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Sphere Transformation

Sphere Transformation
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2023-03-15 20:08
Coordinate GeometrySolid Geometry
In collections geometriadinamica.es
This example demonstrates the transformation of spheres. Key mathematical concepts include geometric transformations and coordinate geometry. Using GeoGebra commands such as Segment, Point, Circle, and Intersect, a series of sphere transformation graphics are constructed. Suitable for high school mathematics teaching and learning of solid geometry transformations.

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, yAxis) → zz1
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz1, 1) → ZZ1
  14. Intersect(zz0, zz1, 2) → ZZ2
  15. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  16. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  17. Point(zElipse1) → XY
  18. Angle(ZZ6, C, XY) → zAngRot
  19. OrthogonalLine(C, xAxis) → zz2
  20. Intersect(zz0, zz2, 2) → ZZ3
  21. Intersect(zz0, zz2, 1) → ZZ4
  22. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  23. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  24. Point(zElipse2) → Z
  25. Angle(ZZ8, C, Z) → zAngInc
  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 * ((((f(u) * sin(u)) * cos(α)) * sin(zAngRot)) + (((f(u) * sin(u)) * sin(α)) * cos(zAngRot)))), (r * ((((((-f(u)) * sin(u)) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + ((((f(u) * sin(u)) * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((f(u) * cos(u)) * cos(zAngInc)))), u, u1, u2) → curva2
  32. CurveCartesian((r * (((f(3.14159 / 2) * cos(v)) * sin(zAngRot)) + ((f(3.14159 / 2) * sin(v)) * cos(zAngRot)))), (r * (((((-f(3.14159 / 2)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((f(3.14159 / 2) * sin(v)) * sin(zAngRot)) * sin(zAngInc)))), v, v1, v2) → curva1
  33. CurveCartesian((r * ((((f(β) * sin(β)) * cos(v)) * sin(zAngRot)) + (((f(β) * sin(β)) * sin(v)) * cos(zAngRot)))), (r * ((((((-f(β)) * sin(β)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((f(β) * sin(β)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((f(β) * cos(β)) * cos(zAngInc)))), v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r * ((((f(u) * sin(u)) * cos(v)) * sin(zAngRot)) + (((f(u) * sin(u)) * sin(v)) * cos(zAngRot)))), (r * ((((((-f(u)) * sin(u)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((f(u) * sin(u)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((f(u) * cos(u)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r * ((((f(u) * sin(u)) * cos(v)) * sin(zAngRot)) + (((f(u) * sin(u)) * sin(v)) * cos(zAngRot)))), (r * ((((((-f(u)) * sin(u)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((f(u) * sin(u)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((f(u) * cos(u)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Create a scene with a center point C. Draw a circle centered at C with radius 0.62. Through C, construct two perpendicular reference lines aligned with the x-axis and y-axis, and intersect them with the circle to obtain two sets of intersection points. Use these points to form two ellipses (with the ellipses determined by the corresponding intersection pairs and a fixed shape parameter of 0.65). Find the intersection points between the two ellipses at the same specified choices and connect some of them by line segments. Construct key angles using C and selected ellipse intersection points to define a rotation and an incremental direction. Finally, plot parametrized Cartesian curves that represent the transformed sphere trajectories for two parameter regions, and generate a sequence/path set by stepping through the parameters to visualize the transformation behavior.
The prompt above was compiled from the original drawing commands and is for reference only.