积极 AI Logo 积极 AI

展开立方体得到八面体

展开立方体得到八面体
几何画板及绘图内容加载中，请稍等……
2020-03-14 17:46
解析几何立体几何变换
①本范例演示将立方体沿特定平面展开成八面体的过程；②涉及的核心数学概念包括立体几何、几何变换、向量与坐标几何；③作图要点提炼自构造摘要，使用了Polygon（多边形）、Point（点）、Cube（立方体）、Midpoint（中点）、Line（直线）、Rotate（旋转）、Plane（平面）、Mirror（镜像）、Intersect（相交）、PointIn（点在对象上）等GeoGebra命令与对象；④适合高中及以上学生学习空间几何变换与三维图形的构建。

GeoGebra 命令及参数

  1. Polygon(A, B, 4, xOyPlane) → poly1, f, g, h, i, C, D
  2. Point(Circle[B, Distance[A, B], Segment[A, B]]) → E
  3. Cube(A, B, E) → a, F, faceEFJI, faceGHIJ, edgeAF, edgeAB, edgeBE, edgeEF, edgeAG, edgeFJ, edgeGJ, edgeGH, G, edgeBH, edgeHI, edgeEI, edgeIJ, H, I, J, faceABEF, faceAFJG, faceABHG, faceBEIH
  4. Midpoint(edgeAB) → K
  5. Midpoint(edgeBE) → L
  6. Midpoint(edgeEF) → M
  7. Midpoint(edgeAF) → N
  8. Midpoint(edgeAG) → O
  9. Midpoint(edgeBH) → P
  10. Midpoint(edgeEI) → Q
  11. Midpoint(edgeFJ) → R
  12. Midpoint(edgeGH) → S
  13. Midpoint(edgeHI) → T
  14. Midpoint(edgeIJ) → U
  15. Midpoint(edgeGJ) → V
  16. Midpoint(S, U) → W
  17. Midpoint(K, M) → Z
  18. Midpoint(U, M) → A_1
  19. Midpoint(P, Q) → B_1
  20. Midpoint(O, P) → C_1
  21. Midpoint(O, R) → D_1
  22. Midpoint(W, U) → E_1
  23. Midpoint(W, T) → F_1
  24. Midpoint(U, A_1) → G_1
  25. Midpoint(M, A_1) → H_1
  26. Midpoint(B_1, Q) → I_1
  27. Midpoint(T, B_1) → J_1
  28. Polygon(F_1, E_1, U, I, T) → poly2, f_1, e_1, u, i_1, t
  29. Midpoint(Q, A_1) → K_1
  30. Polygon(U, G_1, K_1, Q, I) → poly3, u_1, g_1, k_1, q, i_2
  31. Polygon(T, J_1, I_1, Q, I) → poly4, t_1, j_1, j, q_1, i_3
  32. Polygon(E_1, G_1, U) → t1, u_2, k, l
  33. Polygon(F_1, J_1, T) → t2, t_2, m, n
  34. Polygon(K_1, I_1, Q) → t3, q_2, p, r
  35. Polygon(E_1, G_1, K_1, I_1, J_1, F_1) → poly5, s, c, d, e, h_1, l_1
  36. Line(Q, P) → m_1
  37. Rotate(poly4, (-α), m_1) → poly4'
  38. Rotate(poly2, (-α), m_1) → poly2'
  39. Rotate(poly3, (-α), m_1) → poly3'
  40. Rotate(t2, (-α), m_1) → t2'
  41. Rotate(poly5, (-α), m_1) → poly5'
  42. Rotate(t1, (-α), m_1) → t1'
  43. Plane(V, D_1, W) → p_1
  44. Plane(R, D_1, A_1) → r_1
  45. Mirror(t2, r_1) → t2'_1
  46. Mirror(t1, r_1) → t1'_1
  47. Mirror(poly2, r_1) → poly2'_1
  48. Mirror(poly3, r_1) → poly3'_1
  49. Mirror(poly4, r_1) → poly4'_1
  50. Mirror(t1', p_1) → t1''
  51. Mirror(t2', p_1) → t2''
  52. Mirror(poly2', p_1) → poly2''
  53. Mirror(poly3', p_1) → poly3''
  54. Mirror(poly4', p_1) → poly4''
  55. Mirror(poly5', p_1) → poly5''
  56. Plane(N, D_1, Z) → p_2
  57. Plane(U, S, K) → r_2
  58. Mirror(poly3'', r_2) → poly3'''
  59. Mirror(poly4'', r_2) → poly4'''
  60. Mirror(poly2'', r_2) → poly2'''
  61. Mirror(t1'', r_2) → t1'''
  62. Rotate(t3, (-α), m_1) → t3'
  63. Mirror(t3', p_1) → t3''
  64. Mirror(poly5'', r_2) → poly5'''
  65. Mirror(t1', r_2) → t1''_1
  66. Mirror(poly3', r_2) → poly3''_1
  67. Mirror(L, r_2) → L'
  68. Mirror(poly5', r_2) → poly5''_1
  69. Mirror(poly2', r_2) → poly2''_1
  70. Mirror(poly4', r_2) → poly4''_1
  71. Midpoint(A_1, R) → L_1
  72. Mirror(t3, r_2) → t3'_1
  73. Mirror(poly3'_1, r_2) → poly6
  74. Midpoint(B_1, L) → M_1
  75. Midpoint(L', Z) → N_1
  76. Midpoint(Z, L) → O_1
  77. Midpoint(M, Z) → P_1
  78. Polygon(K_1, H_1, P_1, O_1, M_1, I_1) → poly7, n_1, s_1, a_1, o_1, b_1, c_1
  79. Mirror(t2'_1, r_2) → t4
  80. Mirror(poly7, r_2) → poly7'
  81. Mirror(poly2'_1, r_2) → poly8
  82. Mirror(t3'_1, p_2) → t5
  83. Mirror(poly7', p_2) → poly7''
  84. Mirror(poly8, p_2) → poly8'
  85. Mirror(poly6, p_2) → poly6'
  86. Mirror(poly4'_1, p_2) → poly9
  87. Mirror(poly3'_1, p_2) → poly10
  88. Mirror(poly2'_1, p_2) → poly11
  89. Mirror(poly7, p_1) → poly7'_1
  90. Line(Z, L') → d_1
  91. Midpoint(K, Z) → Q_1
  92. Polygon(K, Q_1, O_1, L, B) → poly12, k_2, f_2, g_2, l_2, b_2
  93. Polygon(L, O_1, P_1, M, E) → poly13, l_3, h_2, j_2, m_2, e_2
  94. Polygon(D, M, P_1, N_1, L') → poly14, d_2, m_3, n_2, s_2, l'
  95. Polygon(L', N_1, Q_1, K, A) → poly15, a_2, c_2, f_3, k_3, a_3
  96. Mirror(poly4'_1, r_2) → poly16
  97. Mirror(poly9, r_2) → poly9'
  98. Mirror(t3, p_1) → t3'_2
  99. Mirror(p_1, p_2) → p_3
  100. Mirror(t3'', r_2) → t3'''
  101. Mirror(t3', r_2) → t3''_1
  102. Rotate(poly5', β, d_1) → poly5''_2
  103. Rotate(t1', β, d_1) → t1''_2
  104. Rotate(poly3', β, d_1) → poly3''_2
  105. Rotate(t3', β, d_1) → t3''_2
  106. Rotate(poly4', β, d_1) → poly4''_2
  107. Rotate(poly2', β, d_1) → poly2''_2
  108. Rotate(t3, β, d_1) → t3'_3
  109. Rotate(poly3'_1, β, d_1) → poly17
  110. Rotate(poly7, β, d_1) → poly7'_2
  111. Rotate(poly13, β, d_1) → poly13'
  112. Mirror(poly5''_2, p_1) → poly18
  113. Mirror(t1''_2, p_1) → t6
  114. Mirror(poly3''_2, p_1) → poly19
  115. Mirror(poly17, p_1) → poly17'
  116. Mirror(t3''_2, p_1) → t7
  117. Mirror(t3'_3, p_1) → t8
  118. Mirror(poly7'_2, p_1) → poly20
  119. Mirror(poly18, r_2) → poly18'
  120. Mirror(t6, r_2) → t6'
  121. Mirror(t7, r_2) → t7'
  122. Mirror(t8, r_2) → t8'
  123. Mirror(poly19, r_2) → poly19'
  124. Mirror(poly17', r_2) → poly17''
  125. Mirror(poly20, r_2) → poly20'
  126. Mirror(poly3''_2, r_2) → poly21
  127. Mirror(poly17, r_2) → poly17'_1
  128. Mirror(t3''_2, r_2) → t9
  129. Mirror(t3'_3, r_2) → t10
  130. Mirror(poly7'_2, r_2) → poly22
  131. Mirror(poly5''_2, r_2) → poly23
  132. Mirror(t1''_2, r_2) → t11
  133. Mirror(t1'', r_2) → t1'''_1
  134. Mirror(t1', r_2) → t1''_3
  135. Rotate(poly2''', (-β), d_1) → poly24
  136. Rotate(poly15, (-β), d_1) → poly15'
  137. Rotate(poly12, (-β), d_1) → poly12'
  138. Rotate(poly2'', (-β), d_1) → poly2'''_1
  139. Rotate(poly2''_1, β, d_1) → poly25
  140. Rotate(poly14, β, d_1) → poly14'
  141. Rotate(poly4'_1, β, d_1) → poly26
  142. Rotate(poly9, (-β), d_1) → poly9'_1
  143. Rotate(poly4'', (-β), d_1) → poly4'''_1
  144. Rotate(poly16, β, d_1) → poly16'
  145. Rotate(poly4''_1, β, d_1) → poly27
  146. Rotate(poly9', (-β), d_1) → poly9''
  147. Rotate(poly4''', (-β), d_1) → poly28
  148. Point(zAxis) → R_1
  149. Line(R_1, m_1) → g_3
  150. Rotate(poly20, (-γ), g_3) → poly20'_1
  151. Rotate(t8, (-γ), g_3) → t8'_1
  152. Rotate(t7, (-γ), g_3) → t7'_1
  153. Rotate(poly18, (-γ), g_3) → poly18'_1
  154. Rotate(poly17', (-γ), g_3) → poly17''_1
  155. Rotate(poly19, (-γ), g_3) → poly19'_1
  156. Rotate(poly17, (-γ), g_3) → poly17'_2
  157. Rotate(poly3''_2, (-γ), g_3) → poly29
  158. Rotate(t6, (-γ), g_3) → t6'_1
  159. Rotate(t1''_2, (-γ), g_3) → t12
  160. Rotate(t8', γ, g_3) → t8''
  161. Rotate(poly20', γ, g_3) → poly20''
  162. Rotate(t7', γ, g_3) → t7''
  163. Rotate(poly18', γ, g_3) → poly18''
  164. Rotate(poly17'', γ, g_3) → poly17'''
  165. Rotate(poly19', γ, g_3) → poly19''
  166. Rotate(poly17'_1, γ, g_3) → poly30
  167. Rotate(poly21, γ, g_3) → poly21'
  168. Rotate(t6', γ, g_3) → t6''
  169. Rotate(t11, γ, g_3) → t11'
  170. Rotate(poly5''_2, (-γ), g_3) → poly31
  171. Rotate(poly7'_2, (-γ), g_3) → poly32
  172. Rotate(t3''_2, (-γ), g_3) → t13
  173. Rotate(t3'_3, (-γ), g_3) → t14
  174. Rotate(poly22, γ, g_3) → poly22'
  175. Rotate(t10, γ, g_3) → t10'
  176. Rotate(t9, γ, g_3) → t9'
  177. Rotate(poly23, γ, g_3) → poly23'
  178. Midpoint(E, L) → S_1
  179. Midpoint(L', A) → T_1
  180. Midpoint(L, B) → U_1
  181. Midpoint(L', S) → V_1
  182. Midpoint(D, L') → W_1
  183. Mirror(N_1, L') → Z_1
  184. Mirror(O_1, L) → A_2
  185. Polygon(Z_1, T_1, N_1, W_1) → q1, z_1, h_3, j_3, w_1
  186. Polygon(O_1, U_1, A_2, S_1) → q2, n_3, q_3, r_3, s_3
  187. Rotate(q2, (-γ), g_3) → q2'
  188. Rotate(q2, (-γ), g_3) → q2'_1
  189. Rotate(q1, γ, g_3) → q1'_1
  190. Rotate(q2, (-γ), g_3) → q2'_2
  191. Intersect(d_1, poly32) → D_2
  192. Intersect(d_1, poly4'''_1) → E_2
  193. Mirror(D_2, E_2) → F_2
  194. Mirror(N_1, L') → B_2
  195. PointIn(poly31) → C_2
  196. PointIn(poly20'_1) → G_2
  197. Polygon(F_2, D_2, G_2) → t15, t_3, b_3, c_3
  198. Polygon(F_2, C_2, D_2) → t16, d_3, e_3, f_4
  199. Mirror(t15, r_2) → t15'
  200. Mirror(t16, r_2) → t16'
  201. Mirror(q2'_2, r_2) → q3
  202. Intersect(g_3, poly22') → H_2
  203. PointIn(poly32) → I_2
  204. PointIn(poly22') → J_2
  205. PointIn(poly20'') → K_2
  206. Polygon(I_2, K_2, H_2) → t17, g_4, h_4, i_4
  207. Polygon(K_2, J_2, H_2) → t18, j_4, k_4, l_4
  208. Mirror(q2'_2, r_2) → q4
  209. Mirror(q2'_2, r_2) → q5