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Spherical Coordinates Brick Region Demonstration Tool

Spherical Coordinates Brick Region Demonstration Tool
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2015-05-10 15:05
Coordinate GeometrySolid GeometryParametric EquationGeometric ConstructionSlider Animation
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
This GeoGebra demonstration tool visualizes a “brick”-shaped region in spherical coordinates, defined by the inequalities ρ₁ ≤ ρ ≤ ρ₂, φ₁ ≤ φ ≤ φ₂, and θ₁ ≤ θ ≤ θ₂. It illustrates how radial distance, polar angle, and azimuthal angle jointly bound a three-dimensional solid in space. Core mathematical concepts include spherical coordinate systems, 3D region inequalities, parametric surfaces, and coordinate transformations. The construction relies on GeoGebra commands such as Point, PointIn, If, Surface, CurveCartesian, Intersect, Segment, CircumcircleArc, and CircumcircleSector, together with objects like o, A, B, C, D, α₁, α₂, β₁, β₂, Δα, and Δβ to assemble the bounding faces, arcs, and sectors of the spherical brick. Suitable for high school advanced mathematics, university calculus, analytic geometry, and multivariable integration courses, this tool supports classroom demonstration and independent inquiry into 3D coordinate regions.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a spherical‑coordinate brick region bounded by ρ₁ ≤ ρ ≤ ρ₂, φ₁ ≤ φ ≤ φ₂, and θ₁ ≤ θ ≤ θ₂. Start with the origin, then create two concentric spheres for the inner and outer radial limits. Add two cones around the z‑axis for the constant‑φ boundaries, and two half‑planes through the z‑axis for the constant‑θ boundaries. Plot the intersection curves where the spheres meet the cones and half‑planes, and draw radial segments connecting the inner and outer spheres along those curves. The six curved surfaces together enclose the solid, with visible coordinate lines showing variations in ρ, φ, and θ.
The prompt above was compiled from the original drawing commands and is for reference only.