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椭圆镶嵌图案

椭圆镶嵌图案
几何画板及绘图内容加载中，请稍等……
2025-08-19 01:45
所属专题 Teselaciones regulares
这个范例展示了椭圆镶嵌图案的构造过程。它涉及圆、三角函数和变换等核心数学概念。通过使用旋转、镜像、序列等GeoGebra命令，可以构造出复杂的几何图案。适合高中数学教学和学习。

GeoGebra 命令及参数

  1. Rotate(A, ((-pi)) / 2, OO) → G1
  2. Mirror(L2A, Line[(0, 0, (-b)), (0, 1, (-b))]) → L2B
  3. Mirror(L3B, Line[(0, 0, (-b)), (0, 1, (-b))]) → L3C
  4. Sequence(Rotate[L3C, (k * pi / 2), OO], k, 0, 3) → lista3B'
  5. Sequence(Rotate[L3B, (k * pi / 2), OO], k, 0, 3) → lista3B
  6. Sequence(Rotate[L3A, (k * pi / 2), OO], k, 0, 3) → lista3A
  7. Sequence(Rotate[L2B, (k * pi / 2), OO], k, 0, 3) → lista2'
  8. Sequence(Rotate[L4, (k * pi / 2), OO], k, 0, 3) → lista4
  9. Sequence(Rotate[L2A, (k * pi / 2), OO], k, 0, 3) → lista2
  10. Sequence(Rotate[L1, (k * pi / 4), OO], k, 0, 7) → lista1
  11. Line(((-b), (-b), (-b)), ((-b), b, (-b))) → r1
  12. Rotate(mos1, ((-pi)) / 2, r1) → mos2
  13. Line(((-b), (-b), (-b)), (b, (-b), (-b))) → r2
  14. Rotate(mos1, pi / 2, r2) → mos3
  15. Sequence(Polygon[mos1(1, k)], k, 1, Length[mos1(1)]) → p1azul
  16. Sequence(Polygon[mos2(1, k)], k, 1, Length[mos2(1)]) → p2azul
  17. Sequence(Polygon[mos3(1, k)], k, 1, Length[mos3(1)]) → p3azul
  18. Translate(p1azul, Vector[(0, 0, (2 * b))]) → p1azul'
  19. Translate(p2azul, Vector[((2 * b), 0, 0)]) → p2azul'
  20. Translate(p3azul, Vector[(0, (2 * b), 0)]) → p3azul'
  21. Sequence(Polygon[mos1(2, k)], k, 1, Length[mos1(2)]) → p1naranjaA
  22. Sequence(Polygon[mos2(2, k)], k, 1, Length[mos2(2)]) → p2naranjaA
  23. Sequence(Polygon[mos3(2, k)], k, 1, Length[mos3(2)]) → p3naranjaA
  24. Translate(p1naranjaA, Vector[(0, 0, (2 * b))]) → p1naranjaA'
  25. Translate(p2naranjaA, Vector[((2 * b), 0, 0)]) → p2naranjaA'
  26. Translate(p3naranjaA, Vector[(0, (2 * b), 0)]) → p3naranjaA'
  27. Sequence(Polygon[mos1(3, k)], k, 1, Length[mos1(3)]) → p1naranjaB
  28. Translate(p1naranjaB, Vector[(0, 0, (2 * b))]) → p1naranjaB'
  29. Sequence(Polygon[mos2(3, k)], k, 1, Length[mos2(3)]) → p2naranjaB
  30. Translate(p2naranjaB, Vector[((2 * b), 0, 0)]) → p2naranjaB'
  31. Sequence(Polygon[mos3(3, k)], k, 1, Length[mos3(3)]) → p3naranjaB
  32. Translate(p3naranjaB, Vector[(0, (2 * b), 0)]) → p3naranjaB'
  33. Sequence(Polygon[mos1(4, k)], k, 1, Length[mos1(4)]) → p1lilaA
  34. Sequence(Polygon[mos1(5, k)], k, 1, Length[mos1(5)]) → p1lilaB
  35. Sequence(Polygon[mos1(6, k)], k, 1, Length[mos1(6)]) → p1lilaC
  36. Sequence(Polygon[mos1(7, k)], k, 1, Length[mos1(7)]) → p1verde
  37. Translate(p1lilaA, Vector[(0, 0, (2 * b))]) → p1lilaA'
  38. Translate(p1lilaB, Vector[(0, 0, (2 * b))]) → p1lilaB'
  39. Translate(p1lilaC, Vector[(0, 0, (2 * b))]) → p1lilaC'
  40. Sequence(Polygon[mos2(4, k)], k, 1, Length[mos2(4)]) → p2lilaA
  41. Sequence(Polygon[mos2(5, k)], k, 1, Length[mos2(5)]) → p2lilaB
  42. Sequence(Polygon[mos2(6, k)], k, 1, Length[mos2(6)]) → p2lilaC
  43. Translate(p2lilaA, Vector[((2 * b), 0, 0)]) → p2lilaA'
  44. Translate(p2lilaB, Vector[((2 * b), 0, 0)]) → p2lilaB'
  45. Translate(p2lilaC, Vector[((2 * b), 0, 0)]) → p2lilaC'
  46. Sequence(Polygon[mos3(4, k)], k, 1, Length[mos3(4)]) → p3lilaA
  47. Sequence(Polygon[mos3(5, k)], k, 1, Length[mos3(5)]) → p3lilaB
  48. Sequence(Polygon[mos3(6, k)], k, 1, Length[mos3(6)]) → p3lilaC
  49. Translate(p3lilaA, Vector[(0, (2 * b), 0)]) → p3lilaA'
  50. Translate(p3lilaB, Vector[(0, (2 * b), 0)]) → p3lilaB'
  51. Translate(p3lilaC, Vector[(0, (2 * b), 0)]) → p3lilaC'
  52. Translate(p1verde, Vector[(0, 0, (2 * b))]) → p1verde'
  53. Sequence(Polygon[mos2(7, k)], k, 1, Length[mos2(7)]) → p2verde
  54. Translate(p2verde, Vector[((2 * b), 0, 0)]) → p2verde'
  55. Sequence(Polygon[mos3(7, k)], k, 1, Length[mos3(7)]) → p3verde
  56. Translate(p3verde, Vector[(0, (2 * b), 0)]) → p3verde'
  57. Sequence(Segment[segExtr0(k, 1), segExtr0(k, 2)], k, 1, Length[segExtr0]) → negro01
  58. Sequence(Rotate[negro01, (k * pi / 2), OO], k, 0, 3) → negro1
  59. Translate(negro1, Vector[(0, 0, (2 * b))]) → negro1'
  60. Rotate(negro1, ((-pi)) / 2, Line[((-b), (-b), (-b)), ((-b), b, (-b))]) → negro2
  61. Translate(negro2, Vector[((2 * b), 0, 0)]) → negro2'
  62. Rotate(negro1, pi / 2, Line[((-b), (-b), (-b)), (b, (-b), (-b))]) → negro3
  63. Translate(negro3, Vector[(0, (2 * b), 0)]) → negro3'
  64. Rotate(cara1, ((-pi)) / 2, Line[((-b), (-b), (-b)), ((-b), b, (-b))]) → cara2
  65. Rotate(cara1, pi / 2, Line[((-b), (-b), (-b)), (b, (-b), (-b))]) → cara3
  66. Translate(cara1, Vector[(0, 0, (2 * b))]) → cara1'
  67. Translate(cara2, Vector[((2 * b), 0, 0)]) → cara2'
  68. Translate(cara3, Vector[(0, (2 * b), 0)]) → cara3'
  69. Sphere(O, r) → a
  70. Join(Join[Sequence[Rotate[segExtr0, (k * pi / 2), OO], k, 0, 3]]) → segExtr1
  71. Vector((0, 0, (2 * b))) → v1
  72. Translate(segExtr1, v1) → segExtr1'
  73. Sequence(Ray[O, segExtr1'(k)], k, 1, Length[segExtr1']) → rayos
  74. Sequence(Intersect[Element[rayos, k], a], k, 1, Length[rayos]) → inter1'
  75. Sequence(CircumcircleArc[inter1'(k), Intersect[Ray[O, (inter1'(k) + inter1'(k + 1)) / 2], a], inter1'(k + 1)], k, 1, Length[inter1'], 2) → arc1'
  76. Translate(Rotate[arc1', pi / 2, Line[((-b), (-b), (-b)), ((-b), b, (-b))]], Vector[(0, 0, (2 * b))]) → arc2
  77. Mirror(arc2, Plane[yAxis, zAxis]) → arc2'
  78. Translate(Rotate[arc1', ((-pi)) / 2, Line[((-b), (-b), (-b)), (b, (-b), (-b))]], Vector[(0, 0, (2 * b))]) → arc3
  79. Mirror(arc3, Plane[xAxis, zAxis]) → arc3'
  80. Mirror(arc1', xOyPlane) → arc1
  81. Sequence(Segment[Vcubo2(k, 1), Vcubo2(k, 2)], k, 1, 12) → segCubo
  82. Sequence(Segment[Vdode2(k, 1), Vdode2(k, 2)], k, 1, 30) → segDode
  83. Sequence(Segment[Vico2(k, 1), Vico2(k, 2)], k, 1, 30) → segIco
  84. Sequence(Segment[Vocta2(k, 1), Vocta2(k, 2)], k, 1, 12) → segOcta
  85. Sequence(Segment[Vtetra2(k, 1), Vtetra2(k, 2)], k, 1, 6) → segTetra
  86. Sequence(CircleArc[(0, 0, 0), Vtetra2(k, 1), Vtetra2(k, 2)], k, 1, 6) → arcosTetra
  87. Sequence(CircleArc[(0, 0, 0), Vocta2(k, 1), Vocta2(k, 2)], k, 1, 12) → arcosOcta
  88. Sequence(CircleArc[(0, 0, 0), Vico2(k, 1), Vico2(k, 2)], k, 1, 30) → arcosIco
  89. Sequence(CircleArc[(0, 0, 0), Vdode2(k, 1), Vdode2(k, 2)], k, 1, 30) → arcosDode
  90. Sequence(CircleArc[(0, 0, 0), Vcubo2(k, 1), Vcubo2(k, 2)], k, 1, 12) → arcosCubo
  91. Sphere(O, r) → a'
  92. Vector(((2 * b), 0, 0)) → v2
  93. Vector((0, (2 * b), 0)) → v3
  94. Translate(mos1, v1) → mos1'
  95. Translate(mos2, v2) → mos2'
  96. Translate(mos3, v3) → mos3'
  97. Length(L) → n
  98. Mirror(inter1', xOyPlane) → inter1
  99. Translate(Rotate[inter1', pi / 2, Line[((-b), (-b), (-b)), ((-b), b, (-b))]], Vector[(0, 0, (2 * b))]) → inter2
  100. Translate(Rotate[inter1', ((-pi)) / 2, Line[((-b), (-b), (-b)), (b, (-b), (-b))]], Vector[(0, 0, (2 * b))]) → inter3
  101. Mirror(inter2, Plane[yAxis, zAxis]) → inter2'
  102. Mirror(inter3, Plane[xAxis, zAxis]) → inter3'
  103. Sphere(O, r) → aa
  104. PolyLine(L1(6), L1(1), L1(2)) → poligonal01A1
  105. PolyLine(L1(5), L1(4), L1(3)) → poligonal01A2
  106. Point(poligonal01A1, pinta) → P01A1
  107. Point(poligonal01A2, pinta) → P01A2
  108. Intersect(Ray[O, P01A1], a) → Q01A1,
  109. Intersect(Ray[O, P01A2], a) → Q01A2,
  110. Intersect(Ray[O, P01AM], a) → Q01AM,
  111. CircumcircleArc(Q01A1, Q01AM, Q01A2) → seg01arc0
  112. PolyLine(Sequence[Point[seg01arc0, k], k, 0, 1, 0.1]) → seg01arc
  113. Sequence(Rotate[seg01arc, (k * pi / 4)], k, 0, 7) → seg01A1
  114. Translate(Rotate[seg01A1, ((-pi)) / 2, r1], v2) → seg01A2
  115. Translate(Rotate[seg01A1, pi / 2, r2], v3) → seg01A3
  116. Rotate(seg01A2, pi) → seg01A2'
  117. Rotate(seg01A3, pi) → seg01A3'
  118. Rotate(seg01A1, pi, xAxis) → seg01A1'
  119. Rotate(segExtr1, ((-pi)) / 2, r1) → segExtr2
  120. Rotate(segExtr1, pi / 2, r2) → segExtr3
  121. Translate(segExtr2, v2) → segExtr2'
  122. Translate(segExtr3, v3) → segExtr3'
  123. PolyLine(L2A(4), L2A(3), L2A(2)) → poligonal02A1
  124. PolyLine(L2A(5), L2A(1)) → poligonal02A2
  125. PolyLine(L2B(4), L2B(3), L2B(2)) → poligonal02B1
  126. PolyLine(L2B(5), L2B(1)) → poligonal02B2
  127. Point(poligonal02A1, pinta) → P02A1
  128. Point(poligonal02A2, pinta) → P02A2
  129. Intersect(Ray[O, P02A1], a) → Q02A1,
  130. Intersect(Ray[O, P02A2], a) → Q02A2,
  131. Intersect(Ray[O, P02AM], a) → Q02AM,
  132. CircumcircleArc(Q02A1, Q02AM, Q02A2) → seg02Aarc0
  133. PolyLine(Sequence[Point[seg02Aarc0, k], k, 0, 1, 0.1]) → seg02Aarc
  134. Point(poligonal02B1, pinta) → P02B1
  135. Point(poligonal02B2, pinta) → P02B2
  136. Intersect(Ray[O, P02B1], a) → Q02B1,
  137. Intersect(Ray[O, P02B2], a) → Q02B2,
  138. Intersect(Ray[O, P02BM], a) → Q02BM,
  139. CircumcircleArc(Q02B1, Q02BM, Q02B2) → seg02Barc0
  140. PolyLine(Sequence[Point[seg02Barc0, k], k, 0, 1, 0.1]) → seg02Barc
  141. Sequence(Rotate[seg02Aarc, (k * pi / 2)], k, 0, 3) → seg02A1
  142. Translate(Rotate[seg02A1, ((-pi)) / 2, r1], v2) → seg02A2
  143. Translate(Rotate[seg02A1, pi / 2, r2], v3) → seg02A3
  144. Rotate(seg02A1, pi, xAxis) → seg02A1'
  145. Rotate(seg02A2, pi) → seg02A2'
  146. Rotate(seg02A3, pi) → seg02A3'
  147. Sequence(Rotate[seg02Barc, (k * pi / 2)], k, 0, 3) → seg02B1
  148. Translate(Rotate[seg02B1, ((-pi)) / 2, r1], v2) → seg02B2
  149. Translate(Rotate[seg02B1, pi / 2, r2], v3) → seg02B3
  150. Rotate(seg02B1, pi, xAxis) → seg02B1'
  151. Rotate(seg02B2, pi) → seg02B2'
  152. Rotate(seg02B3, pi) → seg02B3'
  153. PolyLine(L3A(1), L3A(2), L3A(3), L3A(4)) → poligonal03A1
  154. PolyLine(L3A(1), L3A(6), L3A(5), L3A(4)) → poligonal03A2
  155. Point(poligonal03A1, pinta) → P03A1
  156. Point(poligonal03A2, pinta) → P03A2
  157. Intersect(Ray[O, P03A1], a) → Q03A1,
  158. Intersect(Ray[O, P03A2], a) → Q03A2,
  159. Intersect(Ray[O, P03AM], a) → Q03AM,
  160. CircumcircleArc(Q03A1, Q03AM, Q03A2) → seg03Aarc0
  161. PolyLine(Sequence[Point[seg03Aarc0, k], k, 0, 1, 0.1]) → seg03Aarc
  162. Sequence(Rotate[seg03Aarc, (k * pi / 2)], k, 0, 3) → seg03A1
  163. Translate(Rotate[seg03A1, ((-pi)) / 2, r1], v2) → seg03A2
  164. Translate(Rotate[seg03A1, pi / 2, r2], v3) → seg03A3
  165. Rotate(seg03A1, pi, xAxis) → seg03A1'
  166. Rotate(seg03A2, pi) → seg03A2'
  167. Rotate(seg03A3, pi) → seg03A3'
  168. PolyLine(L3B(1), L3B(4)) → poligonal03B1
  169. PolyLine(L3B(2), L3B(3)) → poligonal03B2
  170. Point(poligonal03B1, pinta) → P03B1
  171. Point(poligonal03B2, pinta) → P03B2
  172. Intersect(Ray[O, P03B1], a) → Q03B1,
  173. Intersect(Ray[O, P03B2], a) → Q03B2,
  174. Intersect(Ray[O, P03BM], a) → Q03BM,
  175. CircumcircleArc(Q03B1, Q03BM, Q03B2) → seg03Barc0
  176. PolyLine(Sequence[Point[seg03Barc0, k], k, 0, 1, 0.1]) → seg03Barc
  177. Sequence(Rotate[seg03Barc, (k * pi / 2)], k, 0, 3) → seg03B1
  178. Translate(Rotate[seg03B1, ((-pi)) / 2, r1], v2) → seg03B2
  179. Translate(Rotate[seg03B1, pi / 2, r2], v3) → seg03B3
  180. Rotate(seg03B1, pi, xAxis) → seg03B1'
  181. Rotate(seg03B2, pi) → seg03B2'
  182. Rotate(seg03B3, pi) → seg03B3'
  183. PolyLine(L3C(1), L3C(4)) → poligonal03C1
  184. PolyLine(L3C(2), L3C(3)) → poligonal03C2
  185. Point(poligonal03C1, pinta) → P03C1
  186. Point(poligonal03C2, pinta) → P03C2
  187. Intersect(Ray[O, P03C1], a) → Q03C1,
  188. Intersect(Ray[O, P03C2], a) → Q03C2,
  189. Intersect(Ray[O, P03CM], a) → Q03CM,
  190. CircumcircleArc(Q03C1, Q03CM, Q03C2) → seg03Carc0
  191. PolyLine(Sequence[Point[seg03Carc0, k], k, 0, 1, 0.1]) → seg03Carc
  192. Sequence(Rotate[seg03Carc, (k * pi / 2)], k, 0, 3) → seg03C1
  193. Translate(Rotate[seg03C1, ((-pi)) / 2, r1], v2) → seg03C2
  194. Translate(Rotate[seg03C1, pi / 2, r2], v3) → seg03C3
  195. Rotate(seg03C1, pi, xAxis) → seg03C1'
  196. Rotate(seg03C2, pi) → seg03C2'
  197. Rotate(seg03C3, pi) → seg03C3'
  198. PolyLine(L4(1), L4(2), L4(3), L4(4)) → poligonal04A1
  199. PolyLine(L4(7), L4(6), L4(5), L4(4)) → poligonal04A2
  200. Point(poligonal04A1, pinta) → P04A1
  201. Point(poligonal04A2, pinta) → P04A2
  202. Intersect(Ray[O, P04A1], a) → Q04A1,
  203. Intersect(Ray[O, P04A2], a) → Q04A2,
  204. Intersect(Ray[O, P04AM], a) → Q04AM,
  205. CircumcircleArc(Q04A1, Q04AM, Q04A2) → seg04arc0
  206. PolyLine(Sequence[Point[seg04arc0, k], k, 0, 1, 0.1]) → seg04arc
  207. Sequence(Rotate[seg04arc, (k * pi / 2)], k, 0, 3) → seg04A1
  208. Translate(Rotate[seg04A1, ((-pi)) / 2, r1], v2) → seg04A2
  209. Translate(Rotate[seg04A1, pi / 2, r2], v3) → seg04A3
  210. Rotate(seg04A1, pi, xAxis) → seg04A1'
  211. Rotate(seg04A2, pi) → seg04A2'
  212. Rotate(seg04A3, pi) → seg04A3'