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球坐标系中体积区域（砖块）演示工具

球坐标系中体积区域（砖块）演示工具
几何画板及绘图内容加载中，请稍等……
2015-05-10 15:05
解析几何立体几何参数方程几何作图滑块动画
所属专题 surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
本范例演示在球坐标系下由 ρ₁ ≤ ρ ≤ ρ₂、φ₁ ≤ φ ≤ φ₂、θ₁ ≤ θ ≤ θ₂ 所界定的“砖块”型三维区域，展示球半径、方位角与极角如何共同切出立体图形。核心数学概念包括球坐标系、三维空间区域不等式、曲面参数化以及坐标变换。作图要点基于构造摘要，使用了 Point、PointIn、If、Surface、CurveCartesian、Intersect、Segment、CircumcircleArc、CircumcircleSector 等命令，并涉及 o、A、B、C、D、α₁、α₂、β₁、β₂、Δα、Δβ 等对象，通过分段曲面与圆弧扇面拼装出球坐标砖块的六个面。适合高中选修及大学微积分、解析几何课堂教学，也可用于多元函数积分区域的直观辅助演示与自主探究。

GeoGebra 命令及参数

  1. PointIn(o) → A
  2. If(x(A) > 0, atan(y(A) / x(A)), atan(y(A) / x(A)) + π) → α_1
  3. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, 3.141592653589793) → c
  4. Point(c) → B
  5. If(x(B) > 0, atan(y(B) / x(B)), atan(y(B) / x(B)) + π) → α_2
  6. CurveCartesian((cos(α_1) * sin(β_1 + t)), (sin(α_1) * sin(β_1 + t)), cos(β_1 + t), t, 0, π - β_1) → d
  7. Point(d) → C
  8. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, Δα) → e
  9. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, β_1, β_2, t, α_1, α_2) → a
  10. Intersect(yAxis, zAxis) → E
  11. Segment(A, E) → f
  12. Point(f) → F
  13. Segment(E, F) → h
  14. Segment(F, A) → i
  15. Surface(((h * sin(u)) * cos(t)), ((h * sin(u)) * sin(t)), (h * cos(u)), u, β_1, β_2, t, α_1, α_2) → b
  16. Surface(((t * sin(β_1)) * cos(u)), ((t * sin(β_1)) * sin(u)), (t * cos(β_1)), u, α_1, α_2, t, h, 1) → j
  17. Surface(((t * sin(β_2)) * cos(u)), ((t * sin(β_2)) * sin(u)), (t * cos(β_2)), u, α_1, α_2, t, h, 1) → k
  18. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_1, β_2, t, h, 1) → l
  19. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_1, β_2, t, h, 1) → m
  20. CurveCartesian((cos(α_1 + t) * sin(β_2)), (sin(α_1 + t) * sin(β_2)), cos(β_2), t, 0, Δα) → g
  21. CurveCartesian((cos(α_1) * sin(β_1 + t)), (sin(α_1) * sin(β_1 + t)), cos(β_1 + t), t, 0, Δβ) → n
  22. CurveCartesian((cos(α_2) * sin(β_1 + t)), (sin(α_2) * sin(β_1 + t)), cos(β_1 + t), t, 0, Δβ) → p
  23. Segment(B, B') → q
  24. Segment(C, C') → r
  25. Segment(D, D') → s
  26. CurveCartesian(((h * cos(α_1 + t)) * sin(β_1)), ((h * sin(α_1 + t)) * sin(β_1)), (h * cos(β_1)), t, 0, Δα) → t
  27. CurveCartesian(((h * cos(α_1 + t)) * sin(β_2)), ((h * sin(α_1 + t)) * sin(β_2)), (h * cos(β_2)), t, 0, Δα) → u
  28. CurveCartesian(((h * cos(α_1)) * sin(β_1 + t)), ((h * sin(α_1)) * sin(β_1 + t)), (h * cos(β_1 + t)), t, 0, Δβ) → v
  29. CurveCartesian(((h * cos(α_2)) * sin(β_1 + t)), ((h * sin(α_2)) * sin(β_1 + t)), (h * cos(β_1 + t)), t, 0, Δβ) → w
  30. CurveCartesian((cos(α_1 + t) * sin(β_1)), (sin(α_1 + t) * sin(β_1)), cos(β_1), t, 0, 6.283185307179586) → a_1
  31. CurveCartesian((cos(α_1 + t) * sin(β_2)), (sin(α_1 + t) * sin(β_2)), cos(β_2), t, 0, 6.28319) → b_1
  32. CurveCartesian((cos(α_1) * sin(t)), (sin(α_1) * sin(t)), cos(t), t, 0, 3.141592653589793) → c_1
  33. CurveCartesian((cos(α_2) * sin(t)), (sin(α_2) * sin(t)), cos(t), t, 0, 3.141592653589793) → d_1
  34. Segment(B, E) → f_1
  35. Segment(C, E) → g_1
  36. Segment(D, E) → h_1
  37. CircumcircleArc(H, C, G) → k_1
  38. CircumcircleArc(H, B, G) → p_1
  39. Segment(I, E) → i_1
  40. Segment(J, E) → j_1
  41. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, 0, β_1, t, α_1, α_2) → m_1
  42. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_1, β_2, t, 0, h) → n_1
  43. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, 0, β_1, t, 0, 1) → o_1
  44. Surface(((t * sin(u)) * cos(α_2)), ((t * sin(u)) * sin(α_2)), (t * cos(u)), u, β_2, 1.5707963267948966, t, 0, 1) → q_1
  45. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, 0, β_1, t, 0, 1) → r_1
  46. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_2, 1.5708, t, 0, 1) → s_1
  47. Surface(((t * sin(u)) * cos(α_1)), ((t * sin(u)) * sin(α_1)), (t * cos(u)), u, β_1, β_2, t, 0, h) → t_1
  48. CurveCartesian(cos(t), sin(t), 0, t, 0, 1.5707963267948966) → e_1
  49. Surface((sin(u) * cos(t)), (sin(u) * sin(t)), cos(u), u, β_2, 3.141592653589793, t, α_1, α_2) → l_1
  50. Surface((t * cos(α_1)), (t * sin(α_1)), u, t, 0, 1, u, -1, 1) → u_1
  51. Surface((t * cos(α_2)), (t * sin(α_2)), u, t, 0, 1, u, -1, 1) → v_1
  52. CircumcircleSector(I, K, J) → c_2

AI 绘图提示语参考

绘制一个球坐标“砖块”区域：从原点出发，设定径向距离 ρ₁ 到 ρ₂、极角 φ₁ 到 φ₂、方位角 θ₁ 到 θ₂。先画两个同心球面作为内外径向边界，再画两个以 z 轴为轴的圆锥面对应 φ₁、φ₂，接着画两个过 z 轴的半平面对应 θ₁、θ₂。在球面与圆锥面、半平面的交线上绘制弧段，连接内外球面的径向线段，形成六个曲面围成的立体，并标出代表 ρ、φ、θ 变化的坐标线。
以上提示语为根据原始绘图命令整理而来，仅供参考。