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Unwrapping a Cylinder: Dynamic and Modifiable Illustration

Unwrapping a Cylinder: Dynamic and Modifiable Illustration
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2017-08-02 23:21
VectorTrigonometrySolid GeometryGeometric Construction
In collections GeoGebra 3D & AR: Geometry Resources & Challenges
This example demonstrates how to unfold a cylinder into a flat shape, helping students understand how to calculate lateral and total surface areas. Key math concepts include cylinder, 2D vectors, mode, trapezoid, cuboid, and tangent function. Construction highlights the use of GeoGebra commands like Cylinder, Line, Intersect, Sphere, Plane, Rotate, Dilate, Vector, Translate, Polygon, Segment, CircleArc, Angle, CurveCartesian, Surface, Midpoint, Mirror. Suitable for high school students learning solid geometry and area calculations.

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

Draw an unfolded cylinder, starting by creating a cylinder and then unfolding it into a flat shape. Use rotation, translation, and dilation to generate the various parts of the unfolded shape, such as rectangles, trapezoids, and triangles. Finally, use mirroring and translation to complete the unfolded shape.
The prompt above was compiled from the original drawing commands and is for reference only.