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Extreme Point Distribution on a Sphere: Vector-Based Optimization Visualization

Extreme Point Distribution on a Sphere: Vector-Based Optimization Visualization
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2020-09-09 17:04
CircleVectorSolid GeometryOptimizationGeometric Construction
This GeoGebra example demonstrates how to construct and study an “extreme” distribution of points on the surface of a sphere, highlighting how different spherical point configurations affect a chosen objective (e.g., uniformity or spread) and reveal optimal or worst-case patterns. The key math concepts include sphere geometry (Sphere and its surface), 3D vectors, centroid/barycenter ideas to capture average/symmetry properties, and optimization modeling (Optimization Problems) to define objective and constraints that evaluate the distribution quality. Construction details extracted from the command summary show the use of Text and LaTeX for mathematical labels, Sphere to generate the 3D sphere, Textfield for interactive inputs/parameters, and TableText/Append to organize computed values. Logic and aggregation are handled via If and Zip, while Sum is used to accumulate the objective/constraint expressions across points. The setup also reflects parametrization through InputBox1/InputBox2 and variables such as r, n, and related computed quantities (e.g., matrix/mean-like intermediates). The example suits advanced high school through undergraduate learners interested in geometry-based optimization, discrete point placement on manifolds, and interactive mathematical visualization.

GeoGebra Commands & Parameters

  1. Text("" + Um1) → txt_p
  2. LaTeX(Um1) → lll1_1
  3. LaTeX(Um0) → lll0
  4. Sphere(O, R) → p1
  5. Textfield(n) → InputBox1
  6. Textfield(i0) → InputBox2
  7. Text(r) → Txtr
  8. Textfield(k_1) → InputBox4
  9. UnicodeToLetter(34) → u34
  10. Textfield(k_2) → InputBox5
  11. TableText(ltxt0, ltxt1, "v") → textltxt01
  12. Append(A, Sequence[((random(), random(), random()) * R), i, 2, n]) → l2s
  13. Text(("(up\; to\; 10^{-15})\;\\ accuracy= " * ScientificText[cc, 3]), C, true, true) → text5r
  14. Zip(Vector[a, a + UnitVector[a]], a, lP') → lVector1
  15. Textfield(scale) → InputBox6
  16. TableText(Append["i", 1…n], Append[" \fgcolor{orange}{GM :Δφ_i}, rad", l1], Append["\fgcolor{magenta}{GC :Δφ_i} , rad", l1c], "v") → text23
  17. If(bu, "Animation", "Stop") → bue0
  18. Zip(Vector[a, a + (1.5 * UnitVector[b])], a, lP', b, Wi_P) → lVector3
  19. Zip(Vector[a, a + (1.5 * UnitVector[b])], a, lP', b, Wi_Pc) → lVector2
  20. TableText({Join[{{"n"}, Sequence[i, i, 4, 69 + 3]}], Join[{{" p_n=P_n/[R*n*(n-1)/2]"}, lp}]}, "vl") → txt_lp
  21. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  22. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(1)) → ll1
  23. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(2)) → ll2
  24. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(3)) → ll3
  25. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(4)) → ll4
  26. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(5)) → ll5
  27. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(6)) → ll6
  28. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(7)) → ll7
  29. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(8)) → ll8
  30. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(9)) → ll9
  31. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(10)) → ll10
  32. Text("\;\;\;\;\;\;\;\;\;\;\;\;\;\fgcolor{" + (If[c', "orange", "brown"]) + "} \boxed{ " + c' + ("}\;\;\;\; \;\;\;\;\; \\\;\; ( accuracy- up\; to\;\; 2.*10^{" - k_2) + ("\;})\;\\\;\; ➤➤➤\;\;\;\scalebox{1.5}{ " + (ScientificText[cc, 3])) + "}\;█ ", C, true, true) → text9

AI Drawing Prompt Reference

Visualize a sphere with randomly distributed points on its surface. For each point, draw vectors from the sphere's center to the points, with varying colors and thicknesses. Label the sphere's radius, number of points, and vector information for each point. Also, display the analysis results of the points' distribution uniformity.
The prompt above was compiled from the original drawing commands and is for reference only.