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体育中的抛物线：网球发球

体育中的抛物线：网球发球
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2020-02-02 16:57
向量函数解析几何几何作图
①本范例演示网球发球时球运动的抛物线轨迹；②涉及的核心数学概念包括抛物线、解析几何、向量、函数图像；③作图要点提炼自GeoGebra命令：Line, Point, Segment, OrthogonalLine, Intersect, Polygon, Midpoint, Rotate, Angle, Function, Vector, Translate, If, Mirror, CircumcircleArc等，用于构建和分析抛物线及相关几何元素；④适合高中阶段学生学习抛物线及其在物理运动中的应用，也可用于数学建模教学。

GeoGebra 命令及参数

  1. Line(S, xAxis) → n_1
  2. Line(F_1, xAxis) → m
  3. Line(M, m) → k
  4. Point(m) → L
  5. Line(S, I_1) → t
  6. Point(t) → D_1
  7. Line(I_1, xAxis) → s
  8. Point(xAxis) → E_1
  9. Segment(Service, G) → a
  10. Point(a) → F
  11. OrthogonalLine(F, a) → i
  12. Point(i) → hautfilet
  13. Segment(F, hautfilet) → e
  14. Line(L, e) → n
  15. Line(F_1, n) → b_1
  16. Intersect(s, n) → C_1
  17. Polygon(D_1, E_1, C_1, I_1) → poly1, d_1, e_1, c_1, i_1
  18. Point(e_1) → K_1
  19. Midpoint(K_1, C_1) → S_1
  20. Midpoint(S_1, C_1) → U_1
  21. Line(U_1 + (0, 0.2), n_1) → q_1
  22. Line(K_1, q_1) → s_1
  23. Intersect(s_1, n) → A_2
  24. Line(K_1, k) → a_1
  25. Intersect(a_1, b_1) → P_1
  26. Intersect(n, k) → N
  27. Intersect(f, i) → Ballhaute, A
  28. Intersect(f, yAxis) → ball
  29. Line(ball, xAxis) → b
  30. Intersect(f, b) → B, E
  31. Midpoint(E, ball) → J
  32. Midpoint(ball, J) → H
  33. Intersect(f, xAxis) → C, Impact
  34. Rotate(E, α, ball) → E'
  35. Angle(E, ball, E') → β
  36. Segment(Service, G) → c
  37. Point(xAxis) → fond
  38. Segment(G, fond) → d
  39. Intersect(e, xAxis) → D
  40. Line(hautfilet, D) → g
  41. Function(f, 0, x(Impact)) → h
  42. Rotate(H, α, ball) → J'
  43. Vector(ball, J') → u
  44. Point(xAxis) → O
  45. Segment(fond, Service) → j
  46. Segment(W, Z) → l
  47. Translate(W, (x(Impact) / 18.5 * Vector[W, Z])) → W'
  48. If(y(Ballhaute) ≟ y(hautfilet), 1, If[y(Ballhaute) < y(hautfilet), 2, 0]) → mm
  49. Function(f, 0, x(hautfilet)) → p
  50. Segment(W, W') → q
  51. Segment(R, Q) → r
  52. Line(E_1, n) → f_1
  53. Intersect(f_1, a_1) → L_1
  54. Intersect(k, f_1) → M_1
  55. Intersect(f_1, m) → N_1
  56. Intersect(k, b_1) → O_1
  57. Polygon(N, A_2, P_1, M) → poly3, p_1, t_1, a_2, b_2
  58. Polygon(F_1, L, N, O_1) → poly2, g_1, h_1, j_1, k_1
  59. Polygon(N, A_2, P_1, M) → poly4, l_1, m_1, c_2, d_2
  60. Midpoint(d_1) → Q_1
  61. Midpoint(c_1) → R_1
  62. Midpoint(i_1) → T_1
  63. Midpoint(Q_1, R_1) → V_1
  64. Intersect(n_1, e_1) → W_1
  65. Midpoint(S, W_1) → Z_1
  66. Line(Q_1, R_1) → r_1
  67. Segment(Z_1, V_1) → e_2
  68. Intersect(r_1, q_1) → B_2
  69. Intersect(i_1, q_1) → C_2
  70. Line(B_2, xAxis) → f_2
  71. Intersect(e_1, f_2) → D_2
  72. Segment(C_2, D_2) → g_2
  73. Segment(V_1, B_2) → h_2
  74. Segment(W_1, S) → i_2
  75. Midpoint(Q_1, E_1) → E_2
  76. Midpoint(E_2, E_1) → F_2
  77. Midpoint(D_1, Q_1) → G_2
  78. Midpoint(D_1, G_2) → H_2
  79. Segment(H_2, J_1) → k_2
  80. Mirror(H_2, J_1) → I_2
  81. Mirror(J_1, J_1) → J_2
  82. Segment(I_2, J_2) → l_2
  83. Mirror(J_1, R_1) → K_2
  84. Segment(F_2, K_2) → j_2
  85. Intersect(g_2, k_2) → L_2
  86. Intersect(i_2, k_2) → M_2
  87. Intersect(j_2, i_2) → N_2
  88. Intersect(g_2, j_2) → O_2
  89. Segment(Z_1, N_2) → m_2
  90. Segment(Z_1, M_2) → n_2
  91. Segment(L_2, O_2) → p_2
  92. CircumcircleArc(Q, R, P_2) → q_2
  93. Segment(Q_2, R_2) → r_2
  94. Intersect(l, r_2) → S_2
  95. Segment(Q, P_2) → s_2
  96. Translate(Q, (x(Impact) / 18.5 * Vector[Q, P_2])) → Q'
  97. CircumcircleArc(Q', R, Q) → t_2
  98. Midpoint(q) → T_2
  99. Vector(W, W') → v
  100. Vector(Q, Q') → w
  101. Translate(W, (Animation / x(Impact) * v)) → W'_1
  102. Segment(W, W'_1) → a_3
  103. Translate(Q, (Animation / x(Impact) * w)) → Q'_1