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四面体的三参数模型变换与极端分布

四面体的三参数模型变换与极端分布
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2020-03-07 02:16
立体几何变换最值优化参数方程滑块动画
本范例演示了四面体在三参数模型下的几何变换，并探索其极端分布情况。核心数学概念包括重心（质心）、线性规划与优化、空间几何体的体积与表面积、统计特征（如相关性）以及特殊点（如重心）的构造。作图要点：利用GeoGebra命令如Length、Sequence、Zip、Join、If、Text、Sum、KeepIf、Dilate、Sphere、TableText、UnicodeToLetter等，构建了多个参数（α、m、n等）控制的动态模型，包含滑块（button9-12）用于调节参数，并生成文本显示极值（如pmaxq、pmaxt、pmaxα等）。适合高中至大学阶段的数学教学，尤其适用于立体几何、线性规划与优化问题的可视化探究。

GeoGebra 命令及参数

  1. Length(m00) → n00
  2. Length(m00(1)) → n00_1
  3. Sequence({V00(m00(i, 1) + 1), V00(m00(i, 2) + 1), V00(m00(i, 3) + 1)}, i, 1, n00) → V00'
  4. Sequence(Dilate[Flatten[Sequence[V00'(i, j) + ((If[j < n00_1, V00'(i, j + 1), V00'(i, 1)] - V00'(i, j)) * {t, 1 - t}), j, 1, n00_1]], q, Cmi(i)], i, 1, n00) → V0a
  5. Zip(Rotate[a, α, Line[(0, 0), b]], a, V0a, b, Cmi) → V0b
  6. Join(V0b) → V0c
  7. If(t ≟ 0 ∨ t ≟ 0.5 ∨ q ≟ 1 ∨ q ≟ 0, Unique[V0c], V0c) → V0cc
  8. Length(V0cc) → läV'
  9. Text("" + p) → txt_p
  10. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  11. Sequence(Polygon[Sequence[V00(m00(i, k) + 1), k, 1, Length[m00(1)]]], i, 1, Length[m00]) → Poly3
  12. Length(V0c) → läV0c
  13. Sequence(Segment[V0c(m4(1, j)), V0c(m4(1, If[j ≟ 4, 1, j + 1]))], j, 1, 4) → seg4_i
  14. Length(V0cc) → n3
  15. Length(V00(1)) → läV00_1
  16. Sequence(Segment[V0c(m3a(1, j)), V0c(m3a(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3a_i
  17. Sequence(Segment[V0c(m3b(1, j)), V0c(m3b(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3b_i
  18. Sequence(Segment[V0c(m6a(1, j)), V0c(m6a(1, If[j ≟ 6, 1, j + 1]))], j, 1, 6) → seg6a_i
  19. Sequence(Segment[V0c(m6a(1, j)), V0c(m6a(1, If[j ≟ 6, 1, j + 1]))], j, 1, 6) → seg6_i
  20. KeepIf(x > 0, seg6_i) → seg6_i1
  21. KeepIf(x > 0, seg6a_i) → seg6a_i1
  22. KeepIf(x > 0, seg4_i) → seg4_i1
  23. Length(V0c(1)) → läV0c_1
  24. Dilate(q0, q, Cmi(1)) → q0'
  25. Sphere(O, läV0c_1) → sphe
  26. Sequence(Polygon[Sequence[V0c(m6a(i, k)), k, 1, Length[m6a(1)]]], i, 1, Length[m6a]) → Poly6a
  27. Sequence(Polygon[Sequence[V0c(m4(i, k)), k, 1, Length[m4(1)]]], i, 1, Length[m4]) → Poly4
  28. Sequence(Polygon[Sequence[V0c(m3c(i, k)), k, 1, Length[m3c(1)]]], i, 1, Length[m3c]) → Poly3c
  29. Sequence(Segment[V0c(m3c(1, j)), V0c(m3c(1, If[j ≟ 3, 1, j + 1]))], j, 1, 3) → seg3c_i
  30. TableText(Append["segments", 1…6], Append["\fgcolor{#FE6F5E}{ poly3a} ", seg3a_i], Append["\fgcolor{#D19FE8}{ poly3b} ", seg3b_i], Append["\fgcolor{#008000}{poly\;3c\; }", seg3c_i], Append["\fgcolor{#008B8B}{poly\;6\; or\; 3 }", seg6a_i1], "v") → text6c
  31. TableText(Append["segments", 1…10], Append["\fgcolor{red}{ poly4} ", seg4_i1], Append["\fgcolor{green}{poly\; 6\; or\; 3}", seg6_i1], "v") → text6b
  32. Sequence(Polygon[Sequence[If[α < rad, V0c(m3aa(i, k)), V0c(m3a(i, k))], k, 1, Length[m3a(1)]]], i, 1, Length[m3a]) → Poly3a
  33. Sequence(Polygon[Sequence[If[α < rad, V0c(m3bb(i, k)), V0c(m3b(i, k))], k, 1, Length[m3b(1)]]], i, 1, Length[m3b]) → Poly3b
  34. UnicodeToLetter(34) → u34
  35. Text(text18r, A, false, true) → text18s
  36. Sequence(Polygon[Sequence[V0c(m6b(i, k)), k, 1, Length[m6b(1)]]], i, 1, Length[m6b]) → Poly6b