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Torsion Sphere

Torsion Sphere
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2023-03-15 20:05
TrigonometrySolid GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the graph and mathematical concepts of a torsion sphere. It involves the application of geometry and algebra, showing how to use GeoGebra to draw and analyze the properties of a torsion sphere. Key mathematical concepts include geometry, algebra, and trigonometry. Construction points include the use of Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, and CurveCartesian GeoGebra commands and objects. Suitable for advanced math courses and research fields.

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. Segment(ZZ5, ZZ6) → zz4
  27. Segment(ZZ7, ZZ8) → zz3
  28. Intersect(zElipse2, zElipse1, 1) → zPunto1
  29. Intersect(zElipse2, zElipse1, 4) → zPunto2
  30. Segment(zPunto1, zPunto2) → zSeg2
  31. CurveCartesian((r / 3 * (((cos(α) * cos(v)) * sin(zAngRot)) + ((sin(α) * cos(v)) * cos(zAngRot)))), (r / 3 * (((((-cos(α)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(α) * cos(v)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(v) + α) * cos(zAngInc)))) - 4, v, v1, v2) → curva2
  32. CurveCartesian((r / 3 * (((-cos(u)) * sin(zAngRot)) - (sin(u) * cos(zAngRot)))), (r / 3 * (((cos(u) * cos(zAngRot)) * sin(zAngInc)) - ((sin(u) * sin(zAngRot)) * sin(zAngInc)) + (u * cos(zAngInc)))) - 4, u, u1, u2) → curva1
  33. CurveCartesian((r / 3 * (((cos(u) * cos(β)) * sin(zAngRot)) + ((sin(u) * cos(β)) * cos(zAngRot)))), (r / 3 * (((((-cos(u)) * cos(β)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * cos(β)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(β) + u) * cos(zAngInc)))) - 4, u, u1, u2) → a1
  34. Sequence(CurveCartesian[(r / 3 * (((cos(u) * cos(v)) * sin(zAngRot)) + ((sin(u) * cos(v)) * cos(zAngRot)))), (r / 3 * (((((-cos(u)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * cos(v)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(v) + u) * cos(zAngInc)))) - 4, v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 3 * (((cos(u) * cos(v)) * sin(zAngRot)) + ((sin(u) * cos(v)) * cos(zAngRot)))), (r / 3 * (((((-cos(u)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * cos(v)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(v) + u) * cos(zAngInc)))) - 4, u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Create a three-dimensional mathematical visualization of a torsion sphere. Start with a base circle, then draw two perpendicular lines through its center aligned with horizontal and vertical axes to establish reference directions. Using two opposite points on the circle, construct two ellipses with a fixed length, making their planes perpendicular to each other. Find intersection points on both ellipses and measure the angles between these points and the sphere's center as rotation angles. Finally, use parametric equations to draw multiple spatial curves that spiral around the sphere surface, creating a twisted sphere visual effect.
The prompt above was compiled from the original drawing commands and is for reference only.