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Droz-Farny Line on a Sphere

Droz-Farny Line on a Sphere
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2018-03-26 00:48
TrigonometrySolid GeometryGeometric Construction
This example demonstrates the construction and properties of the Droz-Farny line on a sphere. Key mathematical concepts include spherical geometry, symmetry, centroid, and geometric constructions. The construction involves commands such as Sphere, PointIn, Plane, IntersectPath, OrthogonalLine, Intersect, Line, Circle, Point, Rotate, Midpoint, and Sequence. It is suitable for high school students and mathematics enthusiasts studying spherical geometry and geometric constructions.

GeoGebra Commands & Parameters

  1. Sphere(A, 1) → a
  2. PointIn(a) → B
  3. PointIn(a) → C
  4. PointIn(a) → D
  5. Plane(A, B, C) → b
  6. Plane(A, B, D) → c
  7. Plane(A, C, D) → d
  8. IntersectPath(b, a) → e
  9. IntersectPath(c, a) → f
  10. IntersectPath(d, a) → g
  11. OrthogonalLine(A, c) → i
  12. OrthogonalLine(A, b) → j
  13. Plane(C, i) → k
  14. IntersectPath(k, a) → q
  15. Plane(D, j) → m
  16. IntersectPath(m, a) → r
  17. OrthogonalLine(A, d) → h
  18. Plane(B, h) → l
  19. IntersectPath(l, a) → p
  20. Intersect(r, p, 1) → E
  21. Line(A, E) → n
  22. Circle(E, 1, n) → s
  23. Point(s) → F
  24. Rotate(F, π / 2, n) → F'
  25. Plane(F, n) → o
  26. Plane(F', n) → t
  27. IntersectPath(o, a) → c_1
  28. IntersectPath(t, a) → d_1
  29. Intersect(d_1, e, 1) → G
  30. Intersect(c_1, e, 2) → H
  31. Intersect(g, c_1, 1) → I
  32. Intersect(d_1, g, 2) → J
  33. Midpoint(G, H) → K
  34. Midpoint(I, J) → L
  35. Line(A, K) → f_1
  36. Line(L, A) → g_1
  37. Intersect(f_1, e, 1) → M
  38. Intersect(g_1, g, 2) → N
  39. Intersect(c_1, f, 2) → O
  40. Intersect(f, d_1, 2) → P
  41. Midpoint(O, P) → Q
  42. Line(A, Q) → h_1
  43. Intersect(h_1, f, 1) → R
  44. Circle(M, N, R) → e_1
  45. Sequence(Tool1[A, B, C, D, Rotate[F, ((w * 2) * π / 72)]], w, 1, 72) → L_1

AI Drawing Prompt Reference

On a unit sphere, take the center and three points on the sphere. Three great circles are formed by the center with each pair of the three points. Through the center, draw lines perpendicular to the planes of two of these great circles, and through the remaining sphere point, create planes containing each line, intersecting the sphere to form two new great circles. Similarly, through the center, draw a line perpendicular to the plane of the third great circle, and through the remaining sphere point, create a plane containing this line, intersecting the sphere to form a third new great circle. These new great circles intersect the original ones at several points. Take midpoints of two pairs of these intersection points, connect the center to each midpoint, and intersect these lines with the corresponding original great circles to get three points. Draw a circle through these three points, and rotate a point on this circle to generate a series of trajectories.
The prompt above was compiled from the original drawing commands and is for reference only.