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截面与相对位置

截面与相对位置
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2019-05-12 01:44
多边形立体几何几何作图
①本范例演示了三维几何体（如立方体）的截面形状及其与几何体相对位置关系的动态变化；②涉及的核心数学概念包括多边形、相似三角形、立体几何、几何作图；③作图要点提炼自构造摘要，使用了Polygon（多边形）、Intersect（交点）、OrthogonalLine（垂线）、Point（点）、Plane（平面）、CircleArc（圆弧）、Circle（圆）、Line（直线）、Segment（线段）、PointIn（点在对象上）、Cube（立方体）等GeoGebra命令/对象；④适合高中及以上学段学习立体几何、几何作图及空间想象能力的学生，也可用于相关课题的演示与探究。

GeoGebra 命令及参数

  1. Polygon(C, D, B, A) → q1, c, d, b, a
  2. Intersect(b, yAxis) → E
  3. Intersect(c, yAxis) → F
  4. Intersect(xAxis, d) → G
  5. Intersect(a, xAxis) → H
  6. OrthogonalLine(E, xOyPlane) → f
  7. OrthogonalLine(G, xOyPlane) → g
  8. OrthogonalLine(F, xOyPlane) → h
  9. OrthogonalLine(H, xOyPlane) → i
  10. Point(f) → I
  11. Point(yAxis) → J
  12. Plane(E, J, I) → e
  13. CircleArc(E, I, J) → k_2
  14. Point(k_2) → P_2
  15. Plane(A, B, P_2) → j
  16. Circle(A, 4, j) → p
  17. Circle(B, 4, j) → q
  18. Line(P_2, b) → l
  19. Intersect(l, p) → L, M
  20. Intersect(l, q) → N, O
  21. Polygon(A, B, N, P_2, L) → poly1, a_1, b_1, n, k_5, l_1
  22. Point(i) → P
  23. Point(xAxis) → Q
  24. CircleArc(H, Q, P) → k_3
  25. Point(k_3) → P_3
  26. Plane(C, A, P_3) → m
  27. Line(P_3, a) → s
  28. Circle(C, 4, m) → t
  29. Circle(A, 4, m) → c_1
  30. Intersect(s, t) → S, T
  31. Intersect(s, c_1) → U, V
  32. Polygon(C, A, U, P_3, S) → poly2, c_2, a_2, u, r_1, s_1
  33. Point(g) → W
  34. Point(xAxis) → Z
  35. CircleArc(G, W, Z) → k_1
  36. Point(k_1) → P_1
  37. Plane(B, D, P_1) → o
  38. Line(P_1, d) → f_1
  39. Circle(B, 4, o) → e_1
  40. Circle(D, 4, o) → g_1
  41. Intersect(f_1, e_1) → B_1, C_1
  42. Intersect(f_1, g_1) → D_1, E_1
  43. Polygon(B, D, D_1, P_1, B_1) → poly3, b_2, d_2, h_1, i_1, j_1
  44. Point(h) → F_1
  45. Point(yAxis) → G_1
  46. CircleArc(F, F_1, G_1) → k_4
  47. Point(k_4) → P_4
  48. Plane(D, C, P_4) → k
  49. Line(P_4, c) → r
  50. Circle(D, 4, k) → d_1
  51. Circle(C, 4, k) → p_1
  52. Intersect(r, d_1) → K, R
  53. Intersect(r, p_1) → A_1, H_1
  54. Point(r) → I_1
  55. Polygon(D, C, A_1, I_1, K) → poly4, d_3, c_3, m_1, n_1, k_6
  56. Point(zAxis) → J_1
  57. Point(zAxis) → K_1
  58. Segment(J_1, K_1) → q_1
  59. PointIn(poly2) → A_2
  60. PointIn(poly2) → B_2
  61. PointIn(poly2) → C_2
  62. PointIn(poly2) → D_2
  63. PointIn(poly1) → E_2
  64. PointIn(poly1) → F_2
  65. PointIn(poly1) → G_2
  66. PointIn(poly1) → H_2
  67. PointIn(poly3) → I_2
  68. PointIn(poly3) → J_2
  69. PointIn(poly3) → K_2
  70. PointIn(poly3) → L_2
  71. Polygon(I_2, J_2, K_2, L_2) → q4, r_2, s_2, t_2, e_2
  72. PointIn(poly4) → M_2
  73. PointIn(poly4) → N_2
  74. PointIn(poly4) → O_2
  75. Point(zAxis) → R_2
  76. PointIn(poly4) → S_2
  77. Polygon(N_2, O_2, S_2, M_2) → q5, f_3, o_2, g_3, h_3
  78. Polygon(F_2, G_2, H_2, E_2) → q6, i_3, j_3, l_3, m_3
  79. Polygon(B_2, C_2, D_2, A_2) → q7, n_3, p_3, q_3, r_3
  80. Point(zAxis) → L_1
  81. Plane(L_1, xOyPlane) → w
  82. PointIn(w) → R_1
  83. PointIn(w) → S_1
  84. Point(Circle[S_1, Distance[R_1, S_1], Segment[R_1, S_1]]) → T_1
  85. Cube(R_1, S_1, T_1) → u_1, U_1, face5, face6, edge2, edge1, edge4, edge6, edge3, edge8, edge10, edge9, V_1, edge5, edge11, edge7, edge12, W_1, Z_1, T_2, face1, face3, face2, face4
  86. PointIn(q1) → U_2
  87. PointIn(q1) → V_2
  88. PointIn(q1) → W_2
  89. PointIn(q1) → Z_2
  90. Point(zAxis) → B_3
  91. Segment(A_3, B_3) → s_3
  92. Point(c_2) → C_3
  93. Line(C_3, s_3) → t_3
  94. Segment(A_3, C_3) → a_3
  95. Point(t_3) → D_3
  96. Line(D_3, a_3) → b_3
  97. Line(B_3, b_3) → e_3
  98. Intersect(t_3, e_3) → E_3
  99. Point(t_3) → F_3
  100. Line(F_3, b_3) → t_1
  101. Point(b_3) → M_1
  102. Intersect(t_1, zAxis) → N_1
  103. Intersect(b_1, j_1) → O_1
  104. Point(b_2) → Q_1
  105. Point(d_3) → Q_2
  106. OrthogonalLine(O_1, q1) → f_2
  107. OrthogonalLine(Q_2, q1) → g_2
  108. OrthogonalLine(Q_1, q1) → h_2
  109. Intersect(e_3, h_2) → G_3
  110. Intersect(t_1, h_2) → H_3
  111. Segment(O_1, A_3) → i_2
  112. Line(N_1, i_2) → j_2
  113. Line(B_3, i_2) → l_2
  114. Intersect(f_2, j_2) → I_3
  115. Intersect(g_2, j_2) → J_3
  116. Intersect(l_2, g_2) → L_3
  117. Segment(D_3, F_3) → m_2
  118. Segment(C_3, E_3) → n_2
  119. Point(m_2) → P_5
  120. Point(n_2) → P_6
  121. Plane(P_5, xOyPlane) → v
  122. PointIn(v) → O_3
  123. Polygon(O_3, P_5, 4, xOyPlane) → poly5, p_2, q_2, f_4, g_4, Q_3, R_3
  124. Plane(P_6, xOyPlane) → o_1
  125. PointIn(o_1) → K_3
  126. Polygon(K_3, P_6, 4, xOyPlane) → poly6, h_4, i_4, j_4, l_4, T_3, U_3
  127. PointIn(poly5) → N_3
  128. PointIn(poly5) → M_3
  129. PointIn(poly5) → V_3
  130. PointIn(poly5) → W_3
  131. Polygon(M_3, N_3, W_3, V_3) → q2, m_4, n_4, w_3, v_3
  132. OrthogonalLine(M_3, poly6) → p_4
  133. OrthogonalLine(V_3, poly6) → q_4
  134. OrthogonalLine(W_3, poly6) → r_4
  135. OrthogonalLine(N_3, poly6) → s_4
  136. Intersect(q_4, poly6) → Z_3
  137. Intersect(p_4, poly6) → A_4
  138. Intersect(r_4, poly6) → B_4
  139. Intersect(s_4, poly6) → C_4
  140. Polygon(A_4, C_4, B_4, Z_3) → q3, a_4, c_4, b_4, z_3
  141. Intersect(edge3, edge2) → D_4
  142. Intersect(edge10, edge12) → E_4
  143. Segment(D_4, E_4) → t_4
  144. Intersect(edge4, edge6) → F_4
  145. Segment(E_4, F_4) → d_4
  146. Point(edge9) → G_4
  147. Segment(F_4, G_4) → e_4
  148. Point(j_3) → H_4
  149. Point(l_3) → I_4
  150. Segment(H_4, I_4) → f_5
  151. Intersect(i_3, j_3) → J_4
  152. Segment(I_4, J_4) → g_5
  153. Intersect(m_3, i_3) → K_4
  154. Segment(J_4, K_4) → h_5
  155. Intersect(r_3, q_3) → L_4
  156. Point(p_3) → M_4
  157. Segment(L_4, M_4) → i_5
  158. Point(p_3) → N_4
  159. Intersect(r_3, n_3) → O_4
  160. Segment(N_4, O_4) → j_5
  161. Segment(N_4, M_4) → l_5
  162. Intersect(r_2, s_2) → Q_4
  163. PointIn(poly3) → R_4
  164. Segment(Q_4, R_4) → m_5
  165. Intersect(s_2, t_2) → S_4
  166. Point(r_2) → T_4
  167. Segment(S_4, T_4) → n_5
  168. Segment(R_4, S_4) → p_5
  169. Point(v_3) → U_4
  170. Point(v_3) → V_4
  171. Segment(U_4, V_4) → q_5
  172. Intersect(n_4, w_3) → W_4
  173. Segment(W_4, V_4) → r_5
  174. Intersect(m_4, n_4) → Z_4
  175. Segment(W_4, Z_4) → s_5
  176. Intersect(a_4, z_3) → A_5
  177. Point(z_3) → B_5
  178. Intersect(c_4, b_4) → C_5
  179. Intersect(a_4, c_4) → D_5
  180. Intersect(h_3, g_3) → E_5
  181. Point(g_3) → F_5
  182. Segment(E_5, F_5) → c_5
  183. Intersect(f_3, h_3) → G_5
  184. Segment(F_5, G_5) → d_5
  185. Intersect(f_3, o_2) → H_5
  186. Segment(G_5, H_5) → e_5
  187. Point(z_3) → I_5
  188. Point(z_3) → J_5
  189. Segment(I_5, J_5) → t_5
  190. Point(b_4) → K_5
  191. Segment(J_5, K_5) → a_5
  192. Point(c_4) → L_5
  193. Segment(K_5, L_5) → b_5
  194. Intersect(edge9, edge3) → M_5
  195. Intersect(edge6, edge2) → N_5
  196. Segment(M_5, N_5) → f_6
  197. Intersect(edge11, edge7) → O_5
  198. Segment(N_5, O_5) → g_6
  199. Point(edge4) → Q_5
  200. Segment(O_5, Q_5) → h_6
  201. Intersect(edge5, edge1) → R_5
  202. Intersect(f_6, edge3) → S_5
  203. Segment(R_5, S_5) → i_6
  204. Intersect(f_6, edge2) → T_5
  205. Segment(S_5, T_5) → j_6
  206. Intersect(h_6, g_6) → U_5
  207. Segment(T_5, U_5) → l_6
  208. Intersect(edge5, edge11) → V_5
  209. Segment(U_5, R_5) → m_6

AI 绘图提示语参考

绘制一个立方体及其三个不同平面的截面。首先，绘制一个立方体，然后分别在三个不同的平面上绘制截面多边形。首先在xOy平面上绘制截面多边形，接着在yOz和平面xOz上绘制截面多边形。最后，确保每个截面多边形都与立方体的相应面相交。
以上提示语为根据原始绘图命令整理而来，仅供参考。