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Nested Spheres

Nested Spheres
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2023-03-14 04:31
CircleSolid GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates a nested spheres graph, showcasing the geometric properties of spheres. Key mathematical concepts include geometric shapes, conic sections, and the relationships between spheres. Construction points include using GeoGebra commands Segment, Point, OrthogonalLine, Circle, Intersect, and Ellipse. Suitable for teaching middle school mathematics and geometry.

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 / 10 * ((((u * cos(cos(u))) * cos(α)) * sin(zAngRot)) + (((u * cos(cos(u))) * sin(α)) * cos(zAngRot)))), (r / 10 * ((((((-u) * cos(cos(u))) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + ((((u * cos(cos(u))) * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((u * sin(cos(u))) * cos(zAngInc)))), u, u1, u2) → curva2
  32. CurveCartesian((r / 10 * ((((β * cos(cos(β))) * cos(v)) * sin(zAngRot)) + (((β * cos(cos(β))) * sin(v)) * cos(zAngRot)))), (r / 10 * ((((((-β) * cos(cos(β))) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((β * cos(cos(β))) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((β * sin(cos(β))) * cos(zAngInc)))), v, v1, v2) → a1
  33. CurveCartesian((r * (((cos(cos(10)) * cos(v)) * sin(zAngRot)) + ((cos(cos(10)) * sin(v)) * cos(zAngRot)))), (r * (((((-cos(cos(10))) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((cos(cos(10)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + (sin(cos(10)) * cos(zAngInc)))), v, v1, v2) → curva1
  34. Sequence(CurveCartesian[(r / 10 * ((((u * cos(cos(u))) * cos(v)) * sin(zAngRot)) + (((u * cos(cos(u))) * sin(v)) * cos(zAngRot)))), (r / 10 * ((((((-u) * cos(cos(u))) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((u * cos(cos(u))) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((u * sin(cos(u))) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r / 10 * ((((u * cos(cos(u))) * cos(v)) * sin(zAngRot)) + (((u * cos(cos(u))) * sin(v)) * cos(zAngRot)))), (r / 10 * ((((((-u) * cos(cos(u))) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((u * cos(cos(u))) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((u * sin(cos(u))) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Create a “nested spheres” geometry illustration: use the same reference center C and first draw two perpendicular axes through C. Construct two circles centered at C and use their intersections with the corresponding perpendicular direction lines to obtain two key point sets. From each point set, form an ellipse, then mark important points from the ellipse intersections with the respective axis directions and connect selected key points with short segments. Finally, generate two families of parametric curves whose placement is governed by the established angular relations (rotation and inclination) so the two layers interweave across the ellipses’ implied cross-sections, conveying a nested-spheres effect.
The prompt above was compiled from the original drawing commands and is for reference only.