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椭圆上的台球反弹轨迹（一）

椭圆上的台球反弹轨迹（一）
几何画板及绘图内容加载中，请稍等……
2018-04-20 22:47
函数变换几何作图轨迹
本范例演示了在椭圆边界内，台球（质点）沿直线运动并在椭圆上发生镜面反射的连续轨迹。核心数学概念是椭圆的光学性质：从椭圆一个焦点发出的光线经椭圆反射后必经过另一个焦点，因此台球轨迹会形成闭合或准周期的路径。作图要点：使用GeoGebra的Point、Ellipse、Line、Tangent、Mirror、Segment、Intersect等命令，依次构造椭圆c、初始点A、入射直线f、切点B、切线g、反射线h，并通过Mirror命令实现反射，再通过Intersect求交点，反复迭代生成多段反射线段（如i、j、k、l、m、n、p、q、r、s、t等），最终形成完整的反弹路径。适合高中及以上学生探究椭圆几何性质、反射定律与动态几何作图，也可用于大学解析几何或物理光学中的反射轨迹模拟。

GeoGebra 命令及参数

  1. Point(xAxis) → A_1
  2. Point(xAxis) → B_1
  3. Ellipse(A_1, B_1, A) → c
  4. Point(c) → B
  5. Line(A, B) → f
  6. Tangent(B, c) → g,
  7. Mirror(f, g) → f'
  8. Segment(A, B) → h
  9. Intersect(c, f', 2) → C
  10. Segment(B, C) → i
  11. Tangent(C, c) → j,
  12. Mirror(f', j) → f''
  13. Intersect(c, f'', 2) → E
  14. Segment(C, E) → k
  15. Tangent(E, c) → l,
  16. Mirror(f'', l) → f'''
  17. Intersect(c, f''', 2) → F
  18. Segment(E, F) → m
  19. Tangent(F, c) → n,
  20. Mirror(f''', n) → p
  21. Intersect(c, p, 2) → G
  22. Segment(G, F) → q
  23. Tangent(G, c) → r,
  24. Mirror(p, r) → p'
  25. Intersect(c, p', 2) → H
  26. Tangent(H, c) → s,
  27. Mirror(p', s) → p''
  28. Segment(G, H) → t
  29. Intersect(c, p'', 2) → I
  30. Tangent(I, c) → a,
  31. Mirror(p'', a) → p'''
  32. Segment(I, H) → b
  33. Intersect(c, p''', 2) → J
  34. Tangent(J, c) → d,
  35. Mirror(p''', d) → e
  36. Intersect(c, e, 2) → K
  37. Segment(K, J) → f_1
  38. Tangent(K, c) → g_1,
  39. Mirror(e, g_1) → e'
  40. Intersect(c, e', 2) → L
  41. Segment(L, K) → h_1
  42. Tangent(L, c) → i_1,
  43. Mirror(e', i_1) → e''
  44. Intersect(c, e'', 2) → M
  45. Segment(M, L) → j_1
  46. Tangent(M, c) → k_1,
  47. Mirror(e'', k_1) → e'''
  48. Intersect(c, e''', 2) → N
  49. Segment(N, M) → l_1
  50. Tangent(N, c) → m_1,
  51. Mirror(e''', m_1) → n_1
  52. Intersect(c, n_1, 2) → O
  53. Segment(I, J) → p_1
  54. Line(M, N) → q_1
  55. Tangent(N, c) → r_1,
  56. Mirror(q_1, r_1) → s_1
  57. Line(O, N) → t_1
  58. Tangent(O, c) → a_1,
  59. Mirror(t_1, a_1) → b_1
  60. Intersect(c, b_1, 1) → Q
  61. Segment(O, N) → c_1
  62. Segment(Q, O) → d_1
  63. Tangent(Q, c) → e_1,
  64. Mirror(b_1, e_1) → f_2
  65. Intersect(c, f_2, 1) → R
  66. Segment(R, Q) → g_2
  67. Segment(Q, R) → h_2
  68. Tangent(R, c) → i_2,
  69. Mirror(f_2, i_2) → j_2
  70. Intersect(c, j_2, 1) → S
  71. Segment(S, R) → k_2
  72. Tangent(S, c) → l_2,
  73. Mirror(j_2, l_2) → m_2
  74. Intersect(c, m_2, 1) → T
  75. Segment(T, S) → n_2
  76. Tangent(T, c) → p_2,
  77. Mirror(m_2, p_2) → q_2
  78. Intersect(c, q_2, 1) → U
  79. Segment(U, T) → r_2
  80. Segment(U, T) → s_2
  81. Tangent(U, c) → t_2,
  82. Mirror(q_2, t_2) → a_2
  83. Intersect(c, a_2, 1) → V
  84. Segment(V, U) → b_2
  85. Tangent(V, c) → c_2,
  86. Mirror(a_2, c_2) → d_2
  87. Intersect(c, d_2, 1) → W
  88. Line(V, U) → e_2
  89. Tangent(V, c) → f_3,
  90. Mirror(e_2, f_3) → g_3
  91. Segment(V, W) → h_3
  92. Tangent(W, c) → i_3,
  93. Mirror(g_3, i_3) → j_3
  94. Intersect(c, j_3, 1) → D
  95. Segment(D, W) → k_3
  96. Tangent(D, c) → l_3,
  97. Mirror(j_3, l_3) → m_3
  98. Intersect(c, m_3, 1) → P
  99. Segment(P, D) → n_3
  100. Tangent(P, c) → p_3,
  101. Mirror(m_3, p_3) → q_3
  102. Intersect(c, q_3, 1) → Z
  103. Segment(Z, P) → r_3
  104. Tangent(Z, c) → s_3,
  105. Mirror(q_3, s_3) → t_3
  106. Intersect(c, t_3, 1) → C_1
  107. Segment(C_1, Z) → a_3
  108. Tangent(C_1, c) → b_3,
  109. Mirror(t_3, b_3) → c_3
  110. Intersect(c, c_3, 1) → D_1
  111. Segment(D_1, C_1) → d_3
  112. Line(D_1, C_1) → e_3
  113. Tangent(D_1, c) → f_4,
  114. Mirror(e_3, f_4) → g_4
  115. Intersect(c, g_4, 1) → E_1
  116. Segment(E_1, D_1) → h_4
  117. Tangent(E_1, c) → i_4,
  118. Mirror(g_4, i_4) → j_4
  119. Intersect(c, j_4, 1) → F_1
  120. Segment(F_1, E_1) → k_4
  121. Tangent(F_1, c) → l_4,
  122. Mirror(j_4, l_4) → m_4
  123. Intersect(c, m_4, 1) → G_1
  124. Segment(G_1, F_1) → n_4
  125. Tangent(G_1, c) → p_4,
  126. Mirror(m_4, p_4) → q_4
  127. Intersect(c, q_4, 1) → H_1
  128. Segment(H_1, G_1) → r_4
  129. Tangent(H_1, c) → s_4,
  130. Mirror(q_4, s_4) → t_4
  131. Intersect(c, t_4, 1) → I_1
  132. Segment(I_1, H_1) → a_4
  133. Tangent(I_1, c) → b_4,
  134. Mirror(t_4, b_4) → c_4
  135. Intersect(c, c_4, 1) → J_1
  136. Segment(J_1, I_1) → d_4
  137. Tangent(J_1, c) → e_4,
  138. Mirror(c_4, e_4) → f_5
  139. Intersect(c, f_5, 1) → K_1
  140. Segment(K_1, J_1) → g_5
  141. Tangent(K_1, c) → h_5,
  142. Mirror(f_5, h_5) → i_5
  143. Intersect(c, i_5, 1) → L_1
  144. Segment(L_1, K_1) → j_5
  145. Tangent(L_1, c) → k_5,
  146. Mirror(i_5, k_5) → l_5
  147. Intersect(c, l_5, 1) → M_1
  148. Segment(M_1, L_1) → m_5
  149. Line(M_1, L_1) → n_5
  150. Tangent(M_1, c) → p_5,
  151. Mirror(n_5, p_5) → q_5
  152. Intersect(c, q_5, 1) → N_1
  153. Segment(M_1, N_1) → r_5
  154. Tangent(N_1, c) → s_5,
  155. Mirror(q_5, s_5) → t_5
  156. Intersect(c, t_5, 1) → O_1
  157. Tangent(O_1, c) → a_5,
  158. Mirror(t_5, a_5) → b_5
  159. Intersect(c, b_5, 1) → P_1
  160. Segment(P_1, O_1) → c_5
  161. Tangent(P_1, c) → d_5,
  162. Mirror(b_5, d_5) → e_5
  163. Intersect(c, e_5, 1) → Q_1
  164. Segment(Q_1, P_1) → f_6
  165. Tangent(Q_1, c) → g_6,
  166. Mirror(e_5, g_6) → h_6
  167. Intersect(c, h_6, 1) → R_1
  168. Segment(R_1, Q_1) → i_6
  169. Tangent(R_1, c) → j_6,
  170. Mirror(h_6, j_6) → k_6
  171. Intersect(c, k_6, 1) → S_1
  172. Segment(S_1, R_1) → l_6
  173. Tangent(S_1, c) → m_6,
  174. Mirror(k_6, m_6) → n_6
  175. Intersect(c, n_6, 1) → T_1
  176. Segment(T_1, S_1) → p_6
  177. Tangent(T_1, c) → q_6,
  178. Mirror(n_6, q_6) → r_6
  179. Intersect(c, r_6, 1) → U_1
  180. Segment(U_1, T_1) → s_6
  181. Tangent(U_1, c) → t_6,
  182. Mirror(r_6, t_6) → a_6
  183. Intersect(c, a_6, 1) → V_1
  184. Segment(V_1, U_1) → b_6
  185. Tangent(V_1, c) → c_6,
  186. Mirror(a_6, c_6) → d_6
  187. Intersect(c, d_6, 1) → W_1
  188. Segment(W_1, V_1) → e_6
  189. Tangent(W_1, c) → f_7,
  190. Mirror(d_6, f_7) → g_7
  191. Intersect(c, g_7, 1) → Z_1
  192. Segment(U_1, T_1) → h_7
  193. Segment(N_1, O_1) → k_7

AI 绘图提示语参考

绘制一个椭圆，在其内部从椭圆上一点出发画一条线段，该线段指向椭圆的一个焦点。在椭圆上找到该线段与椭圆的交点，过该交点作椭圆的切线，将入射线段关于切线反射，得到反射线段，反射线段将经过另一个焦点。重复此过程：每次从新的交点出发，作切线、反射、求下一交点，依次生成多条首尾相连的折线段，形成台球在椭圆内的连续反弹轨迹。最终图形包含椭圆、初始点、一系列切线和反射线段，展示椭圆的光学性质。
以上提示语为根据原始绘图命令整理而来，仅供参考。