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矩形内球的反射轨迹

矩形内球的反射轨迹
几何画板及绘图内容加载中，请稍等……
2012-04-13 06:06
向量解析几何几何作图轨迹
①本范例演示了在矩形内反弹的小球轨迹，模拟无孔台球运动；②涉及核心数学概念包括反射定律、向量、解析几何、几何作图；③构造要点运用了Intersect（交点）、Vector（向量）、Segment（线段）、Ray（射线）、If（条件判断）、Element（元素）等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解轨迹生成与几何变换。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Vector(E, F) → dir
  3. Segment(B, C) → st
  4. Segment(C, D) → sr
  5. Segment(A, D) → sb
  6. Segment(A, B) → sl
  7. Ray(E, dir) → A2
  8. Intersect(A2, st) → B2
  9. Intersect(A2, sr) → C2
  10. Intersect(A2, sb) → D2
  11. Intersect(A2, sl) → E2
  12. If(Defined[B2] ∨ Defined[D2], (1, -1), (-1, 1)) → F2
  13. Element(RemoveUndefined[{B2, C2, D2, E2}], 1) → G2
  14. Ray(G2, Vector[H2]) → A3
  15. If(Intersect[A3, st] ≠ B2, Intersect[A3, st], (NaN, NaN)) → B3
  16. If(Intersect[A3, sr] ≠ C2, Intersect[A3, sr], (NaN, NaN)) → C3
  17. If(Intersect[A3, sb] ≠ D2, Intersect[A3, sb], (NaN, NaN)) → D3
  18. If(Intersect[A3, sl] ≠ E2, Intersect[A3, sl], (NaN, NaN)) → E3
  19. If(Defined[B3] ∨ Defined[D3], (1, -1), (-1, 1)) → F3
  20. Element(RemoveUndefined[{B3, C3, D3, E3}], 1) → G3
  21. Ray(G3, Vector[H3]) → A4
  22. If(Intersect[A4, st] ≠ B3, Intersect[A4, st], (NaN, NaN)) → B4
  23. If(Intersect[A4, sr] ≠ C3, Intersect[A4, sr], (NaN, NaN)) → C4
  24. If(Intersect[A4, sb] ≠ D3, Intersect[A4, sb], (NaN, NaN)) → D4
  25. If(Intersect[A4, sl] ≠ E3, Intersect[A4, sl], (NaN, NaN)) → E4
  26. If(Defined[B4] ∨ Defined[D4], (1, -1), (-1, 1)) → F4
  27. Element(RemoveUndefined[{B4, C4, D4, E4}], 1) → G4
  28. Ray(G4, Vector[H4]) → A5
  29. If(Intersect[A5, st] ≠ B4, Intersect[A5, st], (NaN, NaN)) → B5
  30. If(Intersect[A5, sr] ≠ C4, Intersect[A5, sr], (NaN, NaN)) → C5
  31. If(Intersect[A5, sb] ≠ D4, Intersect[A5, sb], (NaN, NaN)) → D5
  32. If(Intersect[A5, sl] ≠ E4, Intersect[A5, sl], (NaN, NaN)) → E5
  33. If(Defined[B5] ∨ Defined[D5], (1, -1), (-1, 1)) → F5
  34. Element(RemoveUndefined[{B5, C5, D5, E5}], 1) → G5
  35. Ray(G5, Vector[H5]) → A6
  36. If(Intersect[A6, st] ≠ B5, Intersect[A6, st], (NaN, NaN)) → B6
  37. If(Intersect[A6, sr] ≠ C5, Intersect[A6, sr], (NaN, NaN)) → C6
  38. If(Intersect[A6, sb] ≠ D5, Intersect[A6, sb], (NaN, NaN)) → D6
  39. If(Intersect[A6, sl] ≠ E5, Intersect[A6, sl], (NaN, NaN)) → E6
  40. If(Defined[B6] ∨ Defined[D6], (1, -1), (-1, 1)) → F6
  41. Element(RemoveUndefined[{B6, C6, D6, E6}], 1) → G6
  42. Ray(G6, Vector[H6]) → A7
  43. If(Intersect[A7, st] ≠ B6, Intersect[A7, st], (NaN, NaN)) → B7
  44. If(Intersect[A7, sr] ≠ C6, Intersect[A7, sr], (NaN, NaN)) → C7
  45. If(Intersect[A7, sb] ≠ D6, Intersect[A7, sb], (NaN, NaN)) → D7
  46. If(Intersect[A7, sl] ≠ E6, Intersect[A7, sl], (NaN, NaN)) → E7
  47. If(Defined[B7] ∨ Defined[D7], (1, -1), (-1, 1)) → F7
  48. Element(RemoveUndefined[{B7, C7, D7, E7}], 1) → G7
  49. Ray(G7, Vector[H7]) → A8
  50. If(Intersect[A8, st] ≠ B7, Intersect[A8, st], (NaN, NaN)) → B8
  51. If(Intersect[A8, sr] ≠ C7, Intersect[A8, sr], (NaN, NaN)) → C8
  52. If(Intersect[A8, sb] ≠ D7, Intersect[A8, sb], (NaN, NaN)) → D8
  53. If(Intersect[A8, sl] ≠ E7, Intersect[A8, sl], (NaN, NaN)) → E8
  54. If(Defined[B8] ∨ Defined[D8], (1, -1), (-1, 1)) → F8
  55. Element(RemoveUndefined[{B8, C8, D8, E8}], 1) → G8
  56. Ray(G8, Vector[H8]) → A9
  57. If(Intersect[A9, st] ≠ B8, Intersect[A9, st], (NaN, NaN)) → B9
  58. If(Intersect[A9, sr] ≠ C8, Intersect[A9, sr], (NaN, NaN)) → C9
  59. If(Intersect[A9, sb] ≠ D8, Intersect[A9, sb], (NaN, NaN)) → D9
  60. If(Intersect[A9, sl] ≠ E8, Intersect[A9, sl], (NaN, NaN)) → E9
  61. If(Defined[B9] ∨ Defined[D9], (1, -1), (-1, 1)) → F9
  62. Element(RemoveUndefined[{B9, C9, D9, E9}], 1) → G9
  63. Ray(G9, Vector[H9]) → A10
  64. If(Intersect[A10, st] ≠ B9, Intersect[A10, st], (NaN, NaN)) → B10
  65. If(Intersect[A10, sr] ≠ C9, Intersect[A10, sr], (NaN, NaN)) → C10
  66. If(Intersect[A10, sb] ≠ D9, Intersect[A10, sb], (NaN, NaN)) → D10
  67. If(Intersect[A10, sl] ≠ E9, Intersect[A10, sl], (NaN, NaN)) → E10
  68. If(Defined[B10] ∨ Defined[D10], (1, -1), (-1, 1)) → F10
  69. Element(RemoveUndefined[{B10, C10, D10, E10}], 1) → G10
  70. Ray(G10, Vector[H10]) → A11
  71. If(Intersect[A11, st] ≠ B10, Intersect[A11, st], (NaN, NaN)) → B11
  72. If(Intersect[A11, sr] ≠ C10, Intersect[A11, sr], (NaN, NaN)) → C11
  73. If(Intersect[A11, sb] ≠ D10, Intersect[A11, sb], (NaN, NaN)) → D11
  74. If(Intersect[A11, sl] ≠ E10, Intersect[A11, sl], (NaN, NaN)) → E11
  75. If(Defined[B11] ∨ Defined[D11], (1, -1), (-1, 1)) → F11
  76. Element(RemoveUndefined[{B11, C11, D11, E11}], 1) → G11
  77. Ray(G11, Vector[H11]) → A12
  78. If(Intersect[A12, st] ≠ B11, Intersect[A12, st], (NaN, NaN)) → B12
  79. If(Intersect[A12, sr] ≠ C11, Intersect[A12, sr], (NaN, NaN)) → C12
  80. If(Intersect[A12, sb] ≠ D11, Intersect[A12, sb], (NaN, NaN)) → D12
  81. If(Intersect[A12, sl] ≠ E11, Intersect[A12, sl], (NaN, NaN)) → E12
  82. If(Defined[B12] ∨ Defined[D12], (1, -1), (-1, 1)) → F12
  83. Element(RemoveUndefined[{B12, C12, D12, E12}], 1) → G12
  84. Ray(G12, Vector[H12]) → A13
  85. If(Intersect[A13, st] ≠ B12, Intersect[A13, st], (NaN, NaN)) → B13
  86. If(Intersect[A13, sr] ≠ C12, Intersect[A13, sr], (NaN, NaN)) → C13
  87. If(Intersect[A13, sb] ≠ D12, Intersect[A13, sb], (NaN, NaN)) → D13
  88. If(Intersect[A13, sl] ≠ E12, Intersect[A13, sl], (NaN, NaN)) → E13
  89. If(Defined[B13] ∨ Defined[D13], (1, -1), (-1, 1)) → F13
  90. Element(RemoveUndefined[{B13, C13, D13, E13}], 1) → G13
  91. If(colno ≥ J2, Segment[F, G2]) → I2
  92. If(colno ≥ J2, Segment[G2, G3]) → I3
  93. If(colno ≥ J3, Segment[G3, G4]) → I4
  94. If(colno ≥ J4, Segment[G4, G5]) → I5
  95. If(colno ≥ J5, Segment[G5, G6]) → I6
  96. If(colno ≥ J6, Segment[G6, G7]) → I7
  97. If(colno ≥ J7, Segment[G7, G8]) → I8
  98. If(colno ≥ J8, Segment[G8, G9]) → I9
  99. If(colno ≥ J9, Segment[G9, G10]) → I10
  100. If(colno ≥ J10, Segment[G10, G11]) → I11
  101. If(colno ≥ J11, Segment[G11, G12]) → I12
  102. If(colno ≥ J12, Segment[G12, G13]) → I13
  103. Ray(G13, Vector[H13]) → A14
  104. If(Intersect[A14, st] ≠ B13, Intersect[A14, st], (NaN, NaN)) → B14
  105. If(Intersect[A14, sr] ≠ C13, Intersect[A14, sr], (NaN, NaN)) → C14
  106. If(Intersect[A14, sb] ≠ D13, Intersect[A14, sb], (NaN, NaN)) → D14
  107. If(Intersect[A14, sl] ≠ E13, Intersect[A14, sl], (NaN, NaN)) → E14
  108. If(Defined[B14] ∨ Defined[D14], (1, -1), (-1, 1)) → F14
  109. Element(RemoveUndefined[{B14, C14, D14, E14}], 1) → G14
  110. If(colno ≥ J13, Segment[G13, G14]) → I14
  111. Ray(G14, Vector[H14]) → A15
  112. If(Intersect[A15, st] ≠ B14, Intersect[A15, st], (NaN, NaN)) → B15
  113. If(Intersect[A15, sr] ≠ C14, Intersect[A15, sr], (NaN, NaN)) → C15
  114. If(Intersect[A15, sb] ≠ D14, Intersect[A15, sb], (NaN, NaN)) → D15
  115. If(Intersect[A15, sl] ≠ E14, Intersect[A15, sl], (NaN, NaN)) → E15
  116. If(Defined[B15] ∨ Defined[D15], (1, -1), (-1, 1)) → F15
  117. Element(RemoveUndefined[{B15, C15, D15, E15}], 1) → G15
  118. If(colno ≥ J14, Segment[G14, G15]) → I15
  119. Ray(G15, Vector[H15]) → A16
  120. If(Intersect[A16, st] ≠ B15, Intersect[A16, st], (NaN, NaN)) → B16
  121. If(Intersect[A16, sr] ≠ C15, Intersect[A16, sr], (NaN, NaN)) → C16
  122. If(Intersect[A16, sb] ≠ D15, Intersect[A16, sb], (NaN, NaN)) → D16
  123. If(Intersect[A16, sl] ≠ E15, Intersect[A16, sl], (NaN, NaN)) → E16
  124. If(Defined[B16] ∨ Defined[D16], (1, -1), (-1, 1)) → F16
  125. Element(RemoveUndefined[{B16, C16, D16, E16}], 1) → G16
  126. If(colno ≥ J15, Segment[G15, G16]) → I16
  127. Ray(G16, Vector[H16]) → A17
  128. If(Intersect[A17, st] ≠ B16, Intersect[A17, st], (NaN, NaN)) → B17
  129. If(Intersect[A17, sr] ≠ C16, Intersect[A17, sr], (NaN, NaN)) → C17
  130. If(Intersect[A17, sb] ≠ D16, Intersect[A17, sb], (NaN, NaN)) → D17
  131. If(Intersect[A17, sl] ≠ E16, Intersect[A17, sl], (NaN, NaN)) → E17
  132. If(Defined[B17] ∨ Defined[D17], (1, -1), (-1, 1)) → F17
  133. Element(RemoveUndefined[{B17, C17, D17, E17}], 1) → G17
  134. If(colno ≥ J16, Segment[G16, G17]) → I17
  135. Ray(G17, Vector[H17]) → A18
  136. If(Intersect[A18, st] ≠ B17, Intersect[A18, st], (NaN, NaN)) → B18
  137. If(Intersect[A18, sr] ≠ C17, Intersect[A18, sr], (NaN, NaN)) → C18
  138. If(Intersect[A18, sb] ≠ D17, Intersect[A18, sb], (NaN, NaN)) → D18
  139. If(Intersect[A18, sl] ≠ E17, Intersect[A18, sl], (NaN, NaN)) → E18
  140. If(Defined[B18] ∨ Defined[D18], (1, -1), (-1, 1)) → F18
  141. Element(RemoveUndefined[{B18, C18, D18, E18}], 1) → G18
  142. If(colno ≥ J17, Segment[G17, G18]) → I18
  143. Ray(G18, Vector[H18]) → A19
  144. If(Intersect[A19, st] ≠ B18, Intersect[A19, st], (NaN, NaN)) → B19
  145. If(Intersect[A19, sr] ≠ C18, Intersect[A19, sr], (NaN, NaN)) → C19
  146. If(Intersect[A19, sb] ≠ D18, Intersect[A19, sb], (NaN, NaN)) → D19
  147. If(Intersect[A19, sl] ≠ E18, Intersect[A19, sl], (NaN, NaN)) → E19
  148. If(Defined[B19] ∨ Defined[D19], (1, -1), (-1, 1)) → F19
  149. Element(RemoveUndefined[{B19, C19, D19, E19}], 1) → G19
  150. If(colno ≥ J18, Segment[G18, G19]) → I19
  151. Ray(G19, Vector[H19]) → A20
  152. If(Intersect[A20, st] ≠ B19, Intersect[A20, st], (NaN, NaN)) → B20
  153. If(Intersect[A20, sr] ≠ C19, Intersect[A20, sr], (NaN, NaN)) → C20
  154. If(Intersect[A20, sb] ≠ D19, Intersect[A20, sb], (NaN, NaN)) → D20
  155. If(Intersect[A20, sl] ≠ E19, Intersect[A20, sl], (NaN, NaN)) → E20
  156. If(Defined[B20] ∨ Defined[D20], (1, -1), (-1, 1)) → F20
  157. Element(RemoveUndefined[{B20, C20, D20, E20}], 1) → G20
  158. If(colno ≥ J19, Segment[G19, G20]) → I20
  159. Ray(G20, Vector[H20]) → A21
  160. If(Intersect[A21, st] ≠ B20, Intersect[A21, st], (NaN, NaN)) → B21
  161. If(Intersect[A21, sr] ≠ C20, Intersect[A21, sr], (NaN, NaN)) → C21
  162. If(Intersect[A21, sb] ≠ D20, Intersect[A21, sb], (NaN, NaN)) → D21
  163. If(Intersect[A21, sl] ≠ E20, Intersect[A21, sl], (NaN, NaN)) → E21
  164. If(Defined[B21] ∨ Defined[D21], (1, -1), (-1, 1)) → F21
  165. Element(RemoveUndefined[{B21, C21, D21, E21}], 1) → G21
  166. If(colno ≥ J20, Segment[G20, G21]) → I21