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截断双五角双锥体：顶点平均距离与截断参数的依赖关系

截断双五角双锥体：顶点平均距离与截断参数的依赖关系
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2020-01-09 17:55
立体几何最值优化参数方程几何作图
本范例演示截断双五角双锥体中，顶点平均距离随截断参数变化的规律及极端分布现象。涉及的核心数学概念包括立体几何、参数方程、最值优化等。作图要点利用了 GeoGebra 的 Join, Unique, Length, Sphere, Ray, Zip, UnicodeToLetter, TableText, KeepIf, Text, Sum 等命令和对象，通过滑块动画展示参数变化对几何形态的影响。适合高中及以上学生及数学爱好者用于理解复杂几何体的性质与优化问题。

GeoGebra 命令及参数

  1. Join(Zip[lP0(x(a)) + ((lP0(y(a)) - lP0(x(a))) * {t, 1 - t}), a, segN]) → lVV
  2. Unique(lVV) → lP'
  3. Length(lP') → n
  4. Length(lVV(1)) → R2
  5. Sphere(O, R1) → sp1
  6. Sphere(O, R2) → sp2
  7. Ray(B, AA) → h
  8. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN1) → seg1
  9. Zip(Segment[lP0(x(a)), lP0(y(a))], a, segN2) → seg2
  10. Zip(Polygon[a], a, v_poly03) → poly03
  11. Zip(Polygon[a], a, v_poly06) → poly06
  12. Zip(Polygon[a], a, v_poly05) → poly05
  13. UnicodeToLetter(34) → u34
  14. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ5) → segΔ5
  15. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ6) → segΔ6
  16. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ31) → segΔ31
  17. Zip(Segment[lVV(x(a)), lVV(y(a))], a, segNΔ32) → segΔ32
  18. TableText(Append["\;\;\; \fgcolor{#CD7F32}{3_1-gons}", segΔ31], Append["\;\;\; \fgcolor{#CD7F32}{3_2-gons}", segΔ32], Append["\;\;\; \fgcolor{red}{5-gons}", segΔ5], Append["\;\;\; \fgcolor{#6E7F80}{6-gons}", segΔ6], "v") → text3a
  19. Unique(round(poly03, 14)) → poly03round
  20. KeepIf(a ≟ poly03round(1), a, poly03) → poly03_2
  21. KeepIf(a ≟ poly03round(2), a, poly03) → poly03_1
  22. Text("" + p3) → txt_p
  23. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  24. Text(text12, C, false, true) → text8