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Geometric Centers on a Sphere Induced by Moving Points in 3D Space

Geometric Centers on a Sphere Induced by Moving Points in 3D Space
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2020-11-30 03:59
VectorSolid GeometryGeometric Construction
This applet demonstrates how the geometric centers on a sphere change when points are moved in three-dimensional space. Key mathematical concepts include geometric transformations, vectors, and solid geometry. The construction uses GeoGebra commands such as Polygon, PointIn, Sphere, Vector, etc. Suitable for high school and above students or teachers.

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
  56. Zip(Segment[AA_2, a], a, alP) → l2

AI Drawing Prompt Reference

Draw a sphere in 3D space with its center at the origin and adjustable radius. Place a quadrilateral plane region inside or on the sphere, and generate several random points within this region. From the sphere's center, draw vectors to each of these points, then compute the sum of these vectors. Based on the direction of the sum vector, mark corresponding points on the sphere's surface and connect them with line segments. Additionally, draw a latitude-longitude grid on the sphere and show an input box to adjust the number of points. Finally, display the points, vectors, sum vector, and their projections on the sphere.
The prompt above was compiled from the original drawing commands and is for reference only.