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螺线二等分立方体

螺线二等分立方体
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2018-12-08 22:42
立体几何变换几何作图
①本范例演示了通过螺线运动将立方体进行二等分的几何现象；②涉及的核心数学概念包括空间几何、螺线运动、二等分原理；③作图要点——从下方『构造摘要』提炼用到了 Point（点）、Cube（立方体）、Surface（曲面）、CurveCartesian（笛卡尔曲线）、Rotate（旋转）、Translate（平移）、Polygon（多边形）、Segment（线段）等 GeoGebra 命令/对象；④适合高中及以上学段学习立体几何、空间变换及参数方程的应用。

GeoGebra 命令及参数

  1. Point(Circle[A, Distance[B, A], Segment[B, A]]) → C
  2. Cube(B, A, C) → a, D, faceCDHG, faceEFGH, edgeBD, edgeAB, edgeAC, edgeCD, edgeBE, edgeDH, edgeEH, edgeEF, E, edgeAF, edgeFG, edgeCG, edgeGH, F, G, H, faceABDC, faceBDHE, faceABEF, faceACGF
  3. Surface((u * cos(t)), (u * sin(t)), (sqrt(2) * t / π), t, 0, π, u, 0, 1.5) → b
  4. CurveCartesian((r(t) * cos(t)), (r(t) * sin(t)), (sqrt(2) * t / π), t, 0, π / 2) → c
  5. CurveCartesian((p(t) * cos(t)), (p(t) * sin(t)), (sqrt(2) * t / π), t, π / 2, π) → d
  6. Rotate(b, π) → b'
  7. Rotate(c, π) → c'
  8. Rotate(d, π) → d'
  9. Surface((u * x(d(t))) + ((1 - u) * x(d'(t))), (u * y(d(t))) + ((1 - u) * y(d'(t))), (u * z(d(t))) + ((1 - u) * z(d'(t))), t, π / 2, π, u, 0, 1) → e
  10. Surface((u * x(c(t))) + ((1 - u) * x(c'(t))), (u * y(c(t))) + ((1 - u) * y(c'(t))), (u * z(c(t))) + ((1 - u) * z(c'(t))), t, 0, π / 2, u, 0, 1) → f
  11. Translate(Rotate[e, n], Vector[(0, 0, (sqrt(2) / π * n))]) → g
  12. Translate(Rotate[f, n], Vector[(0, 0, (sqrt(2) / π * n))]) → h
  13. Translate(Rotate[Polygon[H, F, G], n], Vector[(0, 0, (sqrt(2) / π * n))]) → t1
  14. Translate(Rotate[Polygon[D, B, A], n], Vector[(0, 0, (sqrt(2) / π * n))]) → t2
  15. Translate(Rotate[Surface[x(c(t)), y(c(t)), ((1 - u) * z(c(t))) + (u * sqrt(2)), t, 0, π / 2, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → j
  16. Translate(Rotate[Surface[x(d(t)), y(d(t)), ((1 - u) * z(d(t))) + (u * sqrt(2)), t, π / 2, π, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → i
  17. Translate(Rotate[Surface[x(d'(t)), y(d'(t)), (u * z(d'(t))), t, π / 2, π, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → k
  18. Translate(Rotate[Surface[x(c'(t)), y(c'(t)), ((1 - u) * z(c'(t))), t, 0, π / 2, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → l
  19. Surface(x(c(t)), y(c(t)), ((1 - u) * z(c(t))), t, 0, π / 2, u, 0, 1) → m
  20. Surface(x(d(t)), y(d(t)), ((1 - u) * z(d(t))), t, π / 2, π, u, 0, 1) → o
  21. Surface(x(c'(t)), y(c'(t)), ((1 - u) * z(c'(t))) + (u * sqrt(2)), t, 0, π / 2, u, 0, 1) → q
  22. Surface(x(d'(t)), y(d'(t)), ((1 - u) * z(d'(t))) + (u * sqrt(2)), t, π / 2, π, u, 0, 1) → s
  23. Polygon(F, H, E) → t3, e_1, f_1, h_1
  24. Polygon(A, C, D) → t4, d_1, a_1, c_1
  25. Translate(Rotate[Surface[x(c'(t)), y(c'(t)), ((1 - u) * z(c'(t))), t, 0, π / 2, u, 0, 1], n], Vector[(0, 0, (sqrt(2) / π * n))]) → t
  26. Translate(Rotate[c, n], Vector[(0, 0, (sqrt(2) / π * n))]) → u
  27. Translate(Rotate[c', n], Vector[(0, 0, (sqrt(2) / π * n))]) → v
  28. Translate(Rotate[d', n], Vector[(0, 0, (sqrt(2) / π * n))]) → w
  29. Translate(Rotate[d, n], Vector[(0, 0, (sqrt(2) / π * n))]) → b_1
  30. Translate(Segment[Rotate[A, n], Rotate[F, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → g_1
  31. Translate(Segment[Rotate[G, n], Rotate[F, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → i_1
  32. Translate(Segment[Rotate[G, n], Rotate[H, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → j_1
  33. Translate(Segment[Rotate[F, n], Rotate[H, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → k_1
  34. Translate(Rotate[Segment[G, Midpoint[G, C]], n], Vector[(0, 0, (sqrt(2) / π * n))]) → l_1
  35. Translate(Segment[Rotate[A, n], Rotate[B, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → m_1
  36. Translate(Rotate[Segment[B, Midpoint[B, E]], n], Vector[(0, 0, (sqrt(2) / π * n))]) → n_1
  37. Translate(Segment[Rotate[D, n], Rotate[B, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → p_1
  38. Translate(Segment[Rotate[D, n], Rotate[A, n]], Vector[(0, 0, (sqrt(2) / π * n))]) → q_1
  39. Segment(H, F) → r_1
  40. Segment(H, E) → s_1
  41. Segment(E, F) → t_1
  42. Segment(A, D) → f_2
  43. Segment(A, C) → g_2
  44. Segment(D, C) → h_2
  45. Segment(H, D) → i_2
  46. Segment(E, Midpoint[B, E]) → j_2
  47. Segment(C, Midpoint[C, G]) → k_2

AI 绘图提示语参考

画一个立方体，并展示如何通过螺旋运动将其二等分。首先绘制一个立方体，然后创建一个螺旋面，通过旋转和翻译操作，将螺旋面与立方体结合，形成二等分效果。最终图形应包含立方体、螺旋面以及相关的旋转和翻译几何对象，清晰展示二等分的过程。
以上提示语为根据原始绘图命令整理而来，仅供参考。