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正方形金字塔展开图：动画演示几何拼合

正方形金字塔展开图：动画演示几何拼合
几何画板及绘图内容加载中，请稍等……
2020-12-14 03:32
多边形角立体几何几何作图滑块动画
该范例通过动画演示“正方形金字塔”的展开图如何由若干三角形与边界对应关系拼合成立体。核心概念包括立体几何中的曲面展开、正方锥侧面为等腰三角形、网格（net）与对应边的同长与夹角保持、以及利用中点、角平分线与圆弧进行几何构造。作图要点：从“构造摘要”可见其使用了 Point、Segment、Line、Circle、Intersect 来确定关键点、边界线与交点；借助 Rotate 与 Translate 完成侧面三角形的对称与重复摆放以形成展开网；用 Polygon 生成多边形轮廓；通过 Midpoint、OrthogonalLine（正交线）、Angle（角）与 LineBisector（角平分线）建立几何约束；利用 CircleArc 与 Ray 确定圆弧与射线辅助角度/长度对应。对象如 A,s,B 等用于标定顶点与边界，poly1、d/e/f 等用于构造展开面各三角块。适合中学到高中几何教学，尤其用于立体几何入门、几何展开与拼合推理、以及课堂演示与课后探究。

GeoGebra 命令及参数

  1. Point(Circle[A, s]) → B
  2. Segment(A, B) → a
  3. Line(A, B) → b_2
  4. Circle(B, (3 * s / 4.5)) → c
  5. Intersect(c, b_2, 2) → C
  6. Rotate(C, 30°, B) → C'
  7. Line(C', b_2) → d
  8. Segment(B, C') → e
  9. Line(A, e) → f
  10. Intersect(d, f) → D
  11. Polygon(A, B, C', D) → poly1, a_1, b_1, c', d_1
  12. Segment(A, C') → g
  13. Segment(D, B) → h_1
  14. Intersect(g, h_1) → E
  15. OrthogonalLine(E, a_1) → i
  16. Circle(E, h) → k
  17. Intersect(k, i, 2) → F
  18. Segment(B, F) → j
  19. Segment(C', F) → l_5
  20. Segment(F, D) → m
  21. Segment(F, A) → n
  22. Segment(E, F) → p
  23. Segment(A, D) → q
  24. Midpoint(b_1) → G
  25. Segment(G, F) → r
  26. Angle(G, E, F) → α
  27. Segment(E, G) → t
  28. Line(E, G) → c_1
  29. Circle(G, F) → f_1
  30. Intersect(f_1, c_1, 2) → H
  31. Intersect(c_1, d_1) → I
  32. Circle(I, F) → g_1
  33. Intersect(g_1, c_1, 1) → J
  34. Vector(F, J) → u
  35. Point(c') → L
  36. Circle(L, F) → q_1
  37. Vector(F, H) → w
  38. Midpoint(A, B) → K
  39. Line(K, E) → e_1
  40. Circle(K, h) → p_1
  41. Intersect(e_1, c') → M
  42. Circle(M, h) → r_1
  43. Intersect(r_1, e_1, 2) → N
  44. Intersect(p_1, e_1, 1) → O
  45. Vector(F, N) → v
  46. Vector(F, O) → z
  47. Translate(F, Vector[(z * o)]) → F'_1
  48. Translate(F, Vector[(u * (o - 1))]) → F'_2
  49. Translate(F, Vector[(v * (o - 2))]) → F'_3
  50. Translate(F, Vector[(w * (o - 3))]) → F'
  51. Segment(F, N) → s_{4}
  52. LineBisector(s_{4}) → a_2
  53. Intersect(a_2, c') → P
  54. CircleArc(P, N, F) → c_2
  55. Ray(P, F'_3) → d_2
  56. Intersect(c_2, d_2, 1) → Q
  57. Segment(F, O) → e_2
  58. LineBisector(e_2) → f_2
  59. Intersect(f_2, a_1) → R
  60. CircleArc(R, F, O) → g_2
  61. Ray(R, F'_1) → h_2
  62. Intersect(g_2, h_2, 1) → S
  63. CircleArc(I, F, J) → k_2
  64. CircleArc(G, H, F) → p_2
  65. Ray(I, F'_2) → i_2
  66. Intersect(k_2, i_2, 1) → T
  67. Ray(G, F') → j_2
  68. Intersect(p_2, j_2, 1) → U
  69. Circle(I, E) → s_1
  70. OrthogonalLine(I, a_1) → i_1
  71. Intersect(s_1, i_1, 2) → V
  72. Intersect(s_1, i_1, 1) → W
  73. OrthogonalLine(V, i_1) → j_1
  74. OrthogonalLine(W, i_1) → k_1
  75. OrthogonalLine(G, k_1) → l_1
  76. Intersect(j_1, l_1) → Z
  77. Intersect(k_1, l_1) → A_1
  78. Midpoint(V, Z) → B_1
  79. Midpoint(A_1, W) → C_1
  80. Circle(B_1, r) → t_1
  81. Circle(C_1, r) → q_2
  82. OrthogonalLine(B_1, k_1) → m_1
  83. Intersect(t_1, m_1, 1) → D_1
  84. Intersect(q_2, m_1, 2) → E_1
  85. Vector(C', Z) → u_1
  86. Vector(B, A_1) → v_1
  87. Vector(N, D_1) → w_1
  88. Vector(O, E_1) → z_1
  89. Translate(N, Vector[(w_1 * (o - 4))]) → N'
  90. Translate(O, Vector[(z_1 * (o - 4))]) → O'
  91. Translate(B, Vector[(v_1 * (o - 4))]) → B'
  92. Translate(A, Vector[(v_1 * (o - 4))]) → A'
  93. Translate(C', Vector[(u_1 * (o - 4))]) → C''
  94. Translate(D, Vector[(u_1 * (o - 4))]) → D'
  95. If(o < 1, S, If[o < 4, O, If[o < 5, O', E_1]]) → F_1
  96. If(o < 1, F, If[o < 2, T, J]) → G_1
  97. If(o < 3, F, If[o < 4, U, H]) → H_1
  98. If(o < 2, F, If[o < 3, Q, If[o < 4, N, If[o < 5, N', D_1]]]) → I_1
  99. If(o < 4, A, If[o < 5, A', W]) → J_1
  100. If(o < 4, B, If[o < 5, B', A_1]) → K_1
  101. If(o < 4, C', If[o < 5, C'', Z]) → L_1
  102. If(o < 4, D, If[o < 5, D', V]) → M_1
  103. Polygon(F_1, J_1, K_1) → poly2, k_3, f_3, j_3
  104. Polygon(L_1, H_1, K_1) → poly3, k_4, l_2, h_3
  105. Polygon(L_1, I_1, M_1) → poly4, m_2, l_3, i_3
  106. Polygon(M_1, G_1, J_1) → poly5, j_4, m_3, g_3
  107. Intersect(m, j_3) → N_1
  108. Segment(I_1, N_1) → n_1
  109. Intersect(j_3, i_3) → O_1
  110. Intersect(c', i_3) → Q_1
  111. Intersect(c', i_3) → R_1
  112. Intersect(c', i_3) → S_1
  113. Intersect(c', i_3) → U_1
  114. Intersect(c', i_3) → A_2
  115. Intersect(l_2, i_3) → B_2
  116. Segment(B_2, O_1) → n_2
  117. Segment(B_2, M_1) → r_2
  118. Midpoint(M_1, L_1) → C_2
  119. Midpoint(L_1, K_1) → D_2
  120. Midpoint(K_1, J_1) → E_2
  121. Midpoint(J_1, M_1) → F_2
  122. Segment(G_1, F_2) → s_2
  123. Segment(C_2, I_1) → t_2
  124. Segment(D_2, H_1) → a_3
  125. Segment(E_2, F_1) → b_3
  126. Polygon(J_1, M_1, L_1, K_1) → poly6, j_5, m_4, l_4, k_5
  127. Intersect(j_3, d_1) → P_1
  128. Segment(P_1, M_1) → c_3
  129. Midpoint(p) → V_1
  130. OrthogonalLine(V_1, p) → d_3
  131. Point(d_3) → W_1
  132. Vector(W_1, V_1) → u_2
  133. Midpoint(r) → T_1
  134. Midpoint(a_3) → Z_1
  135. Point(a_2) → G_2
  136. Vector(G_2, T_1) → v_2
  137. Vector(G_2, Z_1) → w_2
  138. Point(Circle[H_2, s]) → I_2
  139. Segment(H_2, I_2) → e_3
  140. Line(H_2, I_2) → b
  141. Circle(I_2, (3 * s / 4.5)) → p_3
  142. Intersect(p_3, b, 2) → J_2
  143. Rotate(J_2, 30°, I_2) → K_2
  144. Segment(I_2, K_2) → n_3
  145. Line(K_2, b) → q_3
  146. Line(H_2, n_3) → r_3
  147. Intersect(q_3, r_3) → L_2
  148. Segment(H_2, K_2) → s_3
  149. Segment(I_2, L_2) → t_3
  150. Intersect(s_3, t_3) → M_2
  151. Circle(M_2, h) → c_4
  152. OrthogonalLine(M_2, b) → a_4
  153. Intersect(c_4, a_4, 2) → N_2
  154. Polygon(H_2, N_2, I_2) → poly7, i_4, h_4, n_4
  155. Polygon(I_2, N_2, K_2) → poly8, k_6, i_5, n_5
  156. Polygon(H_2, L_2, K_2, I_2) → poly9, h_5, l_6, k_7, i_6
  157. Polygon(H_2, N_2, L_2) → poly10, l_7, h_6, n_6
  158. Polygon(L_2, N_2, K_2) → poly11, k_8, l_8, n_7
  159. Midpoint(k_7) → O_2
  160. Segment(M_2, O_2) → b_4
  161. Segment(M_2, N_2) → d_4
  162. Segment(N_2, O_2) → e_4
  163. Angle(O_2, M_2, N_2) → β
  164. Midpoint(N_2, M_2) → P_2
  165. Midpoint(O_2, N_2) → Q_2
  166. Line(P_2, Q_2) → f_4
  167. Circle(P_2, s) → g_4
  168. Intersect(g_4, f_4, 1) → R_2
  169. Intersect(g_4, f_4, 2) → S_2
  170. Vector(R_2, P_2) → z_2
  171. Vector(S_2, Q_2) → u_3
  172. Midpoint(b_3) → V_2

AI 绘图提示语参考

绘制一个正方形金字塔的表面展开图，首先创建一个正方形底面和四个等腰三角形侧面。通过旋转和移动三角形，将它们围绕底面排列，形成展开图。注意保持边长和角度不变，确保展开图的正确性。
以上提示语为根据原始绘图命令整理而来，仅供参考。