A247) → B247
  258. Element(l4, A248) → B248
  259. Element(l4, A249) → B249
  260. Element(l4, A250) → B250
  261. Element(l4, A251) → B251
  262. Element(l4, A252) → B252
  263. Element(l4, A253) → B253
  264. Element(l4, A254) → B254
  265. Element(l4, A255) → B255
  266. Element(l4, A256) → B256
  267. Element(l4, A257) → B257
  268. Element(l4, A258) → B258
  269. Element(l4, A259) → B259
  270. Element(l4, A260) → B260
  271. Element(l4, A261) → B261
  272. Element(l4, A262) → B262
  273. Element(l4, A263) → B263
  274. Element(l4, A264) → B264
  275. Element(l4, A265) → B265
  276. Element(l4, A266) → B266
  277. Element(l4, A267) → B267
  278. Element(l4, A268) → B268
  279. Element(l4, A269) → B269
  280. Element(l4, A270) → B270
  281. Element(l4, A271) → B271
  282. Element(l4, A272) → B272
  283. Element(l4, A273) → B273
  284. Element(l4, A274) → B274
  285. Element(l4, A275) → B275
  286. Element(l4, A276) → B276
  287. Element(l4, A277) → B277
  288. Element(l4, A278) → B278
  289. Element(l4, A279) → B279
  290. Element(l4, A280) → B280
  291. Element(l4, A281) → B281
  292. Element(l4, A282) → B282
  293. Element(l4, A283) → B283
  294. Element(l4, A284) → B284
  295. Element(l4, A285) → B285
  296. Element(l4, A286) → B286
  297. Element(l4, A287) → B287
  298. Element(l4, A288) → B288
  299. Element(l4, A289) → B289
  300. Element(l4, A290) → B290
  301. Element(l4, A291) → B291
  302. Element(l4, A292) → B292
  303. Element(l4, A293) → B293
  304. Element(l4, A294) → B294
  305. Element(l4, A295) → B295
  306. Element(l4, A296) → B296
  307. Element(l4, A297) → B297
  308. Element(l4, A298) → B298
  309. Element(l4, A299) → B299
  310. Element(l4, A300) → B300

AI Drawing Prompt Reference

Draw the relative motion轨迹 of two ellipses to showcase the intricate, symmetric patterns formed by their combination. One ellipse moves around another, with its path traced using parameters and vectors, creating two distinct curves. Connect these curves with line segments to form the final pattern, illustrating vector-driven and parametric concepts, and highlighting the beauty of analytic geometry and geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.