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Rotation of an Ellipse in Space

Rotation of an Ellipse in Space
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2020-07-08 15:38
VectorSequenceSolid GeometryTransformationParametric Equation
① This example demonstrates how to rotate an ellipse in 3D space and visualize the resulting surface using parametric equations. ② Core mathematical concepts include the parametric representation of an ellipse, 3D rotation transformations, vectors, and surface parameterization, along with sequences to generate multiple rotated copies. ③ Construction details: using GeoGebra commands such as Sequence, Vector, Rotate, CurveCartesian, Surface, and Element, with sliders controlling the rotation angle, constructing the ellipse curve, rotation axis vectors, tangent and normal lines, and generating multiple rotated ellipses and surfaces with color coding. ④ Suitable for high school or university courses in analytic geometry, solid geometry, or computer-assisted mathematics, helping students intuitively understand spatial rotations and parametric surfaces.

GeoGebra Commands & Parameters

  1. Sequence(Vector[(cos((137.5° * t)), sin((137.5° * t)))], t, 0, 26) → l1
  2. Vector((sin(Ang), 0, cos(Ang))) → u
  3. Rotate(u, Change, zAxis) → u'
  4. CurveCartesian(((a * cos(t)) * l1(1)) + ((b * sin(t)) * u'), t, 0, (2 * π)) → c
  5. Vector(A) → v
  6. Surface(c(t) + (h * v), t, 0, (2 * π), h, (-e), e) → d
  7. Sequence(Vector[Rotate[u', (137.5° * t), zAxis]], t, 0, 26) → l2
  8. Sequence(Rotate[d, (137.5° * t), zAxis], t, 0, n) → l3
  9. Element(l3, A1) → B1
  10. Element(l3, A2) → B2
  11. Element(l3, A3) → B3
  12. Element(l3, A4) → B4
  13. Element(l3, A5) → B5
  14. Element(l3, A6) → B6
  15. Element(l3, A7) → B7
  16. Element(l3, A8) → B8
  17. Element(l3, A9) → B9
  18. Element(l3, A10) → B10
  19. Element(l3, A11) → B11
  20. Element(l3, A12) → B12
  21. Element(l3, A13) → B13
  22. Element(l3, A14) → B14
  23. Element(l3, A15) → B15
  24. Element(l3, A16) → B16
  25. Element(l3, A17) → B17
  26. Element(l3, A18) → B18
  27. Element(l3, A19) → B19
  28. Element(l3, A20) → B20
  29. Element(l3, A21) → B21
  30. Element(l3, A22) → B22
  31. Element(l3, A23) → B23
  32. Element(l3, A24) → B24
  33. Element(l3, A25) → B25
  34. Element(l3, A26) → B26
  35. Element(l3, A27) → B27
  36. Element(l3, A28) → B28
  37. Element(l3, A29) → B29
  38. Element(l3, A30) → B30
  39. Element(l3, A31) → B31
  40. Element(l3, A32) → B32
  41. Element(l3, A33) → B33
  42. Element(l3, A34) → B34
  43. Element(l3, A35) → B35
  44. Element(l3, A36) → B36
  45. Element(l3, A37) → B37
  46. Element(l3, A38) → B38
  47. Element(l3, A39) → B39
  48. Element(l3, A40) → B40

AI Drawing Prompt Reference

Draw an ellipse in 3D space defined by parametric equations, with its shape controlled by two perpendicular vectors, one of which rotates around the vertical axis. Translate this ellipse along a direction vector to create a surface. Then rotate the entire surface around the vertical axis in steps of about 137.5 degrees, repeating this rotation 27 times to generate multiple colored copies, forming a symmetric 3D arrangement of elliptical rings that illustrates the rotation of the ellipse in space.
The prompt above was compiled from the original drawing commands and is for reference only.