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立方体展开图创建

立方体展开图创建
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2016-12-24 13:09
多边形立体几何几何作图
本范例演示如何通过 GeoGebra 创建立方体的展开图。涉及的核心数学概念包括立体几何、多边形构造和几何变换。作图要点包括使用 Polygon 命令绘制多边形，Rotate 命令进行旋转，Vector 命令创建向量，Midpoint 命令确定中点，OrthogonalLine 命令绘制垂线，Plane 命令定义平面，Mirror 命令进行镜像，以及 Segment 命令绘制线段。适合初中及以上学生学习立体几何和几何作图。

GeoGebra 命令及参数

  1. Polygon(A, B, C, D) → poly1, a, b, c, d
  2. Rotate(Polygon[B, E, F, C], (-α), b) → poly2
  3. Rotate(C, (-α), b) → C'
  4. Rotate(F, (-α), b) → F'
  5. Rotate(E, (-α), b) → E'
  6. Rotate(F, (-α), b) → F'_1
  7. Rotate(B, (-α), b) → B'
  8. Rotate(E, (-α), b) → E'_1
  9. Vector(C, Rotate[F, 90°, C]) → u
  10. Rotate(Polygon[A, B, B - u, A - u], (-α), a) → poly5
  11. Midpoint(A, C') → G
  12. OrthogonalLine(G, poly1) → l
  13. Rotate(poly5, 180°, l) → poly5'
  14. Rotate(poly2, 180°, l) → poly2'
  15. Midpoint(B', F'_1) → H
  16. OrthogonalLine(H, poly2) → m
  17. Rotate(B - u, (-α), a) → I
  18. Midpoint(A, I) → J
  19. OrthogonalLine(J, poly5) → n
  20. Rotate(poly1, (-(180°)), n) → poly1'
  21. Rotate(poly2, (-(180°)), n) → poly2'_1
  22. Rotate(poly5', (-(180°)), n) → poly5''
  23. Vector(B, Rotate[A, 90°, B]) → v
  24. Rotate(Rotate[Polygon[B, E, E + v, B + v], (-α), Segment[B, E]], (-α), b) → poly6
  25. Rotate(poly6, 180°, m) → poly6'
  26. Rotate(poly1, 180°, m) → poly1'_1
  27. Rotate(A, 180°, m) → A'
  28. Midpoint(E'_1, A') → K
  29. OrthogonalLine(K, poly1'_1) → i
  30. Rotate(poly2, 180°, i) → poly2'_2
  31. Vector(A, Rotate[D, -90°, A]) → w
  32. Rotate(Rotate[Rotate[Polygon[E, E + w, E + w + v, E + v], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → poly3
  33. Rotate(Rotate[Rotate[Rotate[Polygon[E + v, E + v + w, E + v + w - u, E + v - u], (-α), Segment[E + v, E + v + w]], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → poly4
  34. Plane(l, m) → e
  35. Mirror(poly3, e) → poly3'
  36. Mirror(poly4, e) → poly4'
  37. Rotate(poly6', 180°, l) → poly6''
  38. Rotate(poly6, 180°, l) → poly6'_1
  39. Rotate(poly3, 180°, l) → poly3'_2
  40. Rotate(poly1', 180°, l) → poly1''
  41. Rotate(poly5'', 180°, l) → poly5'''
  42. Rotate(Rotate[Rotate[E + w + v, α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → L
  43. Midpoint(E'_1, L) → M
  44. OrthogonalLine(M, poly3) → f
  45. Rotate(poly6, (-(180°)), f) → poly6'_2
  46. Rotate(poly6'_2, 180°, l) → poly7
  47. Mirror(poly6'_2, e) → poly8
  48. Rotate(poly8, 180°, l) → poly8'
  49. Rotate(poly3', 180°, l) → poly3''
  50. Mirror(poly2'_1, e) → poly9
  51. Rotate(poly1'_1, 180°, l) → poly10
  52. Segment(C', F'_1) → g
  53. Segment(F'_1, E'_1) → h
  54. Segment(B', E'_1) → j
  55. Rotate(poly6, 180°, i) → poly6'_3
  56. Rotate(poly6', 180°, i) → poly6''_1
  57. Rotate(poly5, 180°, m) → poly5'_1
  58. Rotate(poly5', 180°, m) → poly5''_1
  59. Rotate(poly5'_1, 180°, l) → poly11
  60. Rotate(poly5''_1, 180°, l) → poly12
  61. Rotate(Rotate[Rotate[Rotate[Rotate[Polygon[O, N, N + B - A, O + B - A], α, Segment[O, N]], (-α), Segment[E + v, E + v + w]], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → poly13
  62. Mirror(poly13, e) → poly13'
  63. Rotate(poly6', (-(180°)), n) → poly6''_2
  64. Mirror(poly6''_2, e) → poly14
  65. Rotate(F'_1, 180°, i) → P
  66. Rotate(E'_1, 180°, i) → Q
  67. Segment(P, Q) → h'
  68. Rotate(A, 180°, m) → A'_1
  69. Rotate(B, 180°, m) → B'_1
  70. Segment(A'_1, B'_1) → a'
  71. Rotate(C, 180°, m) → C'_1
  72. Rotate(D, 180°, m) → D'
  73. Segment(C'_1, D') → c'
  74. Rotate(B', 180°, i) → B''
  75. Rotate(E'_1, 180°, i) → R
  76. Segment(B'', R) → j'
  77. Rotate(C', 180°, i) → C''
  78. Rotate(F'_1, 180°, i) → S
  79. Segment(C'', S) → g'
  80. Rotate(C', 180°, l) → C''_1
  81. Rotate(F'_1, 180°, l) → T
  82. Segment(C''_1, T) → g'_1
  83. Rotate(B', 180°, l) → B''_1
  84. Rotate(E'_1, 180°, l) → U
  85. Segment(B''_1, U) → j'_1
  86. Rotate(F'_1, 180°, l) → V
  87. Rotate(E'_1, 180°, l) → W
  88. Segment(V, W) → h'_1
  89. Rotate(C'_1, 180°, l) → Z
  90. Rotate(D', 180°, l) → D''
  91. Segment(Z, D'') → c''
  92. Rotate(A'_1, 180°, l) → A_1
  93. Rotate(B'_1, 180°, l) → B_1
  94. Segment(A_1, B_1) → a''
  95. Rotate(Rotate[Segment[E, E + v], (-α), Segment[B, E]], (-α), b) → k
  96. Mirror(k, e) → k'
  97. Rotate(k, 180°, l) → k'_1
  98. Rotate(k', 180°, l) → k''
  99. Rotate(Rotate[E + v, (-α), Segment[B, E]], (-α), b) → C_1
  100. Midpoint(B, C_1) → D_1
  101. OrthogonalLine(D_1, poly6) → p
  102. Rotate(poly2, (-(180°)), p) → poly2'_3
  103. Rotate(poly1, (-(180°)), p) → poly1'_2
  104. Mirror(poly2'_3, e) → poly15
  105. Mirror(poly1'_2, e) → poly16
  106. Rotate(Rotate[Segment[E + v, B + v], (-α), Segment[B, E]], (-α), b) → q
  107. Mirror(q, e) → q'
  108. Rotate(B, (-(180°)), p) → B'_2
  109. Rotate(C, (-(180°)), p) → C'_2
  110. Segment(B'_2, C'_2) → b'
  111. Mirror(B'_2, e) → E_1
  112. Mirror(C'_2, e) → F_1
  113. Segment(E_1, F_1) → b''
  114. Rotate(Rotate[Rotate[Segment[E + w + v, E + v], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → r
  115. Mirror(B'_2, e) → G_1
  116. Mirror(O, e) → O'
  117. Rotate(Rotate[Rotate[Rotate[Segment[E + v + w, E + v + w - u], (-α), Segment[E + v, E + v + w]], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → s
  118. Rotate(Rotate[Rotate[Segment[E + w, E + w + v], α, Segment[E, E + v]], (-α), Segment[B, E]], (-α), b) → t
  119. Mirror(t, e) → t'
  120. Mirror(r, e) → r'
  121. Mirror(s, e) → s'
  122. Rotate(poly5', (-(180°)), p) → poly5''_2
  123. Mirror(poly5''_2, e) → poly17
  124. Rotate(C, (-(180°)), p) → C'_3
  125. Rotate(D, (-(180°)), p) → D'_1
  126. Segment(C'_3, D'_1) → c'_1
  127. Mirror(C'_3, e) → H_1
  128. Mirror(D'_1, e) → I_1
  129. Segment(H_1, I_1) → f_1
  130. Rotate(Segment[B - u, A - u], (-α), a) → g_1
  131. Rotate(t, 180°, l) → t'_1
  132. Rotate(t', 180°, l) → t''
  133. Rotate(Segment[A, A - u], (-α), a) → poly18
  134. Rotate(Rotate[Polygon[A, A - u, A - u - w, A - w], (-α), Segment[A, A - u]], (-α), a) → poly19
  135. Rotate(Rotate[Rotate[Polygon[A - u, A - u + v, A - u + v - w, A - u - w], α, Segment[A - u, A - u - w]], (-α), Segment[A, A - u]], (-α), a) → poly20
  136. Rotate(Rotate[Segment[A - u - w, A - w], (-α), Segment[A, A - u]], (-α), a) → h_1
  137. Rotate(Rotate[Segment[A - u, A - u - w], (-α), Segment[A, A - u]], (-α), a) → i_1
  138. Rotate(Rotate[Rotate[Segment[A - u + v - w, A - u - w], α, Segment[A - u, A - u - w]], (-α), Segment[A, A - u]], (-α), a) → j_1
  139. Rotate(Rotate[Rotate[Polygon[A - w, A - w - u, (2 * A) - w - u - B, (2 * A) - w - B], (-α), Segment[A - w, A - w - u]], (-α), Segment[A, A - u]], (-α), a) → poly21
  140. Rotate(Rotate[Rotate[Rotate[Polygon[A - u + v - w, A - u - w, A - u - w + B - E, A - u + v - w + B - E], α, Segment[A - u + v - w, A - u - w]], α, Segment[A - u, A - u - w]], (-α), Segment[A, A - u]], (-α), a) → poly22
  141. Mirror(poly19, e) → poly19'
  142. Mirror(h_1, e) → k_1
  143. Mirror(poly21, e) → poly21'
  144. Mirror(poly22, e) → poly22'
  145. Mirror(j_1, e) → l_1
  146. Mirror(poly20, e) → poly20'
  147. Mirror(i_1, e) → m_1
  148. Rotate(C, (-(180°)), n) → C'_4
  149. Rotate(D, (-(180°)), n) → D'_2
  150. Segment(C'_4, D'_2) → c'_2
  151. Rotate(B, (-(180°)), n) → B'_3
  152. Rotate(C, (-(180°)), n) → C'_5
  153. Segment(B'_3, C'_5) → b'_1
  154. Rotate(C', (-(180°)), n) → C''_2
  155. Rotate(F'_1, (-(180°)), n) → J_1
  156. Segment(C''_2, J_1) → g'_2
  157. Mirror(poly18, e) → poly18'
  158. Mirror(g_1, e) → n_1
  159. Mirror(B'_3, e) → K_1
  160. Mirror(C'_5, e) → L_1
  161. Segment(K_1, L_1) → p_1
  162. Mirror(C'_4, e) → M_1
  163. Mirror(D'_2, e) → N_1
  164. Segment(M_1, N_1) → q_1
  165. Mirror(C''_2, e) → O_1
  166. Mirror(J_1, e) → P_1
  167. Segment(O_1, P_1) → r_1

AI 绘图提示语参考

画一个立方体的展开图，包含六个相连的矩形，其中三个矩形在一条直线上，另外三个矩形分别与这三个矩形相邻，形成一个连续的平面图形。先画出三个矩形，然后旋转其中一个矩形并连接相邻的矩形，最后调整位置使所有矩形连接成一个完整的展开图。
以上提示语为根据原始绘图命令整理而来，仅供参考。