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Eight-Point Model on a Sphere with Two Degrees of Freedom (Extended Version)

Eight-Point Model on a Sphere with Two Degrees of Freedom (Extended Version)
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2019-10-25 20:35
VectorSolid GeometryCalculus
In collections ₉Extreme Verteilung der 8 Punkten auf der Kugel
This example demonstrates an eight-point model on a sphere with two degrees of freedom. Key mathematical concepts include spherical geometry, vectors, angles, and rotation. Using GeoGebra commands such as Sphere, Translate, and RotateText, the model can be dynamically constructed and displayed. Suitable for high school students and teachers.

GeoGebra Commands & Parameters

  1. Sphere(O, R) → sphr
  2. Circle(O, r, xOyPlane) → c02
  3. Translate(A01, Vector[(0, 0, (-h))]) → A1
  4. Translate(A02, Vector[(0, 0, (-h))]) → A2
  5. Translate(A03, Vector[(0, 0, (-h))]) → A3
  6. Translate(A04, Vector[(0, 0, (-h))]) → A4
  7. Translate((r; α_1), Vector[(0, 0, h)]) → A5
  8. Translate((r; α_1 + 90°), Vector[(0, 0, h)]) → A6
  9. Translate((r; α_1 + 180°), Vector[(0, 0, h)]) → A7
  10. Translate((r; α_1 + 270°), Vector[(0, 0, h)]) → A8
  11. Ray(O, A1) → g1
  12. Segment(A1, A2) → seg12
  13. Segment(A1, A6) → seg16
  14. Segment(A1, A8) → seg18
  15. Segment(A1, A3) → seg13
  16. Segment(A1, A5) → seg15
  17. Segment(A1, A7) → seg17
  18. Segment(A2, A5) → seg25
  19. PolyLine(A1, A2, A3, A4, A1) → Poly2
  20. PolyLine(A5, A6, A7, A8, A5) → Poly1
  21. UnicodeToLetter(34) → u34
  22. Point(c1) → D
  23. Angle((0, 0, R), O, D) → δ
  24. Segment(O, D) → seg0
  25. Plane(z = 1.480440157243445) → Pl_{Lö1}
  26. Plane(z = 1.481181823885057) → Pl_{Lö2}
  27. Function(GAbstand(u, v), u, -1°, 60°, v, -1°, 90°) → m_{GA}
  28. Plane(z = h) → Obere
  29. PolyLine(A01, A02, A03, A04, A01) → Poly02
  30. Sort(Join[Sequence[Sequence[Segment[lA(i), lA(j)], j, i + 1, 8], i, 1, 8]]) → Länge
  31. Unique(Zip[floor((a * 10000)) / 10000, a, Länge]) → Länge0
  32. Polygon({(-1°, -1°), (60°, -1°), (60°, 90°), (-1°, 90°), (-1°, -1°)}) → Poly_{αθ}
  33. Function(Vol(u, v), u, -1°, 60°, v, -1°, 90°) → m_{vol}
  34. Function(GFläche(u, v), u, -1°, 60°, v, -1°, 90°) → m_{GO}
  35. Textfield(R) → InputBox1
  36. Textfield(scal_v) → InputBox2
  37. Textfield(scal_O) → InputBox3
  38. TableText(Append["Info: ", 1…4], Append["α=45, θ=", lθ], Append["Gesamtabstand", lGA], Append["Gesamtfläche", lGFl], Append["Gesamvolumen", lGVol], "v") → text17
  39. LaTeX(vol_{qA}, false, false) → text18a
  40. LaTeX(flä_{qA}, false, false) → text18b
  41. Sum(lA) → sumA
  42. RotateText(text13, 90°) → Text1
  43. Zip(Ray[O, a], a, lA) → lR
  44. Circle(A8a, A7a, A6a) → f
  45. PolyLine(A5a, A6a, A7a, A8a, A5a) → Poly1a
  46. Sequence(Sum[Sequence[If[abs(Länge0(i) - Länge(k)) < 0.01, 1, 0], k, 1, 28]], i, 1, Length[Länge0]) → m2
  47. TableText(Append["i", 1…Length[Länge0]], Append["n^*", m2], Append["a_i / a_{Würfel}", Länge0 / a_{Würfel}], "v") → text8a