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One-sheet Hyperboloid

One-sheet Hyperboloid
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2023-03-15 20:29
Coordinate GeometrySolid GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the construction of a one-sheet hyperboloid, an important surface in solid geometry. Using GeoGebra commands such as Segment, Point, and OrthogonalLine, the generation process of the hyperboloid is visually illustrated. Suitable for high school mathematics education and students learning 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. 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 * (((sqrt(1 + v^(2)) * cos(α)) * sin(zAngRot)) + ((sqrt(1 + v^(2)) * sin(α)) * cos(zAngRot)))), (r / 2 * (((((-sqrt(1 + v^(2))) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + (((sqrt(1 + v^(2)) * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + (v * cos(zAngInc)))), v, v1, v2) → curva2
  32. Sequence(CurveCartesian[(r / 2 * (((sqrt(1 + v^(2)) * cos(u)) * sin(zAngRot)) + ((sqrt(1 + v^(2)) * sin(u)) * cos(zAngRot)))), (r / 2 * (((((-sqrt(1 + v^(2))) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((sqrt(1 + v^(2)) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (v * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list2
  33. CurveCartesian((r / 2 * ((cos(u) * sin(zAngRot)) + (sin(u) * cos(zAngRot)))), (r / 2 * ((((-cos(u)) * cos(zAngRot)) * sin(zAngInc)) + ((sin(u) * sin(zAngRot)) * sin(zAngInc)))), u, u1, u2) → curva1A
  34. CurveCartesian((r / 2 * (((sqrt(1 + β^(2)) * cos(u)) * sin(zAngRot)) + ((sqrt(1 + β^(2)) * sin(u)) * cos(zAngRot)))), (r / 2 * (((((-sqrt(1 + β^(2))) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((sqrt(1 + β^(2)) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (β * cos(zAngInc)))), u, u1, u2) → a1
  35. Sequence(CurveCartesian[(r / 2 * (((sqrt(1 + v^(2)) * cos(u)) * sin(zAngRot)) + ((sqrt(1 + v^(2)) * sin(u)) * cos(zAngRot)))), (r / 2 * (((((-sqrt(1 + v^(2))) * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((sqrt(1 + v^(2)) * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + (v * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

To draw a one-sheet hyperboloid. First, create a circle and two lines perpendicular to the coordinate axes. The circle intersects the lines at two points, which serve as foci for two ellipses. The ellipses intersect the lines to form intersection points and angles. Then, use parametric curves based on these angles and ellipse parameters to generate the hyperboloid surface. The resulting curves form a continuous, symmetric three-dimensional shape suitable for solid geometry.
The prompt above was compiled from the original drawing commands and is for reference only.