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Cylinder Illusion Demo

Cylinder Illusion Demo
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2020-02-06 22:31
VectorFunctionTrigonometrySolid GeometryGraphing
This example demonstrates the visual illusion of a cylinder in a 2D perspective. It involves core mathematical concepts such as vectors, two-dimensional shapes, sine functions, trigonometry, and piecewise functions. The construction uses GeoGebra commands like Surface, Polygon, and CurveCartesian, along with objects like k, α, f_x, f_y, f_z, h, a, b, c, d, e, f, g, i, j. It is suitable for high school and above students studying solid geometry, coordinate geometry, and function graphing.

GeoGebra Commands & Parameters

  1. Surface((f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → a
  2. Surface((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → b
  3. Surface((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → c
  4. Surface((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → d
  5. Surface((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → e
  6. Surface(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → f
  7. Surface(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → g
  8. Surface((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → i
  9. Surface((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → j
  10. Surface(-9 - ((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → b'
  11. Surface(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → a'
  12. Surface(-9 - ((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → c'
  13. Surface(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → d'
  14. Surface(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → e'
  15. Surface(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → f'
  16. Surface(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → g'
  17. Surface(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → i'
  18. Surface(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + d, s, ((-π)) / 2, (3 * π / 2), d, 0, h) → j'
  19. Polygon((-4.5, (-width), 0), (-4.5, width, 0), (-4.5, width, height), (-4.5, (-width), height)) → poly1, m, n, p, q
  20. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → r
  21. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → o
  22. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → s
  23. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → t
  24. CurveCartesian((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → u
  25. CurveCartesian((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → v
  26. CurveCartesian(-9 - ((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → w
  27. CurveCartesian(-9 - ((2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → a_1
  28. CurveCartesian((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → b_1
  29. CurveCartesian((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → c_1
  30. CurveCartesian(-9 - ((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → d_1
  31. CurveCartesian(-9 - ((-2 * k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), ((-f_x(s)) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → e_1
  32. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → f_1
  33. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → g_1
  34. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → h_1
  35. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → i_1
  36. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → j_1
  37. CurveCartesian((f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → k_1
  38. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → l_1
  39. CurveCartesian(-9 - ((f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-2 * k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → m_1
  40. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → n_1
  41. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → o_1
  42. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → p_1
  43. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → q_1
  44. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → r_1
  45. CurveCartesian(k + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → s_1
  46. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → t_1
  47. CurveCartesian(-9 - (k + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → u_1
  48. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → v_1
  49. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → w_1
  50. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → a_2
  51. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), (-k) - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → b_2
  52. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → c_2
  53. CurveCartesian((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α)), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → d_2
  54. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + h, s, ((-π)) / 2, (3 * π / 2)) → e_2
  55. CurveCartesian(-9 - ((-k) + (f_x(s) * cos(α)) + (f_y(s) * sin(α))), k - (f_x(s) * sin(α)) + (f_y(s) * cos(α)), f_z(s) + 0, s, ((-π)) / 2, (3 * π / 2)) → f_2

AI Drawing Prompt Reference

Draw a cylinder illusion demonstration, showing its visual illusion in 2D perspective. Include multiple surfaces and curves that revolve around a central axis to form different parts of the cylinder. Start by drawing the basic shape of the cylinder, then adjust the height and position of the surfaces to create a visual illusion effect. Finally, add a rectangular base as the ground to make the entire image more complete.
The prompt above was compiled from the original drawing commands and is for reference only.