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Modeling the Cube

Modeling the Cube
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2022-03-02 21:56
Solid GeometryParametric Equation
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This example demonstrates the geometric construction of a cube. Key mathematical concepts include geometric transformations, reflections, and parametric curves. Using GeoGebra commands such as If, OrthogonalLine, and Circle, the cube's faces and edges can be dynamically constructed. Suitable for middle school and above students or teachers for geometry instruction.

GeoGebra Commands & Parameters

  1. If(f ≟ 0° ∨ f ≟ 90° ∨ f ≟ 180° ∨ f ≟ 270° ∨ f ≟ 360°, f + 0.001, f) → f'
  2. If(b ≟ 0° ∨ b ≟ 90° ∨ b ≟ 180° ∨ b ≟ 270° ∨ b ≟ 360°, b + 0.001, b) → b'
  3. If(a ≟ 0° ∨ a ≟ 90° ∨ a ≟ 180° ∨ a ≟ 270° ∨ a ≟ 360°, a + 0.001, a) → a'
  4. If(j ≟ 0° ∨ j ≟ 90° ∨ j ≟ 180° ∨ j ≟ 270° ∨ j ≟ 360°, j + 0.001, j) → j'
  5. If(a ≟ 90°, 2, If[a ≟ 270°, 0, Ny + (cos(b) * Ny) + ((sin(b) * NPy) * sqrt(Nx^(2) + Ny^(2) + Nz^(2)) / sqrt(NPx^(2) + nada^(2) + NPz^(2)))]) → By
  6. If(a ≟ 90° ∨ a ≟ 270°, 0, Nz + (cos(b) * Nz) + ((sin(b) * NPz) * sqrt(Nx^(2) + Ny^(2) + Nz^(2)) / sqrt(NPx^(2) + nada^(2) + NPz^(2)))) → Bz
  7. If(e ≟ 0° ∨ e ≟ 90° ∨ e ≟ 180° ∨ e ≟ 270° ∨ e ≟ 360°, e + 0.001, e) → e'
  8. If((((((sgn(cos(e)) * sgn(cos(a))) * sgn(cos(j))) * sgn(cos(f))) * sgn(cos(d))) * sgn(cos(b))) ≟ 0, (((((sgn(cos(e + 0.01°)) * sgn(cos(a + 0.01°))) * sgn(cos(j + 0.01°))) * sgn(cos(f + 0.01°))) * sgn(cos(d + 0.01°))) * sgn(cos(b + 0.01°))), (((((sgn(cos(e)) * sgn(cos(a))) * sgn(cos(j))) * sgn(cos(f))) * sgn(cos(d))) * sgn(cos(b)))) → signo
  9. If(d ≟ 0° ∨ d ≟ 90° ∨ d ≟ 180° ∨ d ≟ 270° ∨ d ≟ 360°, d + 0.001, d) → d'
  10. OrthogonalLine(C, xAxis) → zz2
  11. Circle(C, 0.62) → zz0
  12. Intersect(zz0, zz2, 1) → ZZ4
  13. Intersect(zz0, zz2, 2) → ZZ3
  14. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  15. Point(zElipse2) → Z
  16. OrthogonalLine(C, yAxis) → zz1
  17. Intersect(zz0, zz1, 2) → ZZ2
  18. Intersect(zz0, zz1, 1) → ZZ1
  19. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  20. Point(zElipse1) → XY
  21. Punto3D(C, XY, Z, LJ1) → J1
  22. Punto3D(C, XY, Z, LJ2) → J2
  23. Punto3D(C, XY, Z, LE) → E
  24. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  25. Angle(ZZ8, C, Z) → zAngInc
  26. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  27. Angle(ZZ6, C, XY) → zAngRot
  28. Segment(ZZ5, ZZ6) → zz4
  29. Segment(ZZ7, ZZ8) → zz3
  30. Intersect(zElipse2, zElipse1, 1) → zPunto1
  31. Intersect(zElipse2, zElipse1, 4) → zPunto2
  32. Segment(zPunto1, zPunto2) → zSeg2
  33. Point(Circle[AuxX, distancia3D]) → AuxY
  34. Segment(AuxX, AuxY) → segmentodistancia3D
  35. Punto3D(C, XY, Z, LU) → U
  36. Punto3D(C, XY, Z, LO) → O
  37. Punto3D(C, XY, Z, LF) → F
  38. Punto3D(C, XY, Z, LA) → A
  39. Polygon(O, E, U) → poly1, poly1c, poly1b, poly1a
  40. Punto3D(C, XY, Z, LB) → B
  41. Segment(O, U) → barraOU
  42. Segment(O, E) → barraOE
  43. Segment(U, F) → barraUF
  44. Segment(F, E) → barraEF
  45. Segment(O, A) → barraOA
  46. Segment(A, B) → barraAB
  47. If(isomero, J1, J2) → J
  48. Segment(B, J) → barraBJ
  49. Segment(U, B) → barraUB
  50. Segment(F, J) → barraFJ
  51. Punto3D(C, XY, Z, LD) → D
  52. Segment(A, D) → barraAD
  53. Segment(E, D) → barraED
  54. Segment(D, J) → barraDJ
  55. Punto3D(C, XY, Z, LM) → M
  56. Punto3D(C, XY, Z, LMP) → MP
  57. Punto3D(C, XY, Z, LNP) → NP
  58. If((((((sgn(cos(e)) * sgn(cos(a))) * sgn(cos(j))) * sgn(cos(f))) * sgn(cos(d))) * sgn(cos(b))) ≟ 0, (((((sgn(cos(e - 0.01°)) * sgn(cos(a - 0.01°))) * sgn(cos(j - 0.01°))) * sgn(cos(f - 0.01°))) * sgn(cos(d - 0.01°))) * sgn(cos(b - 0.01°))), (((((sgn(cos(e)) * sgn(cos(a))) * sgn(cos(j))) * sgn(cos(f))) * sgn(cos(d))) * sgn(cos(b)))) → c
  59. Sequence({((-cos(e)) * arista), (sin(e) * arista), 0}, e, 8°, 300.00000000000006°, 72°) → LCE, {{-4.951340343707852, 0.6958655048003272, 0}, {-0.8682408883346521, 4.92403876506104, 0}, {4.414737964294635, 2.3473578139294533, 0}, {3.5966990016932554, -3.4732918522949867, 0}, {-2.191855733945387, -4.493970231495835, 0}}
  60. Conic(Punto3D[C, XY, Z, Element[LCE, 1]], Punto3D[C, XY, Z, Element[LCE, 2]], Punto3D[C, XY, Z, Element[LCE, 3]], Punto3D[C, XY, Z, Element[LCE, 4]], Punto3D[C, XY, Z, Element[LCE, 5]]) → CE
  61. Sequence({(arista * (Ex / 2 - (((sgn(cos(e)) * dF) * cos(f)) * sin(aF)))), (arista * ((Ey + 1) / 2 + ((dF * cos(f)) * cos(aF)))), ((arista * dF) * sin(f))}, f, 8°, 300.00000000000006°, 72°) → LCFF, {{-4.975667684019047, 4.980645841196502, 0.49229718133007855}, {-2.934118977107226, 2.937054564122473, 3.483561676900716}, {-0.2926308715371856, 0.29292364882176347, 1.660662336901143}, {-0.7016501483281518, 0.7023521495355842, -2.4572159088589807}, {-3.59592606900891, 3.599523794240345, -3.179305286272957}}
  62. Conic(Punto3D[C, XY, Z, Element[LCFF, 1]], Punto3D[C, XY, Z, Element[LCFF, 2]], Punto3D[C, XY, Z, Element[LCFF, 3]], Punto3D[C, XY, Z, Element[LCFF, 4]], Punto3D[C, XY, Z, Element[LCFF, 5]]) → CFF
  63. Sequence({((arista * sin(j)) * cos(a)), (arista * sin(a)), ((arista * cos(j)) * cos(a))}, a, 8°, 300.00000000000006°, 72°) → LCA, {{0, 0.6958655048003272, 4.951340343707852}, {0, 4.92403876506104, 0.8682408883346521}, {0, 2.3473578139294533, -4.414737964294635}, {0, -3.4732918522949867, -3.5966990016932554}, {0, -4.493970231495835, 2.191855733945387}}
  64. Conic(Punto3D[C, XY, Z, Element[LCA, 1]], Punto3D[C, XY, Z, Element[LCA, 2]], Punto3D[C, XY, Z, Element[LCA, 3]], Punto3D[C, XY, Z, Element[LCA, 4]], Punto3D[C, XY, Z, Element[LCA, 5]]) → CA
  65. Sequence({((arista * sin(j)) * cos(a)), (arista * sin(a)), ((arista * cos(j)) * cos(a))}, j, 8°, 300.00000000000006°, 72°) → LCJ, {{0.6958655048003272, 0, 4.951340343707852}, {4.92403876506104, 0, 0.8682408883346521}, {2.3473578139294533, 0, -4.414737964294635}, {-3.4732918522949867, 0, -3.5966990016932554}, {-4.493970231495835, 0, 2.191855733945387}}
  66. Conic(Punto3D[C, XY, Z, Element[LCJ, 1]], Punto3D[C, XY, Z, Element[LCJ, 2]], Punto3D[C, XY, Z, Element[LCJ, 3]], Punto3D[C, XY, Z, Element[LCJ, 4]], Punto3D[C, XY, Z, Element[LCJ, 5]]) → CJ
  67. Sequence({(arista * ((((((sqrt(2) * cos(a)) * sin(d)) * sin(e)) * cos(j)) * sqrt((((-cos(a)) * cos(e)) * sin(j)) + (sin(a) * sin(e)) + 1) / ((2 * sqrt(((cos(a))^(2) * (((sin(e))^(2) * (sin(j))^(2)) + (cos(j))^(2))) + (((((2 * sin(a)) * cos(a)) * sin(e)) * cos(e)) * sin(j)) + ((sin(a))^(2) * (cos(e))^(2)))))) + (cos(a) * ((cos(d) * sin(j) / 2) + sin(j) / 2)) - (cos(d) * cos(e) / 2) - cos(e) / 2)), (arista * ((((((sqrt(2) * cos(a)) * sin(d)) * cos(e)) * cos(j)) * sqrt((((-cos(a)) * cos(e)) * sin(j)) + (sin(a) * sin(e)) + 1) / ((2 * sqrt(((cos(a))^(2) * (((sin(e))^(2) * (sin(j))^(2)) + (cos(j))^(2))) + (((((2 * sin(a)) * cos(a)) * sin(e)) * cos(e)) * sin(j)) + ((sin(a))^(2) * (cos(e))^(2)))))) + (sin(a) * (cos(d) / 2 + 1 / 2)) + (cos(d) * sin(e) / 2) + sin(e) / 2)), (arista * ((cos(a) * ((cos(d) * cos(j) / 2) + cos(j) / 2)) - (((sqrt(2) * sin(d)) * (((cos(a) * sin(e)) * sin(j)) + (sin(a) * cos(e)))) * sqrt((((-cos(a)) * cos(e)) * sin(j)) + (sin(a) * sin(e)) + 1) / ((2 * sqrt(((cos(a))^(2) * (((sin(e))^(2) * (sin(j))^(2)) + (cos(j))^(2))) + (((((2 * sin(a)) * cos(a)) * sin(e)) * cos(e)) * sin(j)) + ((sin(a))^(2) * (cos(e))^(2))))))))}, d, 8°, 300.00000000000006°, 72°) → LCD, {{-4.975670171853926, 0.4920512172381114, 4.975670171853926}, {-2.934120444167326, 3.481821201600095, 2.934120444167326}, {-0.29263101785268253, 1.6598326281007467, 0.29263101785268253}, {-0.7016504991533723, -2.4559882217977695, 0.7016504991533723}, {-3.5959278669726933, -3.177716825141184, 3.5959278669726933}}
  68. Conic(Punto3D[C, XY, Z, Element[LCD, 1]], Punto3D[C, XY, Z, Element[LCD, 2]], Punto3D[C, XY, Z, Element[LCD, 3]], Punto3D[C, XY, Z, Element[LCD, 4]], Punto3D[C, XY, Z, Element[LCD, 5]]) → CD
  69. Sequence({(arista * (((((sqrt(2) * sin(b)) * cos(j)) * sgn(cos(a))) * sqrt(sin(a) + 1) / 2) + (cos(a) * ((cos(b) * sin(j) / 2) + sin(j) / 2)))), (arista * ((sin(a) * (cos(b) / 2 + 1 / 2)) + cos(b) / 2 + 1 / 2)), (arista * ((cos(a) * ((cos(b) * cos(j) / 2) + cos(j) / 2)) - ((((sqrt(2) * sin(b)) * sin(j)) * sgn(cos(a))) * sqrt(sin(a) + 1) / 2)))}, b, 8°, 300.00000000000006°, 72°) → LCB, {{0.4920512172381114, 4.975670171853926, 4.975670171853926}, {3.481821201600095, 2.934120444167326, 2.934120444167326}, {1.6598326281007467, 0.29263101785268253, 0.29263101785268253}, {-2.4559882217977695, 0.7016504991533723, 0.7016504991533723}, {-3.177716825141184, 3.5959278669726933, 3.5959278669726933}}
  70. Conic(Punto3D[C, XY, Z, Element[LCB, 1]], Punto3D[C, XY, Z, Element[LCB, 2]], Punto3D[C, XY, Z, Element[LCB, 3]], Punto3D[C, XY, Z, Element[LCB, 4]], Punto3D[C, XY, Z, Element[LCB, 5]]) → CB
  71. Segment(O, F) → g
  72. Segment(O, D) → h
  73. Segment(O, B) → i

AI Drawing Prompt Reference

Draw a 3D cube construction diagram. First, set up a reference circle together with two perpendicular direction lines, then use their intersections to define key points that generate two ellipse-related loci. From these, pick the corresponding 3D points by using a shared base (same XY projection) with a chosen height to obtain the cube’s top and bottom vertices. Next, connect the appropriate vertex points with straight segments to form the visible cube edges, and trace polygonal faces by linking vertices in sequence. Include the main connecting segment(s) between a selected vertex pair to emphasize the cube’s spatial correspondence, producing a clear perspective view of the cube.
The prompt above was compiled from the original drawing commands and is for reference only.