MathPlotAI Logo MathPlotAI

Butterfly Curve

Butterfly Curve
Loading the math board and drawing, please wait…
2020-03-30 00:49
TransformationParametric EquationGeometric Construction
This example demonstrates the generation process of a butterfly curve. It involves parametric equations and geometric transformations. Using commands like CurveCartesian, Sequence, Line, Rotate, etc., a beautiful butterfly shape can be constructed. Suitable for high school mathematics education and learning about parametric equations and geometric transformations.

GeoGebra Commands & Parameters

  1. CurveCartesian(x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O), t, 0, 1.4) → a
  2. Sequence((x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O)), t, 0, 1.4, b / n) → l1
  3. CurveCartesian(x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O), t, -1.4, 0) → c
  4. CurveCartesian(x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O), t, -2.8, -1.4) → d
  5. CurveCartesian(x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O), t, 1.4, 2.8) → e
  6. Line(O, yAxis) → f
  7. Rotate(l1, g, f) → l1'
  8. Sequence(Segment[O, Element[l1', i]], i, 1, Length[l1']) → l1b
  9. Rotate(a, g, f) → a'
  10. Sequence((x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O)), t, -1.4, 0, b / n) → l2
  11. Rotate(l2, (-g), f) → l2'
  12. Sequence(Segment[O, Element[l2', i]], i, 1, Length[l2']) → l2b
  13. Rotate(c, (-g), f) → c'
  14. Sequence((x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O)), t, -2.8, -1.4, b / n) → l3
  15. Rotate(l3, (-g), f) → l3'
  16. Sequence(Segment[O, Element[l3', i]], i, 1, Length[l3']) → l3b
  17. Rotate(d, (-g), f) → d'
  18. Sequence((x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O)), t, 1.4, 2.8, b / n) → l4
  19. Rotate(l4, g, f) → l4'
  20. Sequence(Segment[O, Element[l4', i]], i, 1, Length[l4']) → l4b
  21. Rotate(e, g, f) → e'
  22. Sequence((x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O)), t, 2.8, 3.2, b / n) → l5
  23. Sequence((x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O)), t, -3.2, -2.8, b / n) → l6
  24. Sequence(Segment[O, Element[l5, i]], i, 1, Length[l5]) → l7
  25. Sequence(Segment[O, Element[l6, i]], i, 1, Length[l6]) → l8
  26. CurveCartesian(x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O), t, -3.2, -2.8) → h
  27. CurveCartesian(x(O) + (sin(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), y(O) + (cos(t) * (ℯ^(cos(t)) - (2 * cos((4 * t))) - sin(t / 12)^(5))), z(O), t, 2.8, 3.2) → i
  28. Element(l1b, A1) → B1
  29. Element(l1b, A2) → B2
  30. Element(l1b, A3) → B3
  31. Element(l1b, A4) → B4
  32. Element(l1b, A5) → B5
  33. Element(l1b, A6) → B6
  34. Element(l1b, A7) → B7
  35. Element(l1b, A8) → B8
  36. Element(l1b, A9) → B9
  37. Element(l1b, A10) → B10
  38. Element(l1b, A11) → B11
  39. Element(l1b, A12) → B12
  40. Element(l1b, A13) → B13
  41. Element(l1b, A14) → B14
  42. Element(l1b, A15) → B15
  43. Element(l1b, A16) → B16
  44. Element(l1b, A17) → B17
  45. Element(l1b, A18) → B18
  46. Element(l1b, A19) → B19
  47. Element(l1b, A20) → B20
  48. Element(l1b, A21) → B21
  49. Element(l1b, A22) → B22
  50. Element(l1b, A23) → B23
  51. Element(l1b, A24) → B24
  52. Element(l1b, A25) → B25
  53. Element(l1b, A26) → B26
  54. Element(l1b, A27) → B27
  55. Element(l1b, A28) → B28
  56. Element(l1b, A29) → B29
  57. Element(l1b, A30) → B30
  58. Element(l1b, A31) → B31
  59. Element(l1b, A32) → B32
  60. Element(l1b, A33) → B33
  61. Element(l1b, A34) → B34
  62. Element(l1b, A35) → B35
  63. Element(l2b, A1) → C1
  64. Element(l2b, A2) → C2
  65. Element(l2b, A3) → C3
  66. Element(l2b, A4) → C4
  67. Element(l2b, A5) → C5
  68. Element(l2b, A6) → C6
  69. Element(l2b, A7) → C7
  70. Element(l2b, A8) → C8
  71. Element(l2b, A9) → C9
  72. Element(l2b, A10) → C10
  73. Element(l2b, A11) → C11
  74. Element(l2b, A12) → C12
  75. Element(l2b, A13) → C13
  76. Element(l2b, A14) → C14
  77. Element(l2b, A15) → C15
  78. Element(l2b, A16) → C16
  79. Element(l2b, A17) → C17
  80. Element(l2b, A18) → C18
  81. Element(l2b, A19) → C19
  82. Element(l2b, A20) → C20
  83. Element(l2b, A21) → C21
  84. Element(l2b, A22) → C22
  85. Element(l2b, A23) → C23
  86. Element(l2b, A24) → C24
  87. Element(l2b, A25) → C25
  88. Element(l2b, A26) → C26
  89. Element(l2b, A27) → C27
  90. Element(l2b, A28) → C28
  91. Element(l2b, A29) → C29
  92. Element(l2b, A30) → C30
  93. Element(l2b, A31) → C31
  94. Element(l2b, A32) → C32
  95. Element(l2b, A33) → C33
  96. Element(l2b, A34) → C34
  97. Element(l2b, A35) → C35
  98. Element(l2b, A36) → C36
  99. Element(l2b, A37) → C37
  100. Element(l2b, A38) → C38
  101. Element(l2b, A39) → C39
  102. Element(l2b, A40) → C40
  103. Element(l3b, A1) → D16
  104. Element(l3b, A2) → D17
  105. Element(l3b, A3) → D18
  106. Element(l3b, A4) → D19
  107. Element(l3b, A5) → D20
  108. Element(l3b, A6) → D21
  109. Element(l3b, A7) → D22
  110. Element(l3b, A8) → D23
  111. Element(l3b, A9) → D24
  112. Element(l3b, A10) → D25
  113. Element(l3b, A11) → D26
  114. Element(l3b, A12) → D27
  115. Element(l3b, A13) → D28
  116. Element(l3b, A14) → D29
  117. Element(l3b, A15) → D30
  118. Element(l3b, A16) → D31
  119. Element(l3b, A17) → D32
  120. Element(l3b, A18) → D33
  121. Element(l3b, A19) → D34
  122. Element(l3b, A20) → D35
  123. Element(l3b, A21) → D36
  124. Element(l3b, A22) → D37
  125. Element(l3b, A23) → D38
  126. Element(l3b, A24) → D39
  127. Element(l3b, A25) → D40
  128. Element(l4b, A1) → E16
  129. Element(l4b, A2) → E17
  130. Element(l4b, A3) → E18
  131. Element(l4b, A4) → E19
  132. Element(l4b, A5) → E20
  133. Element(l4b, A6) → E21
  134. Element(l4b, A7) → E22
  135. Element(l4b, A8) → E23
  136. Element(l4b, A9) → E24
  137. Element(l4b, A10) → E25
  138. Element(l4b, A11) → E26
  139. Element(l4b, A12) → E27
  140. Element(l4b, A13) → E28
  141. Element(l4b, A14) → E29
  142. Element(l4b, A15) → E30
  143. Element(l4b, A16) → E31
  144. Element(l4b, A17) → E32
  145. Element(l4b, A18) → E33
  146. Element(l4b, A19) → E34
  147. Element(l4b, A20) → E35
  148. Element(l4b, A21) → E36
  149. Element(l4b, A22) → E37
  150. Element(l4b, A23) → E38
  151. Element(l4b, A24) → E39
  152. Element(l4b, A25) → E40
  153. Element(l3b, A26) → D41
  154. Element(l4b, A26) → E41
  155. Element(l3b, A27) → D42
  156. Element(l4b, A27) → E42
  157. Element(l3b, A28) → D43
  158. Element(l4b, A28) → E43
  159. Element(l3b, A29) → D44
  160. Element(l4b, A29) → E44
  161. Element(l3b, A30) → D45
  162. Element(l4b, A30) → E45
  163. Element(l3b, A31) → D46
  164. Element(l4b, A31) → E46
  165. Element(l3b, A32) → D47
  166. Element(l4b, A32) → E47
  167. Element(l1b, A36) → B36
  168. Element(l1b, A37) → B37
  169. Element(l1b, A38) → B38
  170. Element(l1b, A39) → B39
  171. Element(l1b, A40) → B40
  172. Element(l3b, A33) → D48
  173. Element(l4b, A33) → E48
  174. Element(l3b, A34) → D49
  175. Element(l4b, A34) → E49
  176. Element(l3b, A35) → D50
  177. Element(l4b, A35) → E50
  178. Element(l2b, A41) → C41
  179. Element(l1b, A41) → B41
  180. Element(l2b, A42) → C42
  181. Element(l1b, A42) → B42
  182. Element(l2b, A43) → C43
  183. Element(l1b, A43) → B43
  184. Element(l2b, A44) → C44
  185. Element(l1b, A44) → B44
  186. Element(l2b, A45) → C45
  187. Element(l1b, A45) → B45
  188. Element(l2b, A46) → C46
  189. Element(l1b, A46) → B46
  190. Element(l2b, A47) → C47
  191. Element(l1b, A47) → B47
  192. Element(l2b, A48) → C48
  193. Element(l1b, A48) → B48
  194. Element(l2b, A49) → C49
  195. Element(l1b, A49) → B49
  196. Element(l3b, A36) → D51
  197. Element(l4b, A36) → E51
  198. Element(l3b, A37) → D52
  199. Element(l4b, A37) → E52
  200. Element(l3b, A38) → D53
  201. Element(l4b, A38) → E53
  202. Element(l3b, A39) → D54
  203. Element(l4b, A39) → E54
  204. Element(l3b, A40) → D55
  205. Element(l4b, A40) → E55
  206. Element(l3b, A41) → D56
  207. Element(l4b, A41) → E56
  208. Element(l4b, A42) → E57
  209. Element(l4b, A43) → E58
  210. Element(l4b, A44) → E59
  211. Element(l4b, A45) → E60
  212. Element(l4b, A46) → E61
  213. Element(l4b, A47) → E62
  214. Element(l4b, A48) → E63
  215. Element(l4b, A49) → E64
  216. Element(l4b, A50) → E65
  217. Element(l4b, A51) → E66
  218. Element(l3b, A42) → D57
  219. Element(l3b, A43) → D58
  220. Element(l3b, A44) → D59
  221. Element(l3b, A45) → D60
  222. Element(l3b, A46) → D61
  223. Element(l3b, A47) → D62
  224. Element(l3b, A48) → D63
  225. Element(l3b, A49) → D64
  226. Element(l3b, A50) → D65
  227. Element(l3b, A51) → D66
  228. Element(l2b, A50) → C50
  229. Element(l1b, A50) → B50
  230. Element(l2b, A51) → C51
  231. Element(l1b, A51) → B51
  232. Element(l2b, A52) → C52
  233. Element(l1b, A52) → B52
  234. Element(l2b, A53) → C53
  235. Element(l1b, A53) → B53
  236. Element(l2b, A54) → C54
  237. Element(l1b, A54) → B54
  238. Element(l2b, A55) → C55
  239. Element(l1b, A55) → B55
  240. Element(l2b, A56) → C56
  241. Element(l1b, A56) → B56
  242. Element(l2b, A57) → C57
  243. Element(l1b, A57) → B57
  244. Element(l2b, A58) → C58
  245. Element(l1b, A58) → B58
  246. Element(l2b, A59) → C59
  247. Element(l1b, A59) → B59

AI Drawing Prompt Reference

Draw a butterfly curve using the parametric equations x(t) = sin(t) * (ℯ^(cos(t)) - 2 * cos(4 * t) - sin(t / 12)^5) and y(t) = cos(t) * (ℯ^(cos(t)) - 2 * cos(4 * t) - sin(t / 12)^5) from t = 0 to t = 1.4, then from t = -1.4 to t = 0, from t = -2.8 to t = -1.4, from t = 1.4 to t = 2.8, and finally from t = 2.8 to t = 3.2 and t = -3.2 to t = -2.8. Use rotation and sequence to generate multiple segments to form the complete butterfly shape.
The prompt above was compiled from the original drawing commands and is for reference only.