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Two Uniform Points on a Sphere from One Uniform Distribution: Cube V=8

Two Uniform Points on a Sphere from One Uniform Distribution: Cube V=8
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2021-01-11 01:56
CircleVectorSolid GeometryProbability & StatisticsGeometric Construction
This example demonstrates how to generate two different points that are uniformly distributed on a sphere using a single uniform source (e.g., one uniform random parameter mapped through geometry). It visualizes the construction and highlights how uniformity is achieved on the spherical surface rather than inside a volume. Key mathematics includes solid geometry on the sphere, uniform probability on a manifold (sphere surface), vector geometry, angle/length-based relationships, and constrained selection using min/max logic. The GeoGebra construction uses Sphere and Circle to define the spherical setting and relevant regions; PointIn, Polygon, and Vector to form candidate constructions and determine membership; Angle and Length to compute spatial relations between points; Function with If as well as Max/Min to implement the piecewise mapping/feasibility constraints tied to the “Cube V=8” volume-normalization idea; and Zip/Flatten with TableText and Textfield to structure computed outputs. It also employs CurveCartesian for plotting and labels/annotations (RotateText, arrow) to clarify key quantities. Suitable for advanced high-school through early university learners in probability & sampling, geometry construction, and mathematical visualization (Monte Carlo style).

GeoGebra Commands & Parameters

  1. Function(f01, (-π), π, ((-π)) / 2, π / 2) → f0
  2. Zip((x(a), y(a), f0(a)), a, Zmin_2) → fmin3D
  3. Zip((x(a), y(a), f0(a)), a, Zmax_2) → fmax3D
  4. Vector((1.3 * Z3), (1.5 * Z3)) → AVZ3r
  5. Vector((1.3 * Z3), (1.3 * Z3) + UnitVector[Sum[Zip[UnitVector[a - Z3], a, lP]]]) → AVZ3
  6. Polygon({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2)}) → q_2
  7. PointIn(q_2) → Atest_2
  8. Vector((1.3 * Z2), (1.5 * Z2)) → AVZ2r
  9. Vector((1.3 * Z2), (1.3 * Z2) + UnitVector[Sum[Zip[UnitVector[a - Z2], a, lP]]]) → AVZ2
  10. Vector((1.3 * Z1), (1.5 * Z1)) → AVZ1r
  11. Vector((1.3 * Z1), (1.3 * Z1) + UnitVector[Sum[Zip[UnitVector[a - Z1], a, lP]]]) → AVZ1
  12. Vector((1.3 * Atest_R), (1.3 * Atest_R) + UnitVector[Sum[Zip[UnitVector[a - Atest_R], a, lP]]]) → AVtest
  13. Vector((1.3 * Atest_R), (1.5 * Atest_R)) → AVtestr
  14. Sphere(O, R) → sphera
  15. Circle(O, R, xOyPlane) → c
  16. PointIn(q_2) → RZ1_2
  17. PointIn(q_2) → RZ2_2
  18. Angle(AVtest, AVtestr) → αA
  19. Angle(AVZ1, AVZ1r) → αZ1
  20. Angle(AVZ2, AVZ2r) → αZ2
  21. Zip(Segment[Atest_R, p], p, lP) → seg
  22. Sum(seg) → sum_seg
  23. PolyLine({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2), ((-π), ((-π)) / 2)}) → q1
  24. Flatten(Sequence[Sequence[((-π) + ((2 * π / na) * i), ((-π)) / 2 + (π / na * j)), j, 1, na], i, 1, na]) → ltest_2
  25. Zip((R; x(a); y(a)), a, ltest_2) → ltest_3
  26. Length(lZmax') → lälZmax'
  27. Textfield(n) → InputBox11
  28. alfa(R, lP, lZsad') → αsad
  29. If(αZ1 > 100°, 180° - αZ1, αZ1) → αZ1'
  30. Textfield(m_1) → InputBox12
  31. Textfield(t) → InputBox13
  32. Textfield(m_2) → InputBox14
  33. PointIn(q_2) → RZ3_2
  34. Sum(Zip[1 / Length[p - Z3], p, lP]) → ZahZ3
  35. Angle(AVZ3, AVZ3r) → αZ3
  36. Textfield(R) → InputBox1
  37. Textfield(na) → InputBox2
  38. Max(f(Zmax_2)) → max_f
  39. Min(f(Zmin_2)) → min_f
  40. Zip(Text["d_{" + (a) + "}", (Atest_R + Element[lP, a]) / 2], a, 1…n) → lPtxt
  41. Zip(Segment[F_o, p], p, lP) → seg0
  42. Sum(seg0) → sum_seg0
  43. RotateText(txt, 90°) → txt1
  44. Sum(lZ3sad) → sumlZ3sad
  45. Textfield(Points3D) → InputBox7
  46. Textfield(lZsad') → InputBox17
  47. Textfield(lZmin') → InputBox16
  48. Textfield(lZmax') → InputBox15
  49. Textfield(jmin) → InputBox18
  50. Textfield(jmax) → InputBox19
  51. Textfield(jsad) → InputBox20
  52. Zip(Text["" + (b), a - (0.1, 0.1)], a, Zmax_2, b, 1…Length[Zmax_2]) → l2maxtxt
  53. Zip(Text["" + (b), a - (0.1, 0.1)], a, Zmin_2, b, 1…Length[Zmin_2]) → l2mintxt
  54. Zip(Text["" + (b), a - (0.1, 0.1)], a, Zsad_2, b, 1…Length[Zsad_2]) → l2sadtxt
  55. Zip(Vector[r1, (1.153846154 * r1)], r1, (1.3 * Zip[(R * UnitVector[a]), a, lZmax'])) → lVmaxr
  56. Zip(Vector[r1, (1.153846154 * r1)], r1, (1.3 * Zip[(R * UnitVector[a]), a, lZmin'])) → lVminr
  57. Zip(Vector[r1, (1.153846154 * r1)], r1, (1.3 * Zip[(R * UnitVector[a]), a, lZsad'])) → lVsadr
  58. alfa(R, lP, lZmax') → αmax
  59. alfa(R, lP, lZmin') → αmin
  60. arrow(R, lP, lZmax') → lVmax
  61. arrow(R, lP, lZmin') → lVmin
  62. arrow(R, lP, lZsad') → lVsad
  63. CurveCartesian((-π) + ((2 * π) * a), ((-π)) / 2, f0((-π) + ((2 * π) * a), ((-π)) / 2), a, 0, 1) → par1
  64. CurveCartesian((-π) + ((2 * π) * a), π / 2, f0((-π) + ((2 * π) * a), π / 2), a, 0, 1) → par2
  65. CurveCartesian((-π), ((-π)) / 2 + (π * a), f0((-π), ((-π)) / 2 + (π * a)), a, 0, 1) → par3
  66. CurveCartesian(π, ((-π)) / 2 + (π * a), f0(π, ((-π)) / 2 + (π * a)), a, 0, 1) → par4
  67. Polygon({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2)}) → q_3
  68. Zip((x(a), y(a), f0(a)), a, Zsad_2) → fsad3D
  69. Textfield(achs) → InputBox3
  70. Zip(Atest_2 + (0.2; ((2 * π / 10) * k)), k, 0…9) → l0r
  71. Sum(Unique[sgn(f(Atest_2) - f(l0r))]) → sum_r
  72. Sum(Sequence[If[sum00_r(i) ≠ sum00_r(i + 1), 1, 0], i, 1, Length[sum00_r] - 1]) → sum01_r
  73. Circle(Atest_2, 0.2) → ci
  74. TableText(Append["i", Sequence[Length[lZsad']]], Append["\frac{N°}{levels} ", msad], Append["P_{sad }", lZsad'], Append["(f'x,f'y)", lZf'xysad], Append[" Δα≟ π ", αsad], "v") → text2c
  75. TableText(Append["i", Sequence[Length[αmax]]], Append["\frac{N°}{levels} ", mmax], Append["P_{Max }", lZmax'], Append["(f'x,f'y)", lZf'xymax], Append[" Δα≟ π ", αmax], "v") → text2a
  76. TableText(Append["i", Sequence[Length[αmin]]], Append["\frac{N°}{levels} ", mmin], Append["P_{Min }", lZmin'], Append["(f'x,f'y)", lZf'xymin], Append[" Δα≟ π ", αmin], "v") → text2b

AI Drawing Prompt Reference

Generate two uniformly distributed points on a sphere to demonstrate how a single uniform distribution source can be mapped to the spherical surface. Use a sphere and a circle to define the space, calculate the relationships between points using angles and lengths, and apply vector geometry with conditional logic (like min/max) for point selection and distribution. Start by drawing the sphere and reference circle, then create two vectors representing candidate points, compute their angular relationships with the sphere, and finally adjust the vectors based on conditions to ensure the points are uniformly distributed on the sphere.
The prompt above was compiled from the original drawing commands and is for reference only.