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Helicoidal Torus

Helicoidal Torus
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2023-03-15 21:37
Coordinate GeometrySolid GeometryParametric Equation
In collections geometriadinamica.es
This example demonstrates the geometric construction of a helicoidal torus. It involves the properties of circles and parametric equations. Using GeoGebra commands such as Segment, Point, Circle, Intersect, etc., it dynamically shows the changes in the torus as it spirals upwards. Suitable for high school mathematics education, particularly in coordinate geometry and parametric equations.

GeoGebra Commands & Parameters

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

AI Drawing Prompt Reference

Draw a helicoidal torus surface rising spirally upward in three-dimensional space. Start with a base circle and construct two orthogonal lines through its center perpendicular to the x-axis and y-axis respectively. Create two elliptical cross-sections on the base circle, lying in perpendicular planes and intersecting the orthogonal lines. Use parametric equations to draw spiral curves that ascend along the central axis, with parameters controlling angular rotation and radius variation, creating a ribbon-like band that winds upward in a helical pattern. The surface features elliptical cross-sections with continuously changing angles as it spirals around the central axis.
The prompt above was compiled from the original drawing commands and is for reference only.