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Projections of a Rotating 3-Cube

Projections of a Rotating 3-Cube
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2020-10-23 04:38
Solid GeometryGeometric Construction
This example demonstrates the projections of a rotating 3-dimensional cube onto various planes. It involves core mathematical concepts such as solid geometry, projection transformations, and rotational motion. The construction uses GeoGebra commands like Cube, Plane, Segment, Line, and Intersect. It is suitable for high school students and above learning spatial geometry, visualization, and projection transformations, as well as for university-level introductory geometry courses.

GeoGebra Commands & Parameters

  1. Cube(A2, B2, C2) → a, D, faceC2DHG, faceEFGH, edgeA2D, edgeA2B2, edgeB2C2, edgeC2D, edgeA2E, edgeDH, edgeEH, edgeEF, E, edgeB2F, edgeFG, edgeC2G, edgeGH, F, G, H, faceA2B2C2D, faceA2DHE, faceA2B2FE, faceB2C2GF
  2. Plane((1, 0, 0), (0, 1, 0), (1, 1, 0)) → Z
  3. Segment(A2, Ap) → f
  4. Segment(B2, Bp) → g
  5. Segment(C2, Cp) → i
  6. Segment(D, Dp) → j
  7. Segment(E, Ep) → k
  8. Segment(F, Fp) → l
  9. Segment(G, Gp) → m
  10. Segment(H, Hp) → n
  11. Segment(Ap, Bp) → p
  12. Segment(Bp, Cp) → q
  13. Segment(Cp, Dp) → r
  14. Segment(Ap, Dp) → s
  15. Segment(Ep, Fp) → t
  16. Segment(Fp, Gp) → b
  17. Segment(Gp, Hp) → c
  18. Segment(Ep, Hp) → d
  19. Segment(Dp, Hp) → e
  20. Segment(Ep, Ap) → f_1
  21. Segment(Fp, Bp) → g_1
  22. Segment(Cp, Gp) → h_1
  23. Line(S, A2) → LA
  24. Intersect(LA, Z) → ap
  25. Line(S, B2) → LB
  26. Intersect(Z, LB) → bp
  27. Line(S, C2) → LC
  28. Intersect(Z, LC) → cp
  29. Line(S, D) → LD
  30. Intersect(Z, LD) → dp
  31. Line(S, E) → LE
  32. Intersect(LE, Z) → ep
  33. Line(S, F) → LF
  34. Intersect(LF, Z) → fp
  35. Line(S, G) → LG
  36. Intersect(Z, LG) → gp
  37. Line(S, H) → LH
  38. Intersect(Z, LH) → hp
  39. Segment(S, ap) → i_1
  40. Segment(S, bp) → j_1
  41. Segment(S, cp) → k_1
  42. Segment(S, dp) → l_1
  43. Segment(S, ep) → m_1
  44. Segment(S, fp) → n_1
  45. Segment(S, gp) → p_1
  46. Segment(S, hp) → q_1
  47. Segment(ap, bp) → r_1
  48. Segment(bp, cp) → s_1
  49. Segment(cp, dp) → t_1
  50. Segment(dp, ap) → a_1
  51. Segment(hp, ep) → b_1
  52. Segment(ep, fp) → c_1
  53. Segment(fp, gp) → d_1
  54. Segment(gp, hp) → e_1
  55. Segment(ap, ep) → f_2
  56. Segment(dp, hp) → g_2
  57. Segment(gp, cp) → h_2
  58. Segment(fp, bp) → i_2

AI Drawing Prompt Reference

Draw a rotating 3D cube with its eight vertices projected onto a fixed plane, forming a projected octagon. First, construct the cube, then choose a horizontal plane as the projection plane. From each vertex of the cube, draw a perpendicular line to the plane, and mark the foot of each perpendicular as the projected point. Connect these projected points to form the projected polygon, while also keeping the lines from the original vertices to their projections and the lines between projected points to show the projection relationship. The overall effect demonstrates the projection transformation in solid geometry.
The prompt above was compiled from the original drawing commands and is for reference only.