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动态三角形点阵 tessellation

动态三角形点阵 tessellation
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2014-02-18 21:01
测试改名0向量变换几何作图
所属专题 MalinC's GeoGebra-book
①本范例演示通过拖动点在三角形上形成动态点阵 tessellation 图形；②涉及核心数学概念包括三角形、向量变换、平移、矩阵运算；③构造要点运用了 GeoGebra 命令：Intersect（交点）、Point（点）、Polygon（多边形）、Vector（向量）、Translate（平移）；④适合高中数学几何教学，用于理解变换与图形组合。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A1
  2. Point(xAxis) → D1
  3. Polygon(A1, D1, 3) → poly1, a, a_1, b, G1
  4. Vector(A1, G1) → u
  5. Vector(A1, D1) → v
  6. Polygon(A1, B1, C1, D1, E1, F1, G1, H1, I1) → J1, K1, L1, M1, N1, O1, P1, Q1, R1, S1
  7. Translate(A1, u) → A2
  8. Translate(B1, u) → B2
  9. Translate(C1, u) → C2
  10. Translate(D1, u) → D2
  11. Translate(E1, u) → E2
  12. Translate(F1, u) → F2
  13. Translate(G1, u) → G2
  14. Translate(H1, u) → H2
  15. Translate(I1, u) → I2
  16. Translate(A2, u) → A3
  17. Translate(B2, u) → B3
  18. Translate(C2, u) → C3
  19. Translate(D2, u) → D3
  20. Translate(E2, u) → E3
  21. Translate(F2, u) → F3
  22. Translate(G2, u) → G3
  23. Translate(H2, u) → H3
  24. Translate(I2, u) → I3
  25. Translate(A3, u) → A4
  26. Translate(B3, u) → B4
  27. Translate(C3, u) → C4
  28. Translate(D3, u) → D4
  29. Translate(E3, u) → E4
  30. Translate(F3, u) → F4
  31. Translate(G3, u) → G4
  32. Translate(H3, u) → H4
  33. Translate(I3, u) → I4
  34. Polygon(A2, B2, C2, D2, E2, F2, G2, H2, I2) → J2, K2, L2, M2, N2, O2, P2, Q2, R2, S2
  35. Polygon(A3, B3, C3, D3, E3, F3, G3, H3, I3) → J3, K3, L3, M3, N3, O3, P3, Q3, R3, S3
  36. Polygon(A4, B4, C4, D4, E4, F4, G4, H4, I4) → J4, K4, L4, M4, N4, O4, P4, Q4, R4, S4
  37. Translate(A1, v) → A5
  38. Translate(B1, v) → B5
  39. Translate(C1, v) → C5
  40. Translate(D1, v) → D5
  41. Translate(E1, v) → E5
  42. Translate(F1, v) → F5
  43. Translate(G1, v) → G5
  44. Translate(H1, v) → H5
  45. Translate(I1, v) → I5
  46. Polygon(A5, B5, C5, D5, E5, F5, G5, H5, I5) → J5, K5, L5, M5, N5, O5, P5, Q5, R5, S5
  47. Translate(A2, v) → A6
  48. Translate(B2, v) → B6
  49. Translate(C2, v) → C6
  50. Translate(D2, v) → D6
  51. Translate(E2, v) → E6
  52. Translate(F2, v) → F6
  53. Translate(G2, v) → G6
  54. Translate(H2, v) → H6
  55. Translate(I2, v) → I6
  56. Translate(A3, v) → A7
  57. Translate(B3, v) → B7
  58. Translate(C3, v) → C7
  59. Translate(D3, v) → D7
  60. Translate(E3, v) → E7
  61. Translate(F3, v) → F7
  62. Translate(G3, v) → G7
  63. Translate(H3, v) → H7
  64. Translate(I3, v) → I7
  65. Translate(A4, v) → A8
  66. Translate(B4, v) → B8
  67. Translate(C4, v) → C8
  68. Translate(D4, v) → D8
  69. Translate(E4, v) → E8
  70. Translate(F4, v) → F8
  71. Translate(G4, v) → G8
  72. Translate(H4, v) → H8
  73. Translate(I4, v) → I8
  74. Polygon(A6, B6, C6, D6, E6, F6, G6, H6, I6) → J6, K6, L6, M6, N6, O6, P6, Q6, R6, S6
  75. Polygon(A7, B7, C7, D7, E7, F7, G7, H7, I7) → J7, K7, L7, M7, N7, O7, P7, Q7, R7, S7
  76. Polygon(A8, B8, C8, D8, E8, F8, G8, H8, I8) → J8, K8, L8, M8, N8, O8, P8, Q8, R8, S8
  77. Translate(A1, Vector[(2 * v)]) → A10
  78. Translate(B1, Vector[(2 * v)]) → B10
  79. Translate(C1, Vector[(2 * v)]) → C10
  80. Translate(D1, Vector[(2 * v)]) → D10
  81. Translate(E1, Vector[(2 * v)]) → E10
  82. Translate(F1, Vector[(2 * v)]) → F10
  83. Translate(G1, Vector[(2 * v)]) → G10
  84. Translate(H1, Vector[(2 * v)]) → H10
  85. Translate(I1, Vector[(2 * v)]) → I10
  86. Polygon(A10, B10, C10, D10, E10, F10, G10, H10, I10) → J10, K10, L10, M10, N10, O10, P10, Q10, R10, S10
  87. Translate(A10, u) → A11
  88. Translate(B10, u) → B11
  89. Translate(C10, u) → C11
  90. Translate(D10, u) → D11
  91. Translate(E10, u) → E11
  92. Translate(F10, u) → F11
  93. Translate(G10, u) → G11
  94. Translate(H10, u) → H11
  95. Translate(I10, u) → I11
  96. Polygon(A11, B11, C11, D11, E11, F11, G11, H11, I11) → J11, K11, L11, M11, N11, O11, P11, Q11, R11, S11
  97. Translate(A11, u) → A12
  98. Translate(B11, u) → B12
  99. Translate(C11, u) → C12
  100. Translate(D11, u) → D12
  101. Translate(E11, u) → E12
  102. Translate(F11, u) → F12
  103. Translate(G11, u) → G12
  104. Translate(H11, u) → H12
  105. Translate(I11, u) → I12
  106. Translate(A12, u) → A13
  107. Translate(B12, u) → B13
  108. Translate(C12, u) → C13
  109. Translate(D12, u) → D13
  110. Translate(E12, u) → E13
  111. Translate(F12, u) → F13
  112. Translate(G12, u) → G13
  113. Translate(H12, u) → H13
  114. Translate(I12, u) → I13
  115. Polygon(A12, B12, C12, D12, E12, F12, G12, H12, I12) → J12, K12, L12, M12, N12, O12, P12, Q12, R12, S12
  116. Polygon(A13, B13, C13, D13, E13, F13, G13, H13, I13) → J13, K13, L13, M13, N13, O13, P13, Q13, R13, S13
  117. Translate(A10, v) → A10'
  118. Translate(B10, v) → B10'
  119. Translate(C10, v) → C10'
  120. Translate(D10, v) → D10'
  121. Translate(E10, v) → E10'
  122. Translate(F10, v) → F10'
  123. Translate(G10, v) → G10'
  124. Translate(H10, v) → H10'
  125. Translate(I10, v) → I10'
  126. Polygon(A10', B10', C10', D10', E10', F10', G10', H10', I10') → J10', a10', b10', c10', d10', e10', f10', g10', h10', i10'
  127. Translate(A11, v) → A11'
  128. Translate(B11, v) → B11'
  129. Translate(C11, v) → C11'
  130. Translate(D11, v) → D11'
  131. Translate(E11, v) → E11'
  132. Translate(F11, v) → F11'
  133. Translate(G11, v) → G11'
  134. Translate(H11, v) → H11'
  135. Translate(I11, v) → I11'
  136. Polygon(A11', B11', C11', D11', E11', F11', G11', H11', I11') → J11', a11', b11', c11', d11', e11', f11', g11', h11', i11'
  137. Translate(A10', v) → A10''
  138. Translate(B10', v) → B10''
  139. Translate(C10', v) → C10''
  140. Translate(D10', v) → D10''
  141. Translate(E10', v) → E10''
  142. Translate(F10', v) → F10''
  143. Translate(G10', v) → G10''
  144. Translate(H10', v) → H10''
  145. Translate(I10', v) → I10''
  146. Polygon(A10'', B10'', C10'', D10'', E10'', F10'', G10'', H10'', I10'') → J10'', a10'', b10'', c10'', d10'', e10'', f10'', g10'', h10'', i10''
  147. Translate(A12, v) → A12'
  148. Translate(B12, v) → B12'
  149. Translate(C12, v) → C12'
  150. Translate(D12, v) → D12'
  151. Translate(E12, v) → E12'
  152. Translate(F12, v) → F12'
  153. Translate(G12, v) → G12'
  154. Translate(H12, v) → H12'
  155. Translate(I12, v) → I12'
  156. Polygon(A12', B12', C12', D12', E12', F12', G12', H12', I12') → J12', a12', b12', c12', d12', e12', f12', g12', h12', i12'
  157. Translate(A3, negv) → A3'
  158. Translate(B3, negv) → B3'
  159. Translate(C3, negv) → C3'
  160. Translate(D3, negv) → D3'
  161. Translate(E3, negv) → E3'
  162. Translate(F3, negv) → F3'
  163. Translate(G3, negv) → G3'
  164. Translate(H3, negv) → H3'
  165. Translate(I3, negv) → I3'
  166. Polygon(A3', B3', C3', D3', E3', F3', G3', H3', I3') → J3', a3', b3', c3', d3', e3', f3', g3', h3', i3'
  167. Translate(A4, negv) → A4'
  168. Translate(B4, negv) → B4'_1
  169. Translate(C4, negv) → C4'_1
  170. Translate(D4, negv) → D4'
  171. Translate(E4, negv) → E4'
  172. Translate(F4, negv) → F4'
  173. Translate(G4, negv) → G4'
  174. Translate(H4, negv) → H4'
  175. Translate(I4, negv) → I4'
  176. Polygon(A4', B4'_1, C4'_1, D4', E4', F4', G4', H4', I4') → J4', a4', b4', c4', d4', e4', f4', g4', h4', i4'
  177. Polygon(D1, I5, H5, A6, C2, B2, A2, F1, E1) → poly2, d1, i5, h5, a6, c2, b2, a2, f1, e1
  178. Polygon(A10, I10, H10, A11, C6, B6, A6, F5, E5) → poly3, a10, i10, h10, a11, c6, b6, a6_1, f5, e5
  179. Polygon(A10', I10', H10', A11', C11, B11, A11, F10, E10) → poly4, a10'_1, i10'_1, h10'_1, a11'_1, c11, b11, a11_1, f10, e10
  180. Polygon(A10'', I10'', H10'', G10'', C11', B11', A11', F10', E10') → poly5, a10''_1, i10''_1, h10''_1, g10''_1, c11'_1, b11'_1, a11'_2, f10'_1, e10'_1
  181. Polygon(A11', I11', H11', A12', C12, B12, A12, F11, E11) → poly6, a11'_3, i11'_1, h11'_1, a12'_1, c12, b12, a12, f11, e11
  182. Polygon(A11, I11, H11, A12, C7, B7, A7, F6, E6) → poly7, a11_2, i11, h11, a12_1, c7, b7, a7, f6, e6
  183. Polygon(A6, I6, H6, A7, C3, B3, D3', F2, E2) → poly8, a6_2, i6, h6, a7_1, c3, b3, d3'_1, f2, e2
  184. Translate(F2, negv) → F2'
  185. Translate(E2, negv) → E2'
  186. Polygon(A2, I2, H2, D3', C3', B3', A3', F2', E2') → poly9, a2_1, i2, h2, d3'_2, c3'_1, b3'_1, a3'_1, f2', e2'
  187. Polygon(D3', I3, H3, D4', C4'_1, B4'_1, A4', F3', E3') → poly10, d3'_3, i3, h3, d4'_1, c4'_1, b4'_1, a4'_1, f3'_1, e3'_1
  188. Polygon(A7, I7, H7, A8, C4, B4, D4', F3, E3) → poly11, a7_2, i7, h7, a8, c4, b4, d4'_2, f3, e3
  189. Polygon(A12, I12, H12, A13, C8, B8, A8, F7, E7) → poly12, a12_2, i12, h12, a13, c8, b8, a8_1, f7, e7
  190. Polygon(A12', I12', H12', G12', C13, B13, A13, F12, E12) → poly13, a12'_2, i12'_1, h12'_1, g12'_1, c13, b13, a13_1, f12, e12
  191. Translate(H11', v) → H11''
  192. Translate(I11', v) → I11''
  193. Polygon(G10'', I11'', H11'', D12', C12', B12', A12', F11', E11') → poly14, g10''_2, i11'', h11'', d12'_1, c12'_1, b12'_1, a12'_3, f11'_1, e11'_1
  194. Translate(C4'_1, u) → C4''
  195. Translate(B4'_1, u) → B4''
  196. Polygon(D4', I4, H4, G4, C4'', B4'', G4', F4', E4') → poly15, d4'_3, i4, h4, g4, c4'', b4'', g4'_1, f4'_1, e4'_1
  197. Translate(C4, u) → C4'
  198. Translate(B4, u) → B4'
  199. Polygon(A8, I8, H8, G8, C4', B4', G4, F4, E4) → poly16, a8_2, i8, h8, g8, c4'_2, b4'_2, g4_1, f4, e4
  200. Translate(C8, u) → C8'
  201. Translate(B8, u) → B8'
  202. Polygon(A13, I13, H13, G13, C8', B8', G8, F8, E8) → poly17, a13_2, i13, h13, g13, c8', b8', g8_1, f8, e8

AI 绘图提示语参考

在坐标平面上，以原点为起点在x轴上取一个可拖动的点，构造一个等边三角形。从原点分别向该点和三角形另一顶点作两个向量。以这两个向量为平移方向，将原三角形及其内部划分出的九个顶点整体平移，形成多排三角形网格。先沿第一个向量平移三次，再沿第二个向量平移一次，每次平移后都将对应顶点连接成多边形，最终得到由多个等边三角形拼接而成的动态镶嵌图案，拖动x轴上的点可改变整体形状。
以上提示语为根据原始绘图命令整理而来，仅供参考。