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圆柱体错觉的GeoGebra绘本

圆柱体错觉的GeoGebra绘本
几何画板及绘图内容加载中，请稍等……
2020-04-14 11:51
函数三角函数解析几何立体几何函数图像
①本范例演示了圆柱体与三角函数结合的动态几何现象，通过参数化展示其表面与内部三角形的互动关系；②涉及的核心数学概念包括圆柱体构造、三角函数应用、对称性分析、抛物线与梯形曲线绘制、函数图像绘制等；③作图要点提炼自构造摘要，主要运用了Surface（曲面）、Polygon（多边形）、CurveCartesian（笛卡尔曲线）等命令，结合k, α, f_x, f_y, f_z, h, a, b, c, d, e, f, g, i, j, text1, text2, l, b', a', c', d', e', f', g', i', j', width, height, poly1, m, n, p, q, r, o, s, t, u, v等对象进行参数化控制与动态展示；④适合高中及以上数学学习者，可用于立体几何、解析几何、三角函数及函数图像综合应用的探究教学。

GeoGebra 命令及参数

  1. Surface((f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → a
  2. Surface((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → b
  3. Surface((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → c
  4. Surface((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → d
  5. Surface((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → e
  6. Surface(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → f
  7. Surface(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → g
  8. Surface((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → i
  9. Surface((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → j
  10. Surface(-9 - ((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → b'
  11. Surface(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → a'
  12. Surface(-9 - ((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → c'
  13. Surface(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → d'
  14. Surface(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → e'
  15. Surface(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → f'
  16. Surface(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → g'
  17. Surface(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → i'
  18. Surface(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → j'
  19. Polygon((-4.5, (-width), 0), (-4.5, width, 0), (-4.5, width, height), (-4.5, (-width), height)) → poly1, m, n, p, q
  20. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → r
  21. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → o
  22. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → s
  23. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → t
  24. CurveCartesian((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → u
  25. CurveCartesian((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → v
  26. CurveCartesian(-9 - ((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → w
  27. CurveCartesian(-9 - ((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → a_1
  28. CurveCartesian((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → b_1
  29. CurveCartesian((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → c_1
  30. CurveCartesian(-9 - ((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → d_1
  31. CurveCartesian(-9 - ((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → e_1
  32. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → f_1
  33. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → g_1
  34. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → h_1
  35. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → i_1
  36. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → j_1
  37. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → k_1
  38. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → l_1
  39. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → m_1
  40. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → n_1
  41. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → o_1
  42. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → p_1
  43. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → q_1
  44. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → r_1
  45. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → s_1
  46. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → t_1
  47. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → u_1
  48. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → v_1
  49. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → w_1
  50. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → a_2
  51. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → b_2
  52. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → c_2
  53. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → d_2
  54. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → e_2
  55. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → f_2