积极 AI Logo 积极 AI

计算两曲面交点

计算两曲面交点
几何画板及绘图内容加载中，请稍等……
2016-04-28 22:20
函数解析几何立体几何最值优化参数方程
所属专题 intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve My 3D Collection
本范例演示了如何计算两个显式曲面的交点。涉及的核心数学概念包括解析几何和立体几何。作图要点包括使用 GeoGebra 命令如 Function, If, Polygon, PointIn, Segment, Textfield, Point, Sum, Zip, Length, Sequence, RemoveUndefined, Append, PolyLine, ImplicitCurve, IntersectPath 等对象。适合高中及以上学生学习和探索曲面交点问题。

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[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
  4. 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
  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. Zip((x(a), y(a), g(x(a), y(a))), a, Join[{P1xxy, P2yyx}]) → list1
  20. Length(list1) → Lä
  21. Textfield(step) → inputBox12
  22. Textfield(zoom_{3D}) → InputBox1
  23. Sequence((x(Element[lr01, i]), y(Element[lr01, i]), f(x(Element[lr01, i]), y(Element[lr01, i]))), i, 1, n) → lr013
  24. Sequence((x(Element[lr02, i]), y(Element[lr02, i]), f(x(Element[lr02, i]), y(Element[lr02, i]))), i, 1, n) → lr023
  25. RemoveUndefined(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, i, NaN], i, 1, n]) → lr04
  26. 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
  27. 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
  28. 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
  29. 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
  30. RemoveUndefined(Sequence[If[f(x(PTest), y(PTest)) - f(x(Element[lr01, i]), y(Element[lr01, i])) > 0, i, (-i)], i, 1, n]) → lr04c
  31. Append(lr04c, Element[lr04c, 1]) → lr04cc
  32. Sum(Sequence[If[(Element[lr04cc, i] * Element[lr04cc, i + 1]) < 0, 1, 0], i, 1, n + 1]) → lr04ccc
  33. If(ext1 ≟ n, "Max", If[ext1 ≟ 0, "Min", If[(lr04ccc > 2) ∧ (lr04ccc ≤ 4), "saddle ? ", "?"]]) → TT1
  34. 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
  35. PolyLine({(0, 0), (0.2, 0), (0.3, 0.6), (0.4, 1), (0.7, 1), (0.9, 0.2), (1, 0)}) → Gr
  36. 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
  37. Point(Pxy) → D
  38. Textfield(z0) → InputBox2
  39. ImplicitCurve(g(x, y) - z0) → p1
  40. ImplicitCurve(g(x, y) - u(x, y)) → pp
  41. Textfield(u) → InputBox4
  42. ImplicitCurve(u(x, y) - z0) → p2
  43. IntersectPath(o, g(x, y)) → s1
  44. IntersectPath(o, u(x, y)) → s2