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Revolution Paraboloid

Revolution Paraboloid
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2023-03-15 21:01
Coordinate GeometrySolid GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the generation process of a revolution paraboloid. It involves key mathematical concepts such as the rotation of a parabola, coordinate geometry, and solid geometry. Using GeoGebra commands like Segment, Point, OrthogonalLine, Circle, Intersect, etc., a geometric model of the revolution paraboloid can be constructed. This example is suitable for high school mathematics education, particularly in coordinate geometry and solid geometry.

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, xAxis) → zz2
  10. Circle(C, 0.62) → zz0
  11. Intersect(zz0, zz2, 1) → ZZ4
  12. Intersect(zz0, zz2, 2) → ZZ3
  13. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  14. Point(zElipse2) → Z
  15. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  16. Angle(ZZ8, C, Z) → zAngInc
  17. OrthogonalLine(C, yAxis) → zz1
  18. Intersect(zz0, zz1, 2) → ZZ2
  19. Intersect(zz0, zz1, 1) → ZZ1
  20. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  21. Point(zElipse1) → XY
  22. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  23. Angle(ZZ6, C, XY) → zAngRot
  24. Segment(Q5, Q6) → p5
  25. Point(p5) → P5
  26. CurveCartesian((r * (((0.75 * cos(u)) * sin(zAngRot)) + ((0.75 * sin(u)) * cos(zAngRot)))), (r * ((((-0.75 * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((0.75 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((0.75^(2) - 0.6) * cos(zAngInc)))), u, u1, u2) → curva1
  27. Segment(ZZ5, ZZ6) → zz4
  28. Segment(ZZ7, ZZ8) → zz3
  29. Intersect(zElipse2, zElipse1, 1) → zPunto1
  30. Intersect(zElipse2, zElipse1, 4) → zPunto2
  31. Segment(zPunto1, zPunto2) → zSeg2
  32. CurveCartesian((r * (((v * cos(α)) * sin(zAngRot)) + ((v * sin(α)) * cos(zAngRot)))), (r * (((((-v) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + (((v * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((v^(2) - 0.6) * cos(zAngInc)))), v, v1, v2) → curva2
  33. Sequence(CurveCartesian[(r * (((v * cos(u)) * sin(zAngRot)) + ((v * sin(u)) * cos(zAngRot)))), (r * (((((-v) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((v * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((v^(2) - 0.6) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1A
  34. CurveCartesian((r * (((β * cos(u)) * sin(zAngRot)) + ((β * sin(u)) * cos(zAngRot)))), (r * (((((-β) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((β * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((β^(2) - 0.6) * cos(zAngInc)))), u, u1, u2) → a1
  35. Sequence(CurveCartesian[(r * (((v * cos(u)) * sin(zAngRot)) + ((v * sin(u)) * cos(zAngRot)))), (r * (((((-v) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((v * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((v^(2) - 0.6) * cos(zAngInc)))), v, v1, v2], u, u1, u2, (3 * paso)) → list1

AI Drawing Prompt Reference

Draw a revolution paraboloid to show its generation process. Start by sketching a parabola, then rotate it around the x-axis and y-axis to form the paraboloid. Pay attention to the intersection points and angle relationships between the parabola and the rotation axes, as well as the elliptical cross-sections formed during rotation. Finally, combine all the rotated curves to create a complete model of the revolution paraboloid.
The prompt above was compiled from the original drawing commands and is for reference only.