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飞行模拟器：双曲线与割线的三维几何演示

飞行模拟器：双曲线与割线的三维几何演示
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2015-06-24 02:26
多边形解析几何立体几何变换几何作图
所属专题 My 3D Collection 3D Geometry & Cross Sections
该范例通过一个名为“飞行模拟器”的交互式三维场景，直观演示了双曲线、割线、相似三角形以及矩形与正方形等几何元素在空间中的组合关系。核心数学概念包括双曲线的定义与性质、割线与曲线的交点、相似三角形的比例关系，以及平面图形（矩形、正方形）在三维空间中的构造。作图要点方面，使用了GeoGebra的Point、Cylinder、Cone、Sphere、Polygon、Intersect、Segment、Rotate、PointIn、Line等命令，创建了包括点A、B、C、D、E、F、G、H、I，线段a、b、c、d、e、f、g、h、i、j、k、l、m、n、p、q、r、s、t，以及多边形poly1、poly2、poly3和多个曲面对象（如圆柱、圆锥、球体）在内的复杂几何结构。该范例适合高中或大学低年级学生，用于理解解析几何与立体几何的交叉应用，也可作为数学建模或计算机辅助几何教学的生动案例，帮助学习者将抽象的圆锥曲线概念与直观的三维视觉模拟相结合。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Cylinder(A, B, 0.5) → a, c, d, b
  3. Cone(B, C, 0.5) → e, f, g
  4. Sphere(A, 0.5) → h
  5. Polygon(B, (4, 0, 3), (6, 0, 3), (6, 0, 2.5), (5, 0, 2.5), (5, 0, 0.5), (6, 0, 0.5), C, B) → poly1, b_1, i, j, k, l, m, n, c_1, b_2
  6. Intersect(k, l) → D
  7. Intersect(l, m) → E
  8. Intersect(j, k) → F
  9. Intersect(n, m) → G
  10. Polygon(D, F, G, E, D) → poly2, d_1, f_1, g_1, e_1, d_2
  11. Segment(A, B) → p
  12. Point(p) → H
  13. Point(p) → I
  14. Polygon(H, (-1, -8), (0, -8), (0, -4), (1, -4), I, H) → poly3, q, r, s, t, a_1, h_1, i_1
  15. Intersect(r, s) → J
  16. Point(poly3) → K
  17. Intersect(t, s) → L
  18. Intersect(t, a_1) → M
  19. Polygon(J, M, (1, -8), J) → poly4, j_1, m_1, k_1, j_2
  20. Rotate(poly3, 3.141592653589793, xAxis) → poly3'
  21. Rotate(poly4, 3.141592653589793, xAxis) → poly4'
  22. Polygon(C, (6, -0.5), (5, -0.5), (5, -4), (4, -4), (4, 4), (5, 4), (5, 0.5), (6, 0.5), C) → poly5, c_2, c_3, l_1, n_1, p_1, q_1, r_1, s_1, t_1, a_2
  23. Intersect(p_1, n_1) → N
  24. PointIn(poly5) → O
  25. Polygon(N, O, (6, -0.5), (6, -4), N) → poly6, n_2, o, e_2, f_2, n_3
  26. Rotate(poly6, 3.141592653589793, xAxis) → poly6'
  27. Rotate(a, up, yAxis) → a'
  28. Rotate(e, up, yAxis) → e'
  29. Rotate(h, up, yAxis) → h'
  30. Rotate(poly3, up, yAxis) → poly3'_1
  31. Rotate(poly4, up, yAxis) → poly4'_1
  32. Rotate(poly1, up, yAxis) → poly1'
  33. Rotate(poly5, up, yAxis) → poly5'
  34. Rotate(poly6, up, yAxis) → poly6'_1
  35. Rotate(poly2, up, yAxis) → poly2'
  36. Rotate(poly3', up, yAxis) → poly3''
  37. Rotate(poly4', up, yAxis) → poly4''
  38. Rotate(A, up, yAxis) → A'
  39. Rotate(B, up, yAxis) → B'
  40. Segment(A', B') → p'
  41. Rotate(a, ax, p) → a'_1
  42. Rotate(poly1, ax, p) → poly1'_1
  43. Rotate(poly2, ax, p) → poly2'_1
  44. Rotate(poly3, ax, p) → poly3'_2
  45. Rotate(poly4, ax, p) → poly4'_2
  46. Rotate(poly5, ax, p) → poly5'_1
  47. Rotate(poly6, ax, p) → poly6'_2
  48. Rotate(h, ax, p) → h'_1
  49. Rotate(e, ax, p) → e'_1
  50. Rotate(poly3', ax, p) → poly3''_1
  51. Rotate(poly4', ax, p) → poly4''_1
  52. Rotate(poly6', ax, p) → poly6''
  53. Rotate(poly3'_1, ax, p') → poly7
  54. Rotate(poly4'_1, ax, p') → poly8
  55. Rotate(poly3'', ax, p') → poly3'''
  56. Rotate(poly4'', ax, p') → poly4'''
  57. Rotate(poly1', ax, p') → poly1''
  58. Rotate(poly2', ax, p') → poly2''
  59. Rotate(poly5', 3.141592653589793, p') → poly5''
  60. Rotate(poly6'_1, ax, p') → poly9
  61. Rotate(poly5'', ax, p') → poly5'''
  62. Rotate(poly9, 3.141592653589793, p') → poly9'
  63. PointIn(poly9) → P
  64. PointIn(poly9) → Q
  65. Segment(P, Q) → g_2
  66. Rotate(poly9, (-up), g_2) → poly9'_2
  67. PointIn(poly5''') → R
  68. PointIn(poly5''') → S
  69. Segment(S, R) → h_2
  70. Rotate(poly9', up, h_2) → poly9''
  71. PointIn(poly1'') → T
  72. PointIn(poly2'') → U
  73. Segment(T, U) → i_2
  74. Rotate(poly2'', Rudder, i_2) → poly2'''
  75. Rotate(poly9'_2, Elevators, g_2) → poly10
  76. Rotate(poly9'', (-Elevators), h_2) → poly9'''
  77. Segment(J, L) → k_2
  78. PointIn(poly8) → V
  79. Polygon(J, L, M, V) → poly11, j_3, l_2, m_2, v
  80. Rotate(poly11, Ailerons, k_2) → poly11'
  81. Rotate(poly11', up, yAxis) → poly11''
  82. Rotate(M, up, yAxis) → M'
  83. PointIn(poly7) → W
  84. PointIn(poly7) → Z
  85. Line(W, Z) → p_2
  86. PointIn(poly7) → A_1
  87. PointIn(poly7) → B_1
  88. Point(p_2) → C_1
  89. Polygon(Z, A_1, B_1, C_1) → poly12, z_1, a_3, b_3, c_4
  90. Rotate(poly12, Ailerons, z_1) → poly12'
  91. PointIn(poly3''') → D_1
  92. PointIn(poly3''') → E_1
  93. Line(D_1, E_1) → q_2
  94. PointIn(poly3''') → F_1
  95. PointIn(poly3''') → G_1
  96. Point(q_2) → H_1
  97. Polygon(F_1, E_1, H_1, G_1) → poly13, f_3, e_3, h_3, g_3
  98. Rotate(poly13, (-Ailerons), f_3) → poly13'
  99. Rotate(h', (-Turn), zAxis) → h''
  100. Rotate(poly7, (-Turn), zAxis) → poly7'
  101. Rotate(poly11, Turn, zAxis) → poly11'_1
  102. Rotate(poly12', (-Turn), zAxis) → poly12''
  103. Rotate(poly3''', (-Turn), zAxis) → poly14
  104. Rotate(poly13', (-Turn), zAxis) → poly13''
  105. Rotate(poly5''', (-Turn), zAxis) → poly15
  106. Rotate(poly10, (-Turn), zAxis) → poly10'
  107. Rotate(poly9''', (-Turn), zAxis) → poly16
  108. Rotate(poly1'', (-Turn), zAxis) → poly1'''
  109. Rotate(poly2''', (-Turn), zAxis) → poly17
  110. Rotate(a', (-Turn), zAxis) → a''
  111. Rotate(e', (-Turn), zAxis) → e''
  112. PointIn(poly1''') → I_1
  113. PointIn(poly17) → J_1
  114. Segment(I_1, J_1) → r_2
  115. Rotate(poly17, (-Turn), r_2) → poly17'
  116. Rotate(A', Turn, zAxis) → A''
  117. Rotate(B', Turn, zAxis) → B''
  118. Rotate(A', (-Turn), zAxis) → A''_1
  119. Rotate(B', (-Turn), zAxis) → B''_1
  120. Segment(A''_1, B''_1) → p''
  121. Rotate(h'', (-Turn2), zAxis) → h'''
  122. Rotate(a'', (-Turn2), zAxis) → a'''
  123. Rotate(e'', (-Turn2), zAxis) → e'''
  124. Rotate(poly1''', (-Turn2), zAxis) → poly18
  125. Rotate(poly17', (-Turn2), zAxis) → poly17''
  126. Rotate(poly15, (-Turn2), zAxis) → poly15'
  127. Rotate(poly10', (-Turn2), zAxis) → poly10''
  128. Rotate(poly16, (-Turn2), zAxis) → poly16'
  129. Rotate(poly7', (-Turn2), zAxis) → poly7''
  130. Rotate(poly12'', (-Turn2), zAxis) → poly12'''
  131. Rotate(poly14, (-Turn2), zAxis) → poly14'
  132. Rotate(poly12'', (-Turn2), zAxis) → poly12'''_1
  133. Rotate(poly13'', (-Turn2), zAxis) → poly13'''
  134. PointIn(poly12'''_1) → K_1
  135. PointIn(poly12''') → L_1
  136. Segment(K_1, L_1) → s_2
  137. Rotate(poly12''', Turn2, s_2) → poly19
  138. PointIn(poly14') → M_1
  139. PointIn(poly13''') → N_1
  140. Segment(M_1, N_1) → t_2
  141. Rotate(poly13''', (-Turn2), t_2) → poly20
  142. Rotate(poly13''', (-Turn2), t_2) → poly21
  143. PointIn(h''') → O_1
  144. PointIn(poly18) → P_1
  145. Segment(O_1, P_1) → d_3
  146. Rotate(poly7'', (-Turn2), d_3) → poly7'''
  147. Rotate(poly19, (-Turn2), d_3) → poly19'
  148. Rotate(poly14', (-Turn2), d_3) → poly14''
  149. Rotate(poly20, (-Turn2), d_2) → poly20'
  150. Rotate(poly20', (-Turn2), d_2) → poly20''
  151. Rotate(poly18, (-Turn2), d_2) → poly18'
  152. Rotate(poly18, (-Turn2), d_3) → poly18'_1
  153. PointIn(poly14'') → Q_1
  154. PointIn(poly14') → R_1
  155. Rotate(poly17'', (-Turn2), d_3) → poly17'''
  156. Rotate(poly20', (-Turn2), d_3) → poly20''_1
  157. Rotate(poly15', (-Turn2), d_3) → poly15''
  158. Rotate(poly10'', (-Turn2), d_3) → poly10'''
  159. Rotate(poly20''_1, (-Turn2), d_3) → poly22
  160. Rotate(poly16', (-Turn2), d_3) → poly16''
  161. PointIn(poly14'') → S_1
  162. PointIn(poly14'') → T_1
  163. Line(S_1, T_1) → i_3
  164. PointIn(poly14'') → V_1
  165. PointIn(poly14'') → W_1
  166. Point(i_3) → Z_1
  167. Polygon(V_1, W_1, Z_1, T_1) → poly23, v_1, w_1, z_2, t_3
  168. Rotate(poly23, Ailerons, t_3) → poly23'
  169. Rotate(poly23, Turn2, t_3) → poly23'_1
  170. Rotate(poly23', Turn2, t_3) → poly23''
  171. PointIn(poly7''') → A_2
  172. PointIn(poly19') → B_2
  173. Segment(B_2, A_2) → k_3
  174. Rotate(poly19', (0.6 * ax), k_3) → poly19''
  175. PointIn(poly18'_1) → C_2
  176. PointIn(poly17''') → D_2
  177. Segment(C_2, D_2) → l_3
  178. Rotate(poly17''', (-0.6 * ax), l_3) → poly24
  179. PointIn(poly14'') → E_2
  180. PointIn(poly14'') → F_2
  181. Segment(E_2, F_2) → m_3
  182. Segment(E_2, F_2) → p_3
  183. Rotate(poly23'', (0.6 * ax), m_3) → poly23'''

AI 绘图提示语参考

绘制一个三维几何场景，包含一条水平轴上的圆柱、圆锥和球体，以及多个多边形平面。在空间中构造双曲线与割线，通过交点形成相似三角形和矩形。沿轴旋转这些平面，并围绕垂直轴和另一条线段进行旋转复制，生成对称的立体组合。整体呈现飞行模拟器的视觉效果，展示圆锥曲线与立体几何的交互关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。