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曲面与移动平面交线演示

曲面与移动平面交线演示
几何画板及绘图内容加载中，请稍等……
2016-04-28 20:12
向量函数立体几何几何作图
①本范例演示曲面与可移动平面交线的动态变化过程；②涉及的核心数学概念包括三维向量、函数图像、对数函数、多边形构造、几何变换等；③作图要点运用了Function, If, Polygon, PointIn, Segment, Textfield, Point, Sum, Length, Sequence, RemoveUndefined, Zip, Append, PolyLine, ImplicitCurve, IntersectPath等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者用于理解空间几何与函数图像的交互。

GeoGebra 命令及参数

  1. Function(f(a, b), a, x_{min} - 0.2, x_{max} + 0.2, b, y_{min} - 0.2, y_{max} + 0.2) → g
  2. If(Length[RemoveUndefined[Join[Zip[Mirror[{Roots[f0(a, x), y_{min}, y_{max}]}, y = x] + (a, 0), a, Sequence[x_{min} + ((x_{max} - x_{min}) / N_P * i), i, 0, N_P]]]]] ≟ 0, false, true) → lb1
  3. If(lb1, RemoveUndefined[Join[Zip[Mirror[{Roots[f0(a, x), y_{min}, y_{max}]}, y = x] + (a, 0), a, Sequence[x_{min} + ((x_{max} - x_{min}) / N_P * i), i, 0, N_P]]]], Sequence[(10000, 10000), k, 1, 50]) → P1xxy
  4. If(lb1, RemoveUndefined[Join[Zip[Reverse[{Roots[f0(x, a), x_{min}, x_{max}]}] + (0, a), a, Sequence[y_{min} + ((y_{max} - y_{min}) / N_P * i), i, 0, N_P]]]], Sequence[(10000, 10000), k, 1, 50]) → P2yyx
  5. Polygon((x_{min}, y_{min}), (x_{max}, y_{min}), (x_{max}, y_{max}), (x_{min}, y_{max})) → poly1, b, c, d, e
  6. PointIn(poly1) → PTest
  7. Segment(B02, B01) → a
  8. Textfield(v_1) → inputBox2
  9. Textfield(f) → inputBox3
  10. Textfield(x_{min}) → inputBox4
  11. Textfield(x_{max}) → inputBox5
  12. Textfield(y_{max}) → inputBox6
  13. Textfield(z_{max}) → inputBox7
  14. Textfield(y_{min}) → inputBox8
  15. Textfield(z_{min}) → inputBox9
  16. Point(a, 1 - zzii_B) → A
  17. Textfield(N_P) → inputBox10
  18. Sum(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, 1, 0], i, 1, n]) → ext1
  19. Length(list1) → Lä
  20. Textfield(step) → inputBox12
  21. Textfield(zoom_{3D}) → InputBox1
  22. Sequence((x(Element[lr01, i]), y(Element[lr01, i]), f(x(Element[lr01, i]), y(Element[lr01, i]))), i, 1, n) → lr013
  23. Sequence((x(Element[lr02, i]), y(Element[lr02, i]), f(x(Element[lr02, i]), y(Element[lr02, i]))), i, 1, n) → lr023
  24. RemoveUndefined(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, i, NaN], i, 1, n]) → lr04
  25. Zip(Vector[b, a], a, Sequence[Element[lr01, Element[lr04, i]], i, 1, Length[lr04]], b, Sequence[Element[lr02, Element[lr04, i]], i, 1, Length[lr04]]) → lr06
  26. Zip(Vector[b, a], a, Sequence[Element[lr013, Element[lr04, i]], i, 1, Length[lr04]], b, Sequence[Element[lr023, Element[lr04, i]], i, 1, Length[lr04]]) → lr063
  27. Zip(Vector[a, b], a, Sequence[Element[lr01, Element[lr05, i]], i, 1, Length[lr05]], b, Sequence[Element[lr02, Element[lr05, i]], i, 1, Length[lr05]]) → lr07
  28. Zip(Vector[a, b], a, Sequence[Element[lr013, Element[lr05, i]], i, 1, Length[lr05]], b, Sequence[Element[lr023, Element[lr05, i]], i, 1, Length[lr05]]) → lr073
  29. RemoveUndefined(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, i, (-i)], i, 1, n]) → lr04c
  30. Append(lr04c, Element[lr04c, 1]) → lr04cc
  31. Sum(Sequence[If[(Element[lr04cc, i] * Element[lr04cc, i + 1]) < 0, 1, 0], i, 1, n + 1]) → lr04ccc
  32. If(ext1 ≟ n, "Max", If[ext1 ≟ 0, "Min", If[(lr04ccc > 2) ∧ (lr04ccc ≤ 4), "saddle ? ", "?"]]) → TT1
  33. PolyLine({(0, 0.5019607843), (0.1, 0.4980392157), (0.2, 0), (0.3, 0), (0.4, 0.2), (0.5, 0.2), (0.7, 1), (1, 1)}) → Ro
  34. PolyLine({(0, 0), (0.2, 0), (0.3, 0.6), (0.4, 1), (0.7, 1), (0.9, 0.2), (1, 0)}) → Gr
  35. PolyLine({(0, 0.5019607843), (0.1, 1), (0.3, 1), (0.4, 0.8), (0.5, 0), (0.6, 0), (0.7, 0.4), (0.8, 0.2), (0.9, 0.4), (1, 0)}) → Bl
  36. Point(Pxy) → D
  37. Textfield(z0) → InputBox2
  38. ImplicitCurve(g(x, y) - z0) → p
  39. IntersectPath(o, g(x, y)) → s