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八面体展开图

八面体展开图
几何画板及绘图内容加载中，请稍等……
2012-05-03 19:58
立体几何几何作图
所属专题 Area, Surface Area, Volume--Nets
这个GeoGebra范例演示了八面体的展开图，并通过动画展示了其构造过程。核心数学概念包括立体几何和几何作图。使用了点、线段、圆、旋转、镜像等GeoGebra命令。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. Point(Circle[A, s]) → B
  2. Segment(A, B) → a
  3. Line(A, B) → b
  4. Circle(B, 3.0 * s / 4.5) → c
  5. Intersect(c, b, 2.0) → C
  6. Rotate(C, 30.0*°, B) → C'
  7. Line(C', b) → d
  8. Segment(B, C') → e
  9. Line(A, e) → f
  10. Intersect(d, f) → D
  11. Segment(A, C') → g
  12. Segment(D, B) → h
  13. Intersect(g, h) → E
  14. OrthogonalLine(E, a) → i
  15. Rotate(i, 5.0*°, E) → i'
  16. Circle(E, s * sqrt(3.0) / 2.0) → k
  17. Intersect(k, i', 1.0) → F
  18. Intersect(k, i', 2.0) → G
  19. Polygon(D, C', 3.0) → poly1, j, d_1, c', H
  20. Polygon(D, H, 3.0) → poly2, l, d_2, h_1, I
  21. Polygon(I, H, 3.0) → poly3, m, i_1, h_2, J
  22. Polygon(H, C', 3.0) → poly4, n, h_3, c'_1, K
  23. Polygon(C', D, 3.0) → poly5, p, c'_2, d_3, L
  24. Polygon(C', L, 3.0) → poly6, q, c'_3, l_1, M
  25. Polygon(L, D, 3.0) → poly7, r, l_2, d_4, N
  26. Polygon(M, L, 3.0) → poly8, t, m_1, l_3, O
  27. Polygon(G, A, D) → poly9, d_5, g_1, a_1
  28. Mirror(D, d_5) → D'
  29. Vector(B, D') → u
  30. Vector(A, I) → v
  31. Vector(D', J) → w
  32. Circle(J, D', B) → c_1
  33. Translate(B, Vector[u * o]) → B'_1
  34. Center(c_1) → P
  35. Ray(P, B'_1) → b_1
  36. Intersect(c_1, b_1, 1.0) → Q
  37. Translate(D', Vector[w * (o - 1.0)]) → D''
  38. Ray(P, D'') → e_1
  39. Intersect(c_1, e_1, 1.0) → R
  40. If(o < 1.0, Q, If[o < 2.0, R, J]) → S
  41. Segment(I, A) → f_1
  42. LineBisector(f_1) → j_1
  43. Intersect(f, j_1) → T
  44. CircleArc(T, I, A) → k_1
  45. Translate(A, Vector[v * (o - 1.0)]) → A'_1
  46. Ray(T, A'_1) → n_1
  47. Intersect(k_1, n_1, 1.0) → U
  48. If(o < 1.0, A, If[o < 2.0, U, I]) → V
  49. Segment(B, K) → p_1
  50. LineBisector(p_1) → q_1
  51. Intersect(q_1, c') → Z
  52. Vector(G, H) → z
  53. Vector(B, K) → u_1
  54. Translate(G, Vector[z * (o - 1.0)]) → G'
  55. LineBisector(G, H) → r_1
  56. Intersect(b, r_1) → W
  57. CircleArc(W, H, G) → s_1
  58. Ray(W, G') → t_1
  59. Intersect(s_1, t_1) → A_1,
  60. If(o < 1.0, G, If[o < 2.0, A_1, H]) → B_1
  61. Polygon(B_1, D, C') → poly10, c'_4, b_2, d_6
  62. Polygon(V, B_1, S) → poly11, s_2, v_1, b_3
  63. Polygon(V, D, B_1) → poly12, b_4, v_2, d_7
  64. LineBisector(K, B) → a_2
  65. LineBisector(A, B) → c_2
  66. OrthogonalLine(H, b) → e_2
  67. Intersect(b, c_2) → C_1
  68. Intersect(e_2, b) → D_1
  69. Vector(C_1, D_1) → w_1
  70. Translate(c_2, Vector[w_1 * (o - 1.0)]) → f_2
  71. Mirror(V, f_2) → V'
  72. If(o < 1.0, B, If[o < 2.0, V', K]) → E_1
  73. Polygon(B_1, E_1, C') → poly13, c'_5, b_5, e_3
  74. Polygon(B_1, D, C') → poly14, c'_6, b_6, d_8
  75. Midpoint(A, B) → F_1
  76. Vector(B, M) → z_1
  77. Vector(F, L) → u_2
  78. Polygon(C', B, F) → poly15, f_3, c'_7, b_7
  79. Mirror(C', B) → C''
  80. Vector(A, C'') → w_2
  81. Vector(A, N) → g_2
  82. Circle(C'', s / 5.0) → k_2
  83. Line(C', B) → i_2
  84. Intersect(k_2, i_2, 2.0) → G_1
  85. CircumcircleArc(A, G_1, O) → p_2
  86. Center(p_2) → H_1
  87. Translate(A, Vector[w_2 * (o - 2.0)]) → A'
  88. Ray(H_1, A') → j_2
  89. Intersect(p_2, j_2, 1.0) → I_1
  90. Vector(G_1, O) → z_2
  91. Translate(G_1, Vector[z_2 * (o - 3.0)]) → J_1
  92. Ray(H_1, J_1) → m_2
  93. Intersect(p_2, m_2, 1.0) → K_1
  94. If(o < 2.0, A, If[o < 3.0, I_1, If[o < 4.0, K_1, O]]) → L_1
  95. LineBisector(B, M) → n_2
  96. Translate(B, Vector[z_1 * (o - 3.0)]) → B'
  97. Segment(A, H_1) → q_2
  98. Segment(B, M) → r_2
  99. Intersect(n_2, r_2) → M_1
  100. Intersect(n_2, q_2) → N_1
  101. CircleArc(N_1, B, M) → t_2
  102. Ray(N_1, B') → a_3
  103. Intersect(t_2, a_3, 1.0) → O_1
  104. If(o < 3.0, B, If[o < 4.0, O_1, M]) → P_1
  105. LineBisector(F, L) → c_3
  106. Intersect(l_2, c_3) → Q_1
  107. CircleArc(Q_1, F, L) → g_3
  108. Translate(F, Vector[u_2 * (o - 3.0)]) → F'
  109. Ray(Q_1, F') → i_3
  110. Intersect(g_3, i_3, 1.0) → R_1
  111. If(o < 3.0, F, If[o < 4.0, R_1, L]) → S_1
  112. Polygon(S_1, P_1, L_1) → poly16, l_4, s_3, p_3
  113. Polygon(S_1, C', P_1) → poly17, p_4, s_4, c'_8
  114. Polygon(S_1, C', D) → poly18, d_9, s_5, c'_9
  115. Midpoint(A, B) → T_1
  116. OrthogonalLine(T_1, b) → j_3
  117. Intersect(b, e_2) → U_1
  118. Vector(T_1, U_1) → u_3
  119. Translate(j_3, Vector[u_3 * (o - 3.0)]) → k_3
  120. Mirror(P_1, k_3) → V_1
  121. If(o < 3.0, A, If[o < 4.0, V_1, N]) → W_1
  122. Polygon(W_1, S_1, D) → poly19, d_{10}, w_3, s_6
  123. Intersect(b_6, s_5) → Z_1
  124. Intersect(c'_5, s_5) → A_2
  125. Intersect(b_4, s_6) → B_2
  126. Intersect(v_1, s_6) → C_2
  127. Segment(C_2, A) → m_3
  128. Segment(C_2, D) → n_3
  129. Segment(V, D) → q_3
  130. Segment(C', A_2) → r_3
  131. Segment(D, A_2) → t_3
  132. Intersect(v_1, t_3) → D_2
  133. Segment(D_2, A_2) → a_4
  134. Segment(D_2, D) → c_4
  135. Intersect(s_3, c'_9) → E_2
  136. Segment(E_2, S_1) → e_4
  137. Intersect(v_1, c'_9) → F_2
  138. Segment(F_2, E_2) → f_4
  139. Segment(F_2, D) → g_4
  140. Segment(E_2, D) → h_4
  141. Intersect(b_3, c'_9) → G_2
  142. Segment(G_2, D) → i_4
  143. Segment(G_2, E_2) → j_4
  144. Point(Circle[H_2, s]) → I_2
  145. Segment(H_2, I_2) → k_4
  146. Line(H_2, I_2) → m_4
  147. Circle(I_2, 3.0 * s / 4.5) → q_4
  148. Intersect(q_4, m_4, 2.0) → J_2
  149. Rotate(J_2, 30.0*°, I_2) → K_2
  150. Line(K_2, m_4) → n_4
  151. Segment(I_2, K_2) → r_4
  152. Line(H_2, r_4) → t_4
  153. Intersect(n_4, t_4) → L_2
  154. Segment(H_2, K_2) → a_5
  155. Segment(I_2, L_2) → c_5
  156. Intersect(a_5, c_5) → M_2
  157. Circle(M_2, s * sqrt(3.0) / 2.0) → e_5
  158. OrthogonalLine(M_2, m_4) → f_5
  159. Rotate(f_5, 5.0*°, M_2) → g_5
  160. Intersect(e_5, g_5, 2.0) → N_2
  161. Intersect(e_5, g_5, 1.0) → O_2
  162. Polygon(H_2, I_2, N_2) → poly20, n_5, h_5, i_5
  163. Polygon(N_2, K_2, I_2) → poly21, i_6, n_6, k_5
  164. Polygon(I_2, K_2, O_2) → poly22, o_2, i_7, k_6
  165. Polygon(O_2, H_2, I_2) → poly23, i_8, o_1, h_6
  166. Segment(H_2, L_2) → j_5
  167. Segment(L_2, N_2) → l_5
  168. Segment(L_2, K_2) → m_5
  169. Segment(L_2, O_2) → p_5

AI 绘图提示语参考

绘制一个八面体的展开图，并逐步展示其构建过程。首先创建一个圆心为A的圆，从中选取一点B，连接AB，再创建一条平行于AB的线b。接着以B为圆心，半径为3/4.5倍圆的半径画圆，交线b于C点，然后将C点绕B点旋转30度得到C'点，连接BC'，再创建一条平行于BC'的线d，连接AB得到线f，交d于D点，连接AD得到线g，连接BD得到线h，交g和h于E点，创建E点的垂直线i，旋转i得到i'，以E点为圆心，半径为根号3/2倍圆的半径画圆k，交i'于F点和G点，依次连接各点形成八面体的展开图。
以上提示语为根据原始绘图命令整理而来，仅供参考。