Demonstration of a Hyperboloid of Two Sheets as a Locus

GeoGebra Commands & Parameters

1. PointIn(c) → A
2. Segment(F_1, A) → f
3. Segment(A, F_2) → g
4. Sphere(F_1, e) → i
5. Intersect(f, i) → F,
6. Dilate(F_1, If[0 < h ≤ 1, 1 - h, 0], F) → C'
7. Segment(F_1, C') → j
8. Dilate(F, If[1 < h ≤ 2, 2 - h, 0], A) → C'_1
9. Segment(F, C'_1) → k
10. Dilate(F_2, If[2 < h ≤ 3, 3 - h, 0], A) → B'
11. Segment(F_2, B') → l
12. Angle(F_2, A, F_1) → α
13. Plane(F_1, A, F_2) → m
14. Rotate(F_2, If[3 < h ≤ 4, ((h - 3) * α), α], A, m) → B'_1
15. Rotate(B', If[3 < h ≤ 4, ((h - 3) * α), α], A, m) → B''
16. Segment(B'_1, B'') → l'
17. Segment(A, F) → n
18. Dilate(F_1, If[4 < h ≤ 5, 5 - h, 0], V_1) → C'_2
19. Vector(F_1, C'_2) → u
20. Translate(F, u) → F'
21. Angle(V_2, V_1, F') → β
22. Plane(F', V_1, V_2) → o
23. Translate(F_1, u) → C'_3
24. Translate(C', u) → C''
25. Segment(C'_3, C'') → j'
26. Rotate(C'_3, If[5 < h ≤ 6, ((-(h - 5)) * β), (-β)], V_1, o) → G
27. Rotate(C'', If[5 < h ≤ 6, ((-(h - 5)) * β), (-β)], V_1, o) → C'''
28. Segment(G, C''') → j''
29. Segment(F_2, A) → p
30. Segment(F_1, A) → q
31. Sphere(F_2, e) → r
32. Distance(V_1, V_2) → t
33. Sphere(F_2, s) → v
34. Intersect(p, v) → Q, I
35. Dilate(F_2, If[0 < h ≤ 1, 1 - h, 0], I) → B'_2
36. Segment(F_2, B'_2) → f_1
37. Dilate(I, If[1 < h ≤ 2, 2 - h, 0], A) → I'
38. Segment(I, I') → g_1
39. Dilate(F_1, If[2 < h ≤ 3, 3 - h, 0], A) → C'_4
40. Segment(F_1, C'_4) → h_1
41. Angle(F_1, A, F_2) → γ
42. Plane(F_1, A, F_2) → w
43. Rotate(F_1, If[3 < h ≤ 4, h - 3]) → C'_5
44. Rotate(C'_4, If[3 < h ≤ 4, h - 3]) → H
45. Segment(C'_5, H) → i_1
46. Rotate(C'_4, If[3 < h ≤ 4, h - 3]) → J
47. Rotate(F_1, If[3 < h ≤ 4, ((-(h - 3)) * γ), (-γ)], A, w) → C'_7
48. Rotate(C'_4, If[3 < h ≤ 4, ((-(h - 3)) * γ), (-γ)], A, w) → K
49. Segment(C'_7, K) → k_1
50. Segment(I, A) → j_1
51. Vector(F_2, V_2) → u_1
52. Translate(I, u_1) → I'_1
53. Dilate(F_2, If[4 < h ≤ 5, 5 - h, 0], V_2) → B'_3
54. Vector(F_2, B'_3) → v_1
55. Translate(F_2, v_1) → B'_4
56. Translate(B'_2, v_1) → L
57. Segment(B'_4, L) → l_1
58. Angle(I'_1, V_2, V_1) → δ
59. Rotate(B'_4, If[5 < h ≤ 6, h - 5]) → M
60. Rotate(L, If[5 < h ≤ 6, h - 5]) → L'
61. Rotate(B'_4, If[5 < h ≤ 6, h - 5]) → N
62. Rotate(L, If[5 < h ≤ 6, h - 5]) → L'_1
63. Rotate(B'_4, If[5 < h ≤ 6, h - 5]) → O
64. Plane(I'_1, V_2, V_1) → a_1
65. Rotate(B'_4, If[5 < h ≤ 6, ((-(h - 5)) * δ), (-δ)], V_2, a_1) → P
66. Rotate(L, If[5 < h ≤ 6, ((-(h - 5)) * δ), (-δ)], V_2, a_1) → L'_3
67. Segment(P, L'_3) → q_1
68. Ray(V_1, F_1) → m_1
69. Ray(V_2, F_2) → n_1