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Ellipsoid Construction and Transformations

Ellipsoid Construction and Transformations
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2023-03-15 19:44
Coordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the geometric construction and transformation properties of an ellipse. Key mathematical concepts include the definition of an ellipse, geometric transformations, and coordinate geometry. Construction involves using GeoGebra commands such as Segment, OrthogonalLine, Ellipse, and Angle to create ellipse objects, and analyzing their properties with CurveCartesian. Suitable for high school students learning geometric transformations and coordinate geometry applications, or for mathematical teaching demonstrations.

GeoGebra Commands & Parameters

  1. Segment(Q3, Q4) → p2
  2. Point(p2) → P3
  3. Segment(P3, Q4) → p4
  4. Point(p4) → P4
  5. Segment(Q1, Q2) → p1
  6. Point(p1) → P1
  7. Segment(P1, Q2) → p3
  8. Point(p3) → P2
  9. OrthogonalLine(C, yAxis) → zz1
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz1, 1) → ZZ1
  12. Intersect(zz0, zz1, 2) → ZZ2
  13. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  14. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  15. Point(zElipse1) → XY
  16. Angle(ZZ6, C, XY) → zAngRot
  17. OrthogonalLine(C, xAxis) → zz2
  18. Intersect(zz0, zz2, 2) → ZZ3
  19. Intersect(zz0, zz2, 1) → ZZ4
  20. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  21. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  22. Point(zElipse2) → Z
  23. Angle(ZZ8, C, Z) → zAngInc
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  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 / 2 * ((((b * cos(v)) * cos(α)) * sin(zAngRot)) + (((c * cos(v)) * sin(α)) * cos(zAngRot)))), (r / 2 * ((((((-b) * cos(v)) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + ((((c * cos(v)) * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((d * sin(v)) * cos(zAngInc)))), v, v1, v2) → curva2
  32. Sequence(CurveCartesian[(r / 2 * ((((b * cos(v)) * cos(u)) * sin(zAngRot)) + (((c * cos(v)) * sin(u)) * cos(zAngRot)))), (r / 2 * ((((((-b) * cos(v)) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((c * cos(v)) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((d * sin(v)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list2
  33. CurveCartesian((r / 2 * (((b * cos(u)) * sin(zAngRot)) + ((c * sin(u)) * cos(zAngRot)))), (r / 2 * (((((-b) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((c * sin(u)) * sin(zAngRot)) * sin(zAngInc)))), u, u1, u2) → curva1
  34. CurveCartesian((r / 2 * ((((b * cos(β)) * cos(u)) * sin(zAngRot)) + (((c * cos(β)) * sin(u)) * cos(zAngRot)))), (r / 2 * ((((((-b) * cos(β)) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((c * cos(β)) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((d * sin(β)) * cos(zAngInc)))), u, u1, u2) → a1
  35. Sequence(CurveCartesian[(r / 2 * ((((b * cos(v)) * cos(u)) * sin(zAngRot)) + (((c * cos(v)) * sin(u)) * cos(zAngRot)))), (r / 2 * ((((((-b) * cos(v)) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((((c * cos(v)) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((d * sin(v)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, (4 * paso)) → list1

AI Drawing Prompt Reference

Draw an ellipsoid by first constructing line segments through points Q1, Q2, Q3, and Q4 and taking midpoints. Then, use these points to construct an ellipse and further ellipses by their intersections with coordinate axes. Finally, use parametric equations to draw the ellipsoid's surface and demonstrate its transformation properties through rotation and translation.
The prompt above was compiled from the original drawing commands and is for reference only.