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Torus Curve Networks

Torus Curve Networks
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2018-04-15 04:21
Coordinate GeometrySolid GeometryGeometric Construction
In collections surface 3D /3D-Oberflächen 1
This example demonstrates the curve networks of a torus. Key mathematical concepts include solid geometry and coordinate geometry. Using GeoGebra commands such as Intersect, Vector, Sequence, Surface, etc., the curve networks of a torus can be constructed. Suitable for high school and above students or teachers.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Vector(A, B) → u
  3. Vector(A, C) → v
  4. Sequence(CurveCartesian[(k / n * v) + (t * u), t, 0, 1], k, 0, n) → LEG_1
  5. Sequence(CurveCartesian[(k / m * u) + (t * v), t, 0, 1], k, 0, m) → LEG_2
  6. Sequence(CurveCartesian[(k / n * v) + ((Min[k, m] * t) * ((1 / m * u) - (1 / n * v))), t, 0, 1], k, 0, n) → LEGD_{12u}
  7. Sequence(CurveCartesian[v + (k / m * u) + ((Min[m - k, n] * t) * ((1 / m * u) - (1 / n * v))), t, 0, 1], k, 1, m) → LEGD_{12o}
  8. Surface(((R + (r * cos(v))) * cos(u)), ((R + (r * cos(v))) * sin(u)), (r * sin(v)), u, 0, (2 * π), v, 0, (2 * π)) → a
  9. CurveCartesian(a((n * t), (m * t)), t, 0, (2 * π)) → b
  10. Sequence(CurveCartesian[a((k / n * v) + (t * u)), t, 0, 1], k, 0, n) → LTG_1
  11. Sequence(CurveCartesian[a((k / m * u) + (t * v)), t, 0, 1], k, 0, m) → LTG_2
  12. Sequence(CurveCartesian[a((k / n * v) + ((Min[k, m] * t) * ((1 / m * u) - (1 / n * v)))), t, 0, 1], k, 0, n) → LTGD_{12o}
  13. Sequence(CurveCartesian[a(v + (k / m * u) + ((Min[m - k, n] * t) * ((1 / m * u) - (1 / n * v)))), t, 0, 1], k, 1, m) → LTGD_{12u}
  14. Intersect(zAxis, yAxis) → D
  15. Point(zAxis) → E
  16. Line(D, E) → f
  17. Sequence(CurveCartesian[(k / m * u) + ((Min[m - k, n] * t) * ((1 / m * u) + (1 / n * v))), t, 0, 1], k, 1, m) → LEGD_{21o}
  18. Sequence(CurveCartesian[a((k / m * u) + ((Min[m - k, n] * t) * ((1 / m * u) + (1 / n * v)))), t, 0, 1], k, 1, m) → LTGD_{21o}
  19. Sequence(CurveCartesian[(k / n * v) + ((Min[m, n - k] * t) * ((1 / m * u) + (1 / n * v))), t, 0, 1], k, 0, n) → LEGD_{21u}
  20. Sequence(CurveCartesian[a((k / n * v) + ((Min[m, n - k] * t) * ((1 / m * u) + (1 / n * v)))), t, 0, 1], k, 0, n) → LTGD_{21u}

AI Drawing Prompt Reference

Draw a torus to demonstrate its curve networks. Start by plotting the intersection points of coordinate axes and creating vectors. Generate two sets of parallel curves using these vectors, then draw two sets of intersecting curves. Finally, create the surface of the torus and plot two sets of parallel and intersecting curves on it, along with two sets of curves extending from the surface.
The prompt above was compiled from the original drawing commands and is for reference only.