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Helical Spiral Torus

Helical Spiral Torus
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2023-03-15 21:36
SequenceSolid GeometryParametric EquationLocus
In collections geometriadinamica.es
① This example demonstrates a three-dimensional helical spiral torus, a torus (doughnut shape) with a helical curve winding around its surface, creating a spring-like structure around the ring. ② Core mathematical concepts include parametric curves, three-dimensional helix, torus parametric equations, and generating surfaces or loci from families of curves. ③ Construction: Based on the construction summary, GeoGebra commands such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence are used. Multiple auxiliary points (e.g., P and Q series) and parameters (e.g., 'paso' step) are defined to generate the skeleton and surface of the helical torus via parametric curves and sequences. ④ Suitable for high school to university-level mathematics, useful for understanding 3D parametric equations, helices, torus geometry, and parametric modeling in dynamic geometry software, also as a visualization or artistic design resource.

GeoGebra Commands & Parameters

  1. Segment(Q5, Q6) → p5
  2. Point(p5) → P5
  3. Segment(Q3, Q4) → p2
  4. Point(p2) → P3
  5. Segment(P3, Q4) → p4
  6. Point(p4) → P4
  7. Segment(Q1, Q2) → p1
  8. Point(p1) → P1
  9. Segment(P1, Q2) → p3
  10. Point(p3) → P2
  11. OrthogonalLine(C, xAxis) → zz2
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz2, 1) → ZZ4
  14. Intersect(zz0, zz2, 2) → ZZ3
  15. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  16. Point(zElipse2) → Z
  17. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  18. Angle(ZZ8, C, Z) → zAngInc
  19. OrthogonalLine(C, yAxis) → zz1
  20. Intersect(zz0, zz1, 2) → ZZ2
  21. Intersect(zz0, zz1, 1) → ZZ1
  22. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  23. Point(zElipse1) → XY
  24. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  25. Angle(ZZ6, C, XY) → zAngRot
  26. CurveCartesian(((((r / 40 * 2) * 3.14159) * (a + cos(v))) * sin(zAngRot)), (r / 40 * (((((-2 * 3.14159) * (a + cos(v))) * cos(zAngRot)) * sin(zAngInc)) + ((((2 * 3.14159) * sin(v)) - ((b * 2) * 3.14159)) * cos(zAngInc)))) + 5, v, v1, v2) → curva1
  27. Segment(ZZ5, ZZ6) → zz4
  28. Segment(ZZ7, ZZ8) → zz3
  29. Intersect(zElipse2, zElipse1, 1) → zPunto1
  30. Intersect(zElipse2, zElipse1, 4) → zPunto2
  31. Segment(zPunto1, zPunto2) → zSeg2
  32. CurveCartesian((r / 40 * ((((u * cos(u)) * (a + cos(α))) * sin(zAngRot)) + (((u * sin(u)) * (a + cos(α))) * cos(zAngRot)))), (r / 40 * ((((((-u) * cos(u)) * (a + cos(α))) * cos(zAngRot)) * sin(zAngInc)) + ((((u * sin(u)) * (a + cos(α))) * sin(zAngRot)) * sin(zAngInc)) + (((u * sin(α)) - (b * u)) * cos(zAngInc)))) + 5, u, u1, u2) → curva2
  33. CurveCartesian((r / 40 * (((((2 * β) * cos((2 * β))) * (a + cos(v))) * sin(zAngRot)) + ((((2 * β) * sin((2 * β))) * (a + cos(v))) * cos(zAngRot)))), (r / 40 * ((((((-2 * β) * cos((2 * β))) * (a + cos(v))) * cos(zAngRot)) * sin(zAngInc)) + (((((2 * β) * sin((2 * β))) * (a + cos(v))) * sin(zAngRot)) * sin(zAngInc)) + ((((2 * β) * sin(v)) - ((b * 2) * β)) * cos(zAngInc)))) + 5, v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r / 40 * ((((u * cos(u)) * (a + cos(v))) * sin(zAngRot)) + (((u * sin(u)) * (a + cos(v))) * cos(zAngRot)))), (r / 40 * ((((((-u) * cos(u)) * (a + cos(v))) * cos(zAngRot)) * sin(zAngInc)) + ((((u * sin(u)) * (a + cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((u * sin(v)) - (b * u)) * cos(zAngInc)))) + 5, v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 40 * ((((u * cos(u)) * (a + cos(v))) * sin(zAngRot)) + (((u * sin(u)) * (a + cos(v))) * cos(zAngRot)))), (r / 40 * ((((((-u) * cos(u)) * (a + cos(v))) * cos(zAngRot)) * sin(zAngInc)) + ((((u * sin(u)) * (a + cos(v))) * sin(zAngRot)) * sin(zAngInc)) + (((u * sin(v)) - (b * u)) * cos(zAngInc)))) + 5, u, u1, u2], v, v1, v2, (3 * paso)) → list1

AI Drawing Prompt Reference

Draw a helical spiral torus, resembling a doughnut with a spiral curve winding around its surface, creating a spring-like structure. The spiral is generated by parametric equations and rotates continuously on the torus. First, define the torus center and radius, then create the spiral path using parametric curves, and finally combine the spiral with the torus surface to form a three-dimensional solid.
The prompt above was compiled from the original drawing commands and is for reference only.