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矩形网格上路径长度的动态探究

矩形网格上路径长度的动态探究
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2020-06-06 15:41
多边形数列解析几何概率统计滑块动画
本范例通过GeoGebra动态演示在矩形网格上随机生成路径并计算其总长度的过程。核心数学概念包括矩形网格的坐标表示、路径的逐段累加求和、随机实验以及有理数与无理数的长度对比。作图要点是利用Sequence命令生成一系列随机点，通过Insert、Join和Polygon命令构建路径，并使用Sum命令累加各段长度；对象包括width、length、list0以及A1至A25等点，最终通过texte1显示长度结果。该范例适合中学数学或大学预科阶段，用于理解路径长度计算、随机过程模拟以及几何与代数的结合，也可作为探究勾股定理在网格中应用的拓展材料。

GeoGebra 命令及参数

  1. Sequence(1, i, 1, length + (a * 0)) → list0
  2. Insert(0, A1, Random[1, Length[A1] + 1]) → A2
  3. Insert(0, A2, Random[1, Length[A2 + 1]]) → A3
  4. Insert(0, A3, Random[1, Length[A3 + 1]]) → A4
  5. Insert(0, A4, Random[1, Length[A4 + 1]]) → A5
  6. Insert(0, A5, Random[1, Length[A5 + 1]]) → A6
  7. Insert(0, A6, Random[1, Length[A6 + 1]]) → A7
  8. Insert(0, A7, Random[1, Length[A7 + 1]]) → A8
  9. Insert(0, A8, Random[1, Length[A8 + 1]]) → A9
  10. Insert(0, A9, Random[1, Length[A9 + 1]]) → A10
  11. Insert(0, A10, Random[1, Length[A10 + 1]]) → A11
  12. Insert(0, A11, Random[1, Length[A11 + 1]]) → A12
  13. Insert(0, A12, Random[1, Length[A12 + 1]]) → A13
  14. Insert(0, A13, Random[1, Length[A13 + 1]]) → A14
  15. Insert(0, A14, Random[1, Length[A14 + 1]]) → A15
  16. Insert(0, A15, Random[1, Length[A15 + 1]]) → A16
  17. Insert(0, A16, Random[1, Length[A16 + 1]]) → A17
  18. Insert(0, A17, Random[1, Length[A17 + 1]]) → A18
  19. Insert(0, A18, Random[1, Length[A18 + 1]]) → A19
  20. Insert(0, A19, Random[1, Length[A19 + 1]]) → A20
  21. Insert(0, A20, Random[1, Length[A20 + 1]]) → A21
  22. Insert(0, A21, Random[1, Length[A21 + 1]]) → A22
  23. Insert(0, A22, Random[1, Length[A22 + 1]]) → A23
  24. Insert(0, A23, Random[1, Length[A23 + 1]]) → A24
  25. Insert(0, A24, Random[1, Length[A24 + 1]]) → A25
  26. Element(L_1, width + 1) → list1
  27. Sequence(Sum[Take[list1, 1, i]], i, 1, length + width) → list2
  28. Sequence((0, 0) + (Element[list2, i] * (1, 0)) + ((i - Element[list2, i]) * (0, 1)), i, 1, length + width) → list3
  29. Join({{(0, 0)}, list3}) → list4
  30. Sequence(Vector[Element[list4, i], Element[list4, i + 1]], i, 1, length + width) → list5
  31. Sequence(Segment[(i, 0), (i, width)], i, 0, length) → list6
  32. Sequence(Segment[(0, i), (length, i)], i, 0, width) → list7
  33. Polygon(C, B, D, A) → poly1, c, b_1, d, a_1
  34. Sequence(Segment[Element[list4, i], Element[list4, i + 1]], i, 1, length + width) → liste1
  35. Sum(liste1) → e
  36. Sequence(If[y(Element[list4, i]) ≟ y(Element[list4, i + 1]), Segment[Element[list4, i], Element[list4, i + 1]], 0], i, 1, Length[list4]) → l2
  37. RemoveUndefined(l2) → l1
  38. Sequence(Midpoint[Element[l1, i]], i, 1, Length[l1]) → l3
  39. Sequence((x(Element[l3, i]), 0), i, 1, Length[l3]) → l3b
  40. Sequence(Vector[Element[l3, i], Element[l3b, i]], i, 1, Length[l3]) → l4
  41. Sequence(Translate[Element[l1, i], Vector[(g / 10 * Element[l4, i])]], i, 1, Length[l1]) → l1b
  42. Sequence(If[x(Element[list4, i]) ≟ x(Element[list4, i + 1]), Segment[Element[list4, i], Element[list4, i + 1]], 0], i, 1, Length[list4]) → verti1
  43. RemoveUndefined(verti1) → ver
  44. Sequence(Midpoint[Element[ver, i]], i, 1, Length[ver]) → milver
  45. Sequence((length, y(Element[milver, i])), i, 1, Length[l3]) → verb
  46. Sequence(Vector[Element[milver, i], Element[verb, i]], i, 1, Length[l3]) → l5
  47. Sequence(Translate[Element[ver, i], Vector[(g / 10 * Element[l5, i])]], i, 1, Length[ver]) → l6