MathPlotAI Logo MathPlotAI

Rotated Ellipsoid Construction Example

Rotated Ellipsoid Construction Example
Loading the math board and drawing, please wait…
2023-03-15 20:03
Coordinate GeometrySolid GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates constructing a rotated ellipsoid using GeoGebra. Key mathematical concepts include geometric transformations, conic sections, and solid geometry. Construction involves commands like Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, and Sequence. Suitable for high school students or college undergraduates learning geometric construction and coordinate 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 * (((cos(u) * cos(α)) * sin(zAngRot)) + ((sin(u) * cos(α)) * cos(zAngRot)))), (r * (((((-cos(u)) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * cos(α)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(α) + (0.7 * u)) * cos(zAngInc)))), u, u1, u2) → curva2
  32. CurveCartesian(((r * cos(v)) * sin(zAngRot)), (r * ((((-cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (sin(v) * cos(zAngInc)))), v, v1, v2) → curva1
  33. CurveCartesian((r * (((cos(β) * cos(v)) * sin(zAngRot)) + ((sin(β) * cos(v)) * cos(zAngRot)))), (r * (((((-cos(β)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(β) * cos(v)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(v) + (0.7 * β)) * cos(zAngInc)))), v, v1, v2) → a1
  34. Sequence(CurveCartesian[(r * (((cos(u) * cos(v)) * sin(zAngRot)) + ((sin(u) * cos(v)) * cos(zAngRot)))), (r * (((((-cos(u)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * cos(v)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(v) + (0.7 * u)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list2
  35. Sequence(CurveCartesian[(r * (((cos(u) * cos(v)) * sin(zAngRot)) + ((sin(u) * cos(v)) * cos(zAngRot)))), (r * (((((-cos(u)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + (((sin(u) * cos(v)) * sin(zAngRot)) * sin(zAngInc)) + ((sin(v) + (0.7 * u)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Draw a rotated ellipsoid centered at the origin. Construct two perpendicular axes, one along the x-axis and one along the y-axis, with an ellipse placed on each axis, its major axis aligned with the corresponding axis. The two ellipses intersect at two points, connected by a segment. Then generate a family of surface curves using parametric equations, forming a grid in two directions, and add a helical auxiliary curve to show the rotation. The final figure consists of these curves and intersection points, illustrating the 3D rotated ellipsoid shape.
The prompt above was compiled from the original drawing commands and is for reference only.