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Parallel Vector Transport

Parallel Vector Transport
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2018-07-31 19:44
VectorDerivativeGeometric Construction
This example demonstrates the concept of parallel vector transport in geometric contexts, involving derivatives, geometric means, and isosceles triangles. The construction uses GeoGebra commands like Sphere, Segment, If, and Line to visualize the vectors and their properties. Suitable for high school students learning geometry and calculus fundamentals.

GeoGebra Commands & Parameters

  1. Sphere((0, 0), 1) → a
  2. Segment((1, 0, 0), (1, 0, 1)) → u
  3. If(b > 0 ∧ b < π / 2, Rotate[u, (-b), yAxis]) → u'
  4. If(b > π / 2 ∧ b ≤ π, Rotate[u', (-(b - π / 2)), xAxis]) → u''
  5. Line((1, 0, 0), (1, 0, 1)) → e
  6. If(b > 0 ∧ b > π / 2, Rotate[e, (-b), yAxis]) → e'
  7. If(b > π / 2 ∧ π ≤ π, Rotate[e', (-(b - π / 2)), xAxis]) → e''
  8. If(b > π ∧ b < (3 / 2 * π), Rotate[e'', (-(b - π)), zAxis]) → e'''
  9. Point(e''') → A
  10. Point(e) → B
  11. CircleArc((0, 0, 0), (1, 0, 0), (0, 0, 1), xOyPlane) → f
  12. CircleArc((0, 0, 0), (0, 1, 0), (0, 0, 1), xOyPlane) → g
  13. CircleArc((0, 0, 0), (1, 0, 0), (0, 1, 0), xOyPlane) → h
  14. If(b > π ∧ b < (3 / 2 * π), Rotate[u'', (-(b - π)), zAxis]) → u'''
  15. Point(u''') → D
  16. Angle(B, C, D) → α
  17. Circle((1, 0, 0), 0.05, xOyPlane) → c
  18. Translate(c, Vector[(0, 0, 0), (0, 0, 1)]) → c'
  19. Cone(c', 0.3) → d, k, i
  20. If(b ≤ π / 2, Rotate[i, (-b), yAxis]) → j
  21. If(b > π / 2 ∧ b ≤ π, Rotate[j, (-(b - π / 2)), xAxis]) → l
  22. If(b < π / 2, Rotate[d, (-b), yAxis]) → m
  23. If(b ≥ π / 2 ∧ b < π, Rotate[m, (-(b - π / 2)), xAxis]) → n
  24. If(b > π, Rotate[n, (-(b - π)), zAxis]) → o
  25. Rotate(c, ((-π)) / 2, xAxis) → c'_1
  26. Rotate(c'_1, π / 2, zAxis) → p
  27. Cone(p, 0.3) → q, r, s
  28. Translate(q, Vector[(0, 0, 0), (-1, 0, 0)]) → g'
  29. Rotate(g', π / 2, xAxis) → g''
  30. If(b > π / 2 ∧ b < π, Rotate[g'', (-(b - π / 2)), xAxis]) → t
  31. If(b > π ∧ b < (1.5 * π), Rotate[g', (-(b - π)), zAxis]) → v