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Cube Cross-Section Explorer

Cube Cross-Section Explorer
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2012-08-22 14:53
PolygonVectorSequenceSolid GeometryGeometric Construction
This GeoGebra example demonstrates the dynamic cross-sections formed when a plane intersects a cube, showing how the resulting polygon changes as the cutting plane moves and rotates. It covers core concepts in solid geometry, including polyhedron sections, plane-solid incidence, 3D vector dot products for coplanarity and perpendicularity, and ordered polygon construction. The construction relies on GeoGebra commands such as Odcinek_3D (3D segment), Element, Polygon, IloczynSkalarny3D (3D dot product), If, RemoveUndefined, Sequence, and Sort, together with points A1–C8, A, D1, D2, vectors v, w, u, and parameters a, b, zoom, α, β, z, δ, o, γ, r to generate intersection points and section boundaries. Suitable for high school solid geometry, university analytic geometry, and GeoGebra 3D modeling enthusiasts studying dynamic plane sections of cubes.

GeoGebra Commands & Parameters

  1. Odcinek_3D(D1, D5, D4, D2) → c, d
  2. Odcinek_3D(D1, D2, D5, D4) → e, f
  3. Odcinek_3D(D1, D4, D2, D5) → g, h
  4. Odcinek_3D(D2, D6, D1, D3) → i, j
  5. Odcinek_3D(D2, D3, D6, D1) → k, l
  6. Odcinek_3D(D3, D7, D2, D4) → m, n
  7. Odcinek_3D(D3, D4, D7, D2) → p, q
  8. Odcinek_3D(D4, D8, D3, D1) → s, t
  9. Odcinek_3D(D8, D5, D7, D4) → a_1, b_1
  10. Odcinek_3D(D8, D7, D4, D5) → c_1, d_1
  11. Odcinek_3D(D7, D6, D3, D8) → e_1, f_1
  12. Odcinek_3D(D6, D5, D2, D7) → g_1, h_1
  13. Element(Join[B10], 1) → C10
  14. Element(Join[B10], 2) → D10
  15. Element(Join[B10], 3) → E10
  16. Element(Join[B11], 1) → C11
  17. Element(Join[B11], 2) → D11
  18. Element(Join[B11], 3) → E11
  19. Element(Join[B12], 1) → C12
  20. Element(Join[B12], 2) → D12
  21. Element(Join[B12], 3) → E12
  22. Element(Join[B13], 1) → C13
  23. Element(Join[B13], 2) → D13
  24. Element(Join[B13], 3) → E13
  25. Element(Join[B14], 1) → C14
  26. Element(Join[B14], 2) → D14
  27. Element(Join[B14], 3) → E14
  28. Polygon(F12, F13, F10, F11) → poly1, f12, f13, f10, f11
  29. IloczynSkalarny3D(B14, B15) → C15
  30. IloczynSkalarny3D(B15, C17) → E17
  31. IloczynSkalarny3D(B15, C18) → E18
  32. IloczynSkalarny3D(B15, C19) → E19
  33. IloczynSkalarny3D(B15, C20) → E20
  34. IloczynSkalarny3D(B15, C21) → E21
  35. IloczynSkalarny3D(B15, C22) → E22
  36. IloczynSkalarny3D(B15, C23) → E23
  37. IloczynSkalarny3D(B15, C24) → E24
  38. IloczynSkalarny3D(B15, C25) → E25
  39. IloczynSkalarny3D(B15, C26) → E26
  40. IloczynSkalarny3D(B15, C27) → E27
  41. IloczynSkalarny3D(B15, C28) → E28
  42. If((F17 ≥ 0) ∧ (F17 ≤ 1), F17) → G17
  43. If((F18 ≥ 0) ∧ (F18 ≤ 1), F18) → G18
  44. If((F19 ≥ 0) ∧ (F19 ≤ 1), F19) → G19
  45. If((F20 ≥ 0) ∧ (F20 ≤ 1), F20) → G20
  46. If((F21 ≥ 0) ∧ (F21 ≤ 1), F21) → G21
  47. If((F22 ≥ 0) ∧ (F22 ≤ 1), F22) → G22
  48. If((F23 ≥ 0) ∧ (F23 ≤ 1), F23) → G23
  49. If((F24 ≥ 0) ∧ (F24 ≤ 1), F24) → G24
  50. If((F25 ≥ 0) ∧ (F25 ≤ 1), F25) → G25
  51. If((F26 ≥ 0) ∧ (F26 ≤ 1), F26) → G26
  52. If((F27 ≥ 0) ∧ (F27 ≤ 1), F27) → G27
  53. If((F28 ≥ 0) ∧ (F28 ≤ 1), F28) → G28
  54. RemoveUndefined(list1) → list2
  55. Sequence(Angle[Element[list2, 1], F14, Element[list2, i]], i, 1, Length[list2]) → list3
  56. Sort(list3) → list4
  57. Sequence(Sequence[If[Element[list3, j] ≟ Element[list4, i], Element[list2, j]], j, 1, Length[list2]], i, 1, Length[list4]) → list5
  58. RemoveUndefined(Join[list5]) → list6
  59. Polygon(list6) → wielokąt1, , , ,

AI Drawing Prompt Reference

Draw a cube and show how the cross-section polygon changes as a plane intersects it. Start by drawing the 8 vertices of the cube, then draw the 12 edges. Define a plane and adjust its position and angle to observe the changes in the cross-section polygon. Finally, based on the intersection points of the plane and the cube, draw the cross-section polygon.
The prompt above was compiled from the original drawing commands and is for reference only.