Cone Geometry and Optimization Example

GeoGebra Commands & Parameters

1. Cylinder(A, C, r) → a, c, d, b
2. Line(D, E) → f
3. Intersect(f, d, 1) → F
4. Sphere(F, r) → e
5. Intersect(f, e) → G, H
6. Plane(G, F, C) → g
7. Rotate(C, If[0 < i ≤ 1, (i * π / 2), π / 2], D, g) → C'
8. Rotate(d, If[0 < i ≤ 1, (i * π / 2), π / 2], D, g) → d'
9. Rotate(dd, If[0 < i ≤ 1, (i * π / 2), π / 2], D, g) → dd'
10. Dilate(C, If[1 < i ≤ 2, 2 - i, 0], I) → C'_1
11. Dilate(C, If[1 < i ≤ 2, 2 - i, 0], J) → C'_2
12. Vector(C, C'_1) → u
13. Vector(C, C'_2) → v
14. Translate(dd, u) → dd'_1
15. Translate(dd, v) → dd'_2
16. Translate(K, u) → K'
17. Translate(K, v) → K'_1
18. Rotate(K', If[1 < i ≤ 2, ((i - 1) * π), π], C'_1) → K''
19. Rotate(K'_1, If[1 < i ≤ 2, ((-(i - 1)) * π), (-π)], C'_2) → L
20. Translate(D, u) → D'
21. Translate(D, v) → D'_1
22. Polygon(D', D'_1, N, M) → poly1, d'_1, d'_2, n, m
23. Segment(K'', Q) → j
24. CircleArc(P, Q, M) → k
25. Segment(L, S) → l
26. CircleArc(R, N, S) → p
27. CircleArc(C'_2, D'_1, L) → q
28. CircleArc(C'_1, K'', D') → s
29. Plane(T, A, B) → o
30. Rotate(C', If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → C''
31. Rotate(poly1, If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → poly1'
32. Rotate(d', If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → d'_3
33. Rotate(dd', If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → dd'_3
34. Rotate(K'', If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → K'''
35. Rotate(Q, If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → Q'
36. Segment(K''', Q') → j'
37. Rotate(L, If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → L'
38. Rotate(S, If[2 < i ≤ 3, ((i - 2) * π / 2), π / 2], E, g) → S'
39. Segment(L', S') → l'
40. Dilate(C'', If[3 < i ≤ 4, 4 - i, 0], U) → C'''
41. Vector(C'', C''') → w
42. Dilate(C'', If[3 < i ≤ 4, 4 - i, 0], V) → C'''_1
43. Vector(C'', C'''_1) → t
44. Translate(Z, w) → Z'
45. Translate(Z, t) → Z'_1
46. Translate(W, w) → W'
47. Translate(W, t) → W'_1
48. Rotate(W', If[3 < i ≤ 4, ((-(i - 3)) * π), (-π)], C''') → W''
49. CircleArc(C''', Z', W'') → c_1
50. Segment(Z, Z') → f_1
51. Rotate(W'_1, If[3 < i ≤ 4, ((i - 3) * π), π], C'''_1) → A_1
52. CircleArc(C'''_1, Z'_1, A_1) → d_1
53. Segment(Z, Z'_1) → g_1
54. Segment(R, N) → h_1
55. Angle(N, R, S) → α
56. CurveCartesian(x(C'_2) + (r * cos(t)), y(C'_2) + (r * sin(t)), z(C'_2), t, π / 2, π / 2 + α) → a_1
57. Vector(A, C) → tt
58. Surface(x(C'_2) + (r * cos(t)) - (u * x(tt)), y(C'_2) + (r * sin(t)) - (u * y(tt)), z(C'_2) - (u * z(tt)), u, 0, 1, t, π / 2, π / 2 + α) → b_1
59. Surface((-x(C'_2)) - (r * cos(t)) - (u * x(tt)), y(C'_2) + (r * sin(t)) - (u * y(tt)), z(C'_2) - (u * z(tt)), u, 0, 1, t, π / 2, π / 2 + α) → e_1
60. Midpoint(L', S') → B_1
61. Midpoint(K''', Q') → C_1
62. Line(B_1, C_1) → i_1
63. Mirror(d_1, i_1) → k_1
64. Mirror(c_1, i_1) → p_1
65. Mirror(Z, i_1) → Z'_2
66. Mirror(Z', i_1) → Z''
67. Segment(Z'_2, Z'') → j_1
68. Mirror(Z, i_1) → Z'_3
69. Mirror(Z'_1, i_1) → D_1
70. Segment(Z'_3, D_1) → l_1