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正方体展开图 by Benita

正方体展开图 by Benita
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2017-05-13 17:05
多边形立体几何几何作图
① 此范例演示正方体的网状展开图，通过平面图形直观展示立方体的空间折叠与展开现象。② 核心数学概念涉及空间几何、多面体结构、平面与立体的转换与对称性。③ 作图要点提炼自构造摘要，用到的 GeoGebra 命令包括 Point、Segment、Polygon、Circle、Line、Intersect、Rotate、OrthogonalLine、CircleArc、Vector、Translate、Ray、LineBisector、If、Midpoint，对象涉及顶点、边、多边形、圆、线、交点、旋转、垂线、圆弧、向量、平移、射线、中垂线、中点等，展示了从二维构建三维空间图形的完整过程。④ 适合初中及高中空间几何教学，帮助学生理解立体图形的展开、折叠及几何变换规律。

GeoGebra 命令及参数

  1. Point(Circle[A, s]) → B
  2. Segment(A, B) → a
  3. Polygon(A, B, 4) → poly1, a_1, b, c, d, C, D
  4. Circle(B, e) → f
  5. Line(A, B) → g
  6. Intersect(f, g, 2) → E
  7. Rotate(E, 30°, B) → E'
  8. OrthogonalLine(E', g) → h
  9. Segment(B, E') → i
  10. Line(C, i) → j
  11. Intersect(h, j) → F
  12. Polygon(E', F, 4) → poly2, e', f_1, k, l, G, H
  13. Polygon(A, H, G, D) → poly3, a_2, h_1, g_1, d_1
  14. Polygon(C, F, E', B) → poly4, c_1, f_2, e'_1, b_1
  15. Circle(B, A) → p
  16. Circle(A, B) → q
  17. Intersect(q, g, 2) → I
  18. Intersect(p, g, 2) → J
  19. Circle(J, B) → r
  20. Intersect(r, g, 2) → K
  21. Line(J, e'_1) → m
  22. Line(K, e'_1) → n
  23. Line(I, a_2) → t
  24. Line(H, E') → e_1
  25. Intersect(t, e_1) → L
  26. Intersect(m, e_1) → M
  27. Intersect(n, e_1) → N
  28. Line(B, E') → i_1
  29. Line(A, H) → j_1
  30. Intersect(f, i_1, 2) → O
  31. Line(O, g) → k_1
  32. Circle(E', B) → p_1
  33. Intersect(p_1, i_1, 2) → P
  34. Line(P, e_1) → l_1
  35. Intersect(j_1, l_1) → Q
  36. Line(C, B) → m_1
  37. Circle(C, B) → q_1
  38. Intersect(q_1, m_1, 2) → R
  39. Line(R, c_1) → n_1
  40. Intersect(h, n_1) → S
  41. Intersect(j_1, k_1) → T
  42. CircleArc(C, R, D) → r_1
  43. CircleArc(B, K, R) → s_1
  44. CircleArc(E', N, S) → t_1
  45. CircleArc(E', M, F) → c_2
  46. CircleArc(B, J, C) → d_2
  47. CircleArc(A, B, I) → e_2
  48. CircleArc(H, G, L) → g_2
  49. CircleArc(F, S, G) → h_2
  50. Vector(F, P) → u
  51. Vector(C, O) → v
  52. Vector(D, R) → w
  53. Vector(S, N) → u_1
  54. Vector(D, I) → v_1
  55. Translate(D, Vector[(o * w)]) → D'_1
  56. Ray(C, D'_1) → b_2
  57. Intersect(r_1, b_2, 1) → U
  58. Translate(G, Vector[(o * w)]) → G'_1
  59. Ray(F, G'_1) → i_2
  60. Intersect(h_2, i_2, 1) → V
  61. LineBisector(F, P) → j_2
  62. Intersect(f, j_2, 2) → W
  63. CircleArc(W, P, F) → k_2
  64. Line(W, l_1) → l_2
  65. LineBisector(G, Q) → m_2
  66. Intersect(l_2, m_2) → Z
  67. CircleArc(Z, Q, G) → p_2
  68. Translate(C, Vector[((o - 1) * v)]) → C'
  69. Translate(D, Vector[((o - 1) * v)]) → D'_2
  70. If(o < 1, C, If[o < 2, C', O]) → A_1
  71. If(o < 1, D, If[o < 2, D'_2, T]) → B_1
  72. Translate(D, Vector[((o - 2) * v_1)]) → D'
  73. Translate(G, Vector[((o - 2) * v_1)]) → G'_2
  74. Ray(A, D') → n_2
  75. Ray(H, G'_2) → q_2
  76. Intersect(e_2, n_2, 1) → C_1
  77. Intersect(g_2, q_2) → D_1,
  78. If(o < 2, G, If[o < 3, D_1, L]) → E_1
  79. If(o < 2, D, If[o < 3, C_1, I]) → F_1
  80. Translate(G, Vector[((o - 3) * u)]) → G'
  81. Translate(F, Vector[((o - 3) * u)]) → F'
  82. Ray(H, G') → r_2
  83. Ray(E', F') → s_2
  84. Intersect(p_2, r_2, 1) → G_1
  85. Intersect(k_2, s_2, 1) → H_1
  86. If(o < 3, G, If[o < 4, G_1, Q]) → I_1
  87. If(o < 3, F, If[o < 4, H_1, P]) → J_1
  88. Translate(S, Vector[((o - 4) * u_1)]) → S'
  89. Ray(E', S') → t_2
  90. Intersect(t_1, t_2, 1) → K_1
  91. Intersect(c_2, t_2, 1) → L_1
  92. Translate(R, Vector[((o - 4) * u_1)]) → R'
  93. Ray(B, R') → a_3
  94. Intersect(s_1, a_3, 1) → M_1
  95. Intersect(d_2, a_3, 1) → N_1
  96. If(o < 1, U, If[o < 4, R, If[o < 5, M_1, K]]) → O_1
  97. If(o < 1, V, If[o < 4, S, If[o < 5, K_1, N]]) → P_1
  98. If(o < 4, C, If[o < 5, N_1, J]) → Q_1
  99. If(o < 4, F, If[o < 5, L_1, M]) → R_1
  100. Midpoint(B, E') → S_1
  101. Midpoint(A, H) → T_1
  102. Line(T_1, S_1) → c_3
  103. Circle(S_1, a / 2) → d_3
  104. OrthogonalLine(S_1, c_3) → g_3
  105. Intersect(d_3, g_3) → U_1, V_1
  106. Polygon(U_1, V_1, 4) → poly11, u_2, v_2, k_3, l_3, W_1, Z_1
  107. Polygon(Z_1, W_1, 4) → poly12, z_1, w_1, m_3, n_3, A_2, B_2
  108. Polygon(U_1, Z_1, 4) → poly13, u_3, z_2, s_3, t_3, C_2, D_2
  109. Polygon(W_1, V_1, 4) → poly14, w_2, v_3, c_4, d_4, E_2, F_2
  110. Polygon(V_1, U_1, 4) → poly15, v_4, u_4, e_4, f_4, G_2, H_2
  111. Polygon(H_2, G_2, 4) → poly16, h_6, g_4, i_4, j_4, I_2, J_2
  112. Segment(B, U_1) → m_4
  113. LineBisector(m_4) → n_4
  114. Vector(B, U_1) → z
  115. Vector(T, C_2) → z_4
  116. Vector(L, A_2) → u_5
  117. Vector(P, E_2) → w_5
  118. Translate(I, Vector[(z * (o - 5))]) → I'
  119. Translate(A, Vector[(z * (o - 5))]) → A'
  120. Translate(B, Vector[(z * (o - 5))]) → B'
  121. Translate(J, Vector[(z * (o - 5))]) → J'
  122. Translate(K, Vector[(z * (o - 5))]) → K'
  123. Translate(L, Vector[(u_5 * (o - 5))]) → L'
  124. Translate(H, Vector[(u_5 * (o - 5))]) → H'
  125. Translate(E', Vector[(u_5 * (o - 5))]) → E''
  126. Translate(M, Vector[(u_5 * (o - 5))]) → M'
  127. Translate(N, Vector[(u_5 * (o - 5))]) → N'
  128. Translate(T, Vector[(z_4 * (o - 5))]) → T'
  129. Translate(O, Vector[(z_4 * (o - 5))]) → O'
  130. Translate(P, Vector[(w_5 * (o - 5))]) → P'
  131. Translate(Q, Vector[(w_5 * (o - 5))]) → Q'
  132. If(o < 5, O_1, If[o < 6, K', I_2]) → K_2
  133. If(o < 5, P_1, If[o < 6, N', J_2]) → L_2
  134. If(o < 5, Q_1, If[o < 6, J', G_2]) → M_2
  135. If(o < 5, R_1, If[o < 6, M', H_2]) → N_2
  136. If(o < 5, E', If[o < 6, E'', V_1]) → O_2
  137. If(o < 5, H, If[o < 6, H', W_1]) → P_2
  138. If(o < 5, E_1, If[o < 6, L', A_2]) → Q_2
  139. If(o < 5, J_1, If[o < 6, P', E_2]) → R_2
  140. If(o < 5, I_1, If[o < 6, Q', F_2]) → S_2
  141. If(o < 5, B, If[o < 6, B', U_1]) → T_2
  142. If(o < 5, A, If[o < 6, A', Z_1]) → U_2
  143. If(o < 5, A_1, If[o < 6, O', D_2]) → V_2
  144. If(o < 5, B_1, If[o < 6, T', C_2]) → W_2
  145. If(o < 5, F_1, If[o < 6, I', B_2]) → Z_2
  146. Polygon(Z_2, Q_2, P_2, U_2) → poly5, z_3, q_3, p_3, u_6
  147. Polygon(U_2, T_2, V_2, W_2) → poly6, u_7, t_4, v_5, w_3
  148. Polygon(U_2, P_2, O_2, T_2) → poly7, u_8, p_4, o_2, t_5
  149. Polygon(O_2, N_2, M_2, T_2) → poly8, o_1, n_5, m_5, t_6
  150. Polygon(N_2, L_2, K_2, M_2) → poly9, n_6, l_4, k_4, m_6
  151. Polygon(O_2, R_2, S_2, P_2) → poly10, o_3, r_3, s_4, p_5
  152. Intersect(k, v_5) → B_3
  153. Segment(G, B_3) → b_3
  154. Intersect(v_5, a_2) → A_3
  155. Segment(P_2, A_3) → e_3
  156. Intersect(t_4, p_5) → C_3
  157. Intersect(m_5, p_5) → D_3
  158. Segment(D_3, C_3) → f_3
  159. Segment(D_3, P_2) → h_3
  160. Point(Circle[E_3, s]) → F_3
  161. Segment(E_3, F_3) → j_3
  162. Line(E_3, F_3) → a_4
  163. Circle(F_3, e) → h_4
  164. Intersect(h_4, a_4, 2) → G_3
  165. Rotate(G_3, 30°, F_3) → H_3
  166. Segment(F_3, H_3) → b_4
  167. Line(H_3, j_3) → q_4
  168. Line(E_3, b_4) → r_4
  169. Intersect(q_4, r_4) → I_3
  170. OrthogonalLine(H_3, q_4) → a_5
  171. OrthogonalLine(F_3, j_3) → b_5
  172. OrthogonalLine(E_3, a_4) → c_5
  173. OrthogonalLine(I_3, q_4) → d_5
  174. Circle(E_3, s) → e_5
  175. Intersect(e_5, c_5, 2) → J_3
  176. OrthogonalLine(J_3, c_5) → f_5
  177. Circle(I_3, s) → g_5
  178. Intersect(g_5, d_5, 2) → K_3
  179. OrthogonalLine(K_3, d_5) → h_5
  180. Intersect(a_5, h_5) → L_3
  181. Intersect(b_5, f_5) → M_3
  182. Polygon(J_3, M_3, L_3, K_3) → poly17, j_5, m_7, l_5, k_5
  183. Polygon(J_3, E_3, F_3, M_3) → poly18, j_6, e_6, f_6, m_8
  184. Polygon(F_3, M_3, L_3, H_3) → poly19, f_7, m_9, l_6, h_7
  185. Polygon(I_3, K_3, L_3, H_3) → poly20, i_5, k_6, l_7, h_8
  186. Polygon(E_3, J_3, K_3, I_3) → poly21, e_7, j_7, k_7, i_6
  187. Polygon(E_3, I_3, H_3, F_3) → poly22, e_8, i_7, h_9, f_8