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Ten Moving Points Induce Two Antipodal Geometric Centers on a Sphere

Ten Moving Points Induce Two Antipodal Geometric Centers on a Sphere
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2020-12-01 18:38
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① This applet demonstrates a dynamic geometric phenomenon: ten freely moving points in space induce two geometric centers on a sphere, which are antipodal (opposite ends of a diameter). ② The core mathematical concepts involve spherical geometry, antipodal points, generalized geometric centers (such as centroid or barycenter), and symmetry in multi-point systems. ③ Construction highlights: based on the construction summary, the applet uses GeoGebra commands including Polygon, PointIn, Sphere, Circle, Angle, Zip, Sum, PolyLine, Flatten, Textfield, KeepIf, Function, Sequence, CurveCartesian, Vector, Segment, Translate, and RotateText, creating multiple points (P1 to P10), a sphere, auxiliary circles, and vectors, with dynamic computation yielding the two centers. ④ Suitable for high school to university-level mathematics teaching, useful for exploring geometric properties of point systems, spherical geometry, and symmetry, as well as a demonstration of dynamic geometry and computer-assisted instruction.

GeoGebra Commands & Parameters

  1. Polygon({B01, B02, B04, B03}) → q2
  2. PointIn(q2) → B1
  3. PointIn(q2) → B2
  4. PointIn(q2) → B3
  5. PointIn(q2) → B4
  6. PointIn(q2) → B5
  7. PointIn(q2) → B6
  8. PointIn(q2) → B7
  9. PointIn(q2) → B8
  10. PointIn(q2) → B9
  11. PointIn(q2) → B10
  12. First({P1, P2, P3, P4, P5, P6, P7, P8, P9, P10}, n) → lP
  13. Polygon({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2)}) → q
  14. PointIn(q) → A_2
  15. Sphere(O, R) → b
  16. Circle(O, R, xOyPlane) → c
  17. Angle(sumunitA_q, vA) → αA
  18. Zip(Segment[A, p], p, lP) → seg
  19. Sum(seg) → sum_seg
  20. PolyLine({((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2), ((-π), ((-π)) / 2)}) → q1
  21. Flatten(Sequence[Sequence[((-π) + ((2 * π / na) * i), ((-π)) / 2 + (π / na * j)), j, 1, na - 1], i, 1, na - 1]) → ltest_2
  22. Zip((R; x(a); y(a)), a, ltest_2) → ltest_3
  23. Zip((arg(a), alt(a)), a, lP) → lP_2
  24. Textfield(n) → InputBox11
  25. KeepIf(x > 0, Sequence[If[abs(Element[αsad, i] - π) < 0.001 ∨ abs(Element[αsad, i]) < 0.001, i, 0], i, 1, Length[αsad]]) → Nαsad
  26. Textfield(R) → InputBox1
  27. Textfield(na) → InputBox2
  28. Function(f01, (-π), π, ((-π)) / 2, π / 2) → f0
  29. Sequence(Vector[(1.3 * Point[O, Vector[Element[sumunit_sad, i]]]), (1.5 * Point[O, Vector[Element[sumunit_sad, i]]])], i, 1, Length[sumunit_sad]) → sumunit_sad1
  30. Zip((Length[a]; arg(a); alt(a)), a, lP) → lPsphere
  31. CurveCartesian((-π) + ((2 * π) * a), ((-π)) / 2, f0((-π) + ((2 * π) * a), ((-π)) / 2), a, 0, 1) → par1
  32. CurveCartesian((-π) + ((2 * π) * a), π / 2, f0((-π) + ((2 * π) * a), π / 2), a, 0, 1) → par2
  33. CurveCartesian(π, ((-π)) / 2 + (π * a), f0(π, ((-π)) / 2 + (π * a)), a, 0, 1) → par4
  34. Zip(Text["d_{" + (a) + "}", (A + Element[lP, a]) / 2], a, 1…n) → lPtxt
  35. CurveCartesian((-π), ((-π)) / 2 + (π * a), f0((-π), ((-π)) / 2 + (π * a)), a, 0, 1) → par3
  36. Zip((x(a), y(a)), a, lP) → lP_{Gr1}
  37. First({aP1, aP2, aP3, aP4, aP5, aP6, aP7, aP8, aP9, aP10}, n) → Output
  38. Vector((1.3 * A), (1.5 * A)) → lvA_a
  39. Zip(Segment[F_o, p], p, lP) → segF_o
  40. Sum(segF_o) → sum_segFo
  41. Sum(seg^(2)) → sum_q
  42. Vector((1.3 * A), (1.1 / 1.7 * Point[O, Vector[(-sumunitA_q)]])) → l1A_q
  43. Vector((1.3 * Zq), (1.5 * Zq)) → lvZq
  44. Vector((1.3 * Zq'), (1.5 * Zq')) → lvZq'
  45. Vector((1.3 * Zq), (1.1 / 1.7 * Point[O, Vector[(-sumunitZ_q)]])) → l1Z_q
  46. Vector((1.3 * Zq'), (1.1 / 1.7 * Point[O, Vector[(-sumunitZ_q')]])) → l1Z_q'
  47. Segment(Zq, Zq') → antipodalAchse
  48. Segment(Zq, Cm_0) → segZq
  49. Segment(Zq', Cm_0) → segZq'
  50. Zip(Segment[Cm_0, p], p, lP) → segC_m
  51. Sum(Zip[a^(2), a, segC_m]) → sum_segCmoq
  52. Zip(Text["P_{" + (a) + "}", b + (0.1, 0.1)], a, 1…n, b, lP_2) → l1
  53. Zip(Segment[(x(a), y(a), -1), (x(a), y(a), f0(x(a), y(a)))], a, {((-π), ((-π)) / 2), ((-π), π / 2), (π, π / 2), (π, ((-π)) / 2)}) → lseg
  54. Translate(q, Vector[(0, 0, -1)]) → q'
  55. RotateText(txt, 90°) → txt1

AI Drawing Prompt Reference

Draw a sphere with ten freely moving points on its surface, which generate two antipodal geometric centers. First, create a sphere. Then, place ten points on the sphere. Next, calculate the positions of these points to find the two antipodal geometric centers. Finally, display these two center points.
The prompt above was compiled from the original drawing commands and is for reference only.