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Polyhedron Calculation with a Fixed Vertex on a Sphere

Polyhedron Calculation with a Fixed Vertex on a Sphere
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2020-11-26 01:05
VectorSolid GeometryOptimizationGeometric Construction
This example demonstrates how to construct a polyhedron with one vertex fixed on a sphere. It involves core mathematical concepts such as spherical geometry, polyhedron construction, vector operations, and optimization problems. Key GeoGebra commands and objects used in the construction include Sphere, Text, TableText, Sum, and Sequence. It is suitable for high school students and mathematics enthusiasts exploring solid geometry.

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. Text(("(up\; to\; 10^{-15})\;\\ accuracy= " * ScientificText[cc, 3]), C, true, true) → text5r
  13. Zip(Vector[a, a + UnitVector[a]], a, lP') → lVector1
  14. Textfield(scale) → InputBox6
  15. TableText(Append["i", 1…n], Append[" \fgcolor{orange}{GM :Δφ_i}, rad", l1], Append["\fgcolor{magenta}{GC :Δφ_i} , rad", l1c], "v") → text23
  16. If(bu, "Animation", "Stop") → bue0
  17. Zip(Vector[a, a + (1.5 * UnitVector[b])], a, lP', b, Wi_P) → lVector3
  18. Zip(Vector[a, a + (1.5 * UnitVector[b])], a, lP', b, Wi_Pc) → lVector2
  19. TableText({Join[{{"n"}, Sequence[i, i, 4, 69 + 3]}], Join[{{" p_n=P_n/[R*n*(n-1)/2]"}, lp}]}, "vl") → txt_lp
  20. Sum(Flatten[Sequence[Append[Sequence[Element[txt_p, i], i, m1(k), m2(k)], " "], k, 1, 5]]) → txtUm1
  21. 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
  22. Sequence(Text["" + (i), Wi_Pc1(i) - (0.4, 0.4), true, true], i, 1, n) → l4b
  23. Sequence(Text["" + (i), Wi_Pc2(i) + (0.4, 0.4), true, true], i, 1, n) → l4a
  24. Append(As, Sequence[((random(), random(), random()) * R), i, 2, n]) → l2s
  25. PointIn(p1) → A
  26. Sequence(Text["A" + (i), lP'(i) - (-0.6, 0.6), true, true], i, 1, n) → l4cc
  27. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(1)) → ll1
  28. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(2)) → ll2
  29. Zip(Segment[Element[lP', y(a)], Element[lP', z(a)]], a, ll1bb(3)) → ll3

AI Drawing Prompt Reference

In a 3D scene, draw a sphere centered at O with radius R, and place one fixed vertex A on the sphere. Using given parameters such as n and other inputs (e.g., i0, k_1, k_2, and scale), construct a sequence of additional points on the sphere that extends from A, forming a set of numbered vertices (label them A1 through An). Display the connectivity implied by the construction by drawing the corresponding line segments between consecutive or paired vertices, producing a polyhedron-like outline on the sphere surface. Add on-canvas text and a small table showing values related to Δφ_i with GM and GC, as well as a normalized p_n value, and include a boxed accuracy/precision message indicating an accuracy bound.
The prompt above was compiled from the original drawing commands and is for reference only.