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Cylinder Net Demonstration

Cylinder Net Demonstration
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2018-09-19 19:45
Solid GeometryGeometric ConstructionGraphing
This example demonstrates the net of a cylinder and its geometric properties. It involves core mathematical concepts such as solid geometry, plane figures, area calculation, and spatial visualization. The construction utilizes commands like Cylinder, Line, Intersect, Sphere, Plane, Rotate, Dilate, Vector, Translate, Polygon, Segment, CircleArc, Angle, CurveCartesian, Surface, Midpoint, and Mirror, as outlined in the construction summary. It is suitable for high school students learning solid geometry and for educational demonstrations.

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

AI Drawing Prompt Reference

The figure is an unfolded cylinder. It consists of a rectangle and several arcs, where the rectangle represents the lateral surface of the cylinder and the arcs represent the top and bottom. The construction order: first draw the cylinder, then use rotation and plane operations to unfold it, and finally draw the unfolded figure using polygons, line segments, and arcs. The figure includes a central axis, top and bottom arcs, and a rectangle connecting them. It presents as a planar figure that helps understand the geometric structure of the cylinder.
The prompt above was compiled from the original drawing commands and is for reference only.