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Artformer Helix

Artformer Helix
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2020-03-07 19:43
CircleTrigonometryCoordinate GeometryParametric EquationGeometric Construction
This example demonstrates the construction of an artformer helix. It involves core mathematical concepts such as circles, ellipses, and rotation. Using GeoGebra commands like Line, Sequence, Polygon, and Intersect, the helix shape can be dynamically constructed. Suitable for high school and above students learning coordinate geometry and parametric equations.

GeoGebra Commands & Parameters

  1. Line((0, 0, 0), (0, 0, 1)) → t
  2. Sequence((XB((i * β)), YB((i * β)), ZB((i * β))), i, 0, n) → PontB
  3. Polygon(PontB) → Lap1
  4. Intersect(t, Line[A1, B2]) → Mt
  5. Distance(Mt, A1) → ma
  6. Distance(Mt, B2) → mb
  7. Sequence((XC((i * β)), YC((i * β)), ZC((i * β))), i, 0, n) → PontC
  8. Circle((d, 0, 0), r, xOyPlane) → k0
  9. Line((0, 0, 0), (0, 1, 0)) → ty
  10. Ellipse((0, fe), (0, (-fe)), ae) → e0
  11. Dilate(Translate[e0, Vector[(0, 0, c)]], dd, Mt) → alaplap
  12. Rotate(k0, φ, ty) → kör1
  13. Rotate(kör1, 180°, t) → k1
  14. Dilate(k1, nm, Mt) → kör2
  15. Dilate(kör1, nm^(2), Mt) → kör3
  16. Dilate(kör2, nm^(2), Mt) → kör4
  17. Dilate(kör3, nm^(2), Mt) → kör5
  18. Dilate(PontB, nm^(2), Mt) → PontD
  19. Center(kör1) → K1
  20. OrthogonalLine(K1, kör1) → f1
  21. Dilate(PontC, nm^(2), Mt) → PontE
  22. Polygon(PontC) → Lap2
  23. Dilate(PontD, nm^(2), Mt) → PontF
  24. Polygon(PontD) → Lap3
  25. Polygon(PontE) → Lap4
  26. Polygon(PontF) → Lap5
  27. Segment(A, Mt) → alkotó
  28. Sequence((XA((i * β)), YA((i * β)), ZA((i * β))), i, 0, n) → PontA
  29. Polygon(PontA) → LapA
  30. Sequence(Polygon[Element[PontB, i], Element[PontA, i], Element[PontA, i + 1], Element[PontB, i + 1]], i, 1, n) → QA
  31. Sequence(Polygon[Element[PontC, i], Element[PontB, i], Element[PontB, i + 1], Element[PontC, i + 1]], i, 1, n) → Q1
  32. Sequence(Polygon[Element[PontD, i], Element[PontC, i], Element[PontC, i + 1], Element[PontD, i + 1]], i, 1, n) → Q2
  33. Sequence(Polygon[Element[PontF, i], Element[PontE, i], Element[PontE, i + 1], Element[PontF, i + 1]], i, 1, n) → Q4
  34. Sequence(Polygon[Element[PontF, i], Element[PontF, i + 1], Mt], i, 1, n) → QC
  35. Center(kör2) → K2
  36. Center(kör3) → K3
  37. Center(kör4) → K4
  38. Center(kör5) → K5
  39. OrthogonalLine(K2, kör2) → f2
  40. OrthogonalLine(K3, kör3) → f3
  41. OrthogonalLine(K4, kör4) → f4
  42. OrthogonalLine(K5, kör5) → f5
  43. Rotate(Rotate[Rotate[Rotate[Rotate[Mt, (s5 * α), f5], (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → M
  44. Rotate(Rotate[Rotate[Rotate[Rotate[kör5, (s5 * α), f5], (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → Fkör5
  45. Rotate(Rotate[Rotate[Rotate[kör4, (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → Fkör4
  46. Rotate(Rotate[Rotate[kör3, (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → Fkör3
  47. Rotate(Rotate[kör2, (s2 * α), f2], (s1 * α), f1) → Fkör2
  48. Rotate(Q1, (s1 * α), f1) → FQ1
  49. Rotate(Rotate[Rotate[Rotate[Rotate[QC, (s5 * α), f5], (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → FQC
  50. Rotate(Rotate[Rotate[Rotate[Q4, (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → FQ4
  51. Rotate(Rotate[Q2, (s2 * α), f2], (s1 * α), f1) → FQ2
  52. Rotate(Rotate[Rotate[Rotate[Rotate[Lap5, (s5 * α), f5], (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → FLAP5
  53. Rotate(Rotate[Rotate[Rotate[Lap4, (s4 * α), f4], (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → FLAP4
  54. Rotate(Rotate[Rotate[Lap3, (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → FLAP3
  55. Rotate(Rotate[Lap2, (s2 * α), f2], (s1 * α), f1) → FLAP2
  56. If(Cs, Locus[M, α], NaN) → vm
  57. Sequence(Polygon[Element[PontE, i], Element[PontD, i], Element[PontD, i + 1], Element[PontE, i + 1]], i, 1, n) → Q3
  58. Rotate(Rotate[Rotate[Q3, (s3 * α), f3], (s2 * α), f2], (s1 * α), f1) → FQ3
  59. If(n ≤ 12, (360°) / n, 5°) → dα
  60. Polygon(G, F, H) → t1, h, g, f
  61. Polygon(K, J, I, L) → q1, k, j, i, l
  62. Polygon(N, O, P, Q) → q2, n_1, o, p, q
  63. Polygon(R, S, T, U) → q3, r_1, s_1, t_1, u
  64. Polygon(V, E_1, D_1, C_1) → q4, v, e_1, d_1, c_1
  65. Polygon(B_1, A_1, Z, W) → q5, b_1, a_1, z_1, w

AI Drawing Prompt Reference

Create an art helix with a series of ellipses and circles, gradually building it through rotation and scaling. Start by drawing a line, then generate a sequence of points to form the helix's base. Next, draw a series of circles and ellipses, and use rotation and scaling to make them rotate around a central point and gradually increase in size. Finally, use these circles and ellipses to create polygons, forming the helix's appearance.
The prompt above was compiled from the original drawing commands and is for reference only.