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Eight-Point Model with Two Degrees of Freedom on a Sphere

Eight-Point Model with Two Degrees of Freedom on a Sphere
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2019-10-21 21:34
VectorCoordinate GeometrySolid GeometryCalculusOptimization
In collections ₉Extreme Verteilung der 8 Punkten auf der Kugel
This example demonstrates the construction of an eight-point model with two degrees of freedom on a spherical surface. It involves core mathematical concepts such as vectors, coordinate geometry, solid geometry, calculus, and optimization problems. The construction utilizes GeoGebra commands and objects like Sphere, Circle, Translate, Ray, Segment, PolyLine, Zip, UnicodeToLetter, Point, Angle, Plane, Function, Sort, Unique, Polygon, Sequence, TableText, and LaTeX. It is suitable for high school and above learners studying spatial geometry and computational geometry applications.

GeoGebra Commands & Parameters

  1. Sphere(O, R) → a
  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. PolyLine({A1, A5, A2, A6, A3, A7, A4, A8, A1}) → Poly3
  22. Zip((A1 + a) / 2, a, {A2, A5, A8, A3, A7, A6}) → l6
  23. UnicodeToLetter(34) → u34
  24. Point(c1) → D
  25. Angle((0, 0, R), O, D) → δ
  26. Segment(O, D) → seg0
  27. Plane(z = 1.480440157243445) → Pl_{Lö1}
  28. Plane(z = 1.481181823885057) → Pl_{Lö2}
  29. Function(P(u, v), u, -1°, 60°, v, -1°, 90°) → m
  30. Plane(z = h) → Obere
  31. PolyLine(A01, A02, A03, A04, A01) → Poly02
  32. Zip(Segment[A1, a], a, {A2, A5, A8, A3, A7, A6}) → l5
  33. Sort(Join[Sequence[Sequence[Segment[lA(i), lA(j)], j, i + 1, 8], i, 1, 8]]) → Länge
  34. Unique(Zip[floor((a * 10000)) / 10000, a, Länge]) → Länge0
  35. Polygon({(-1°, -1°), (60°, -1°), (60°, 90°), (-1°, 90°), (-1°, -1°)}) → Poly_{αθ}
  36. Sequence(Sum[Sequence[If[abs(Länge0(i) - Länge(k)) < 0.01, 1, 0], k, 1, 28]], i, 1, Length[Länge0]) → m2
  37. TableText(Append["i", 1…Length[Länge0]], Append["n^*", m2], Append["a_i", Länge0], "v") → text8a
  38. Zip(Ray[O, a], a, lA) → lR
  39. Sequence((0, 0, 0) + ((1.5 * R) * UnitVector[Sum[Zip[UnitVector[lA(i) - a], a, lA \ {lA(i)}]]]), i, 1, 8) → lB
  40. Zip(Vector[a, b], a, lA, b, lB) → lU
  41. Zip(Text[a, P, true, true], P, l6, a, 1…8) → l7
  42. LaTeX(seg16, true, true) → text8
  43. Zip(Angle[a, O, b], a, lA, b, lB) → lβ
  44. Sequence((0, 0, 0) - ((1.5 * R) * UnitVector[(1 / 7 * Sum[lA \ {lA(i)}])]), i, 1, 8) → lGC
  45. Zip(Angle[a, O, b], a, lA, b, lGC) → lβGZ
  46. Sequence(Vector[lA(i), lGC(i)], i, 1, 8) → lUGZ1
  47. TableText(Append["i", 1…8], Append["\fgcolor{orange}{βi-GM}", lβ], Append["\fgcolor{magenta}{βi-GC}", lβGZ], "v") → text4

AI Drawing Prompt Reference

Draw an eight-point model with two degrees of freedom on a sphere. Start by sketching a sphere and a circle on its surface. Translate points on the circle vertically to get eight points. Connect these points to form two pentagons and one octagon. Then, draw segments between specific pairs of points to create intricate geometric shapes. Label angles and lengths, and differentiate key segments with colors.
The prompt above was compiled from the original drawing commands and is for reference only.