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Pseudosphere

Pseudosphere
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2023-03-15 21:21
CircleCoordinate GeometryGeometric Construction
In collections geometriadinamica.es
This example demonstrates the construction of a pseudosphere, a surface in hyperbolic geometry with negative curvature, similar to a sphere. Using GeoGebra commands such as Segment, Point, and OrthogonalLine, dynamic representations of points, lines, and curves on the pseudosphere are shown. Suitable for high school mathematics education, particularly in coordinate geometry and hyperbolic 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(α) / cosh(u) * sin(zAngRot)) + (sin(α) / cosh(u) * cos(zAngRot)))), (r * (((((-cos(α))) / cosh(u) * cos(zAngRot)) * sin(zAngInc)) + ((sin(α) / cosh(u) * sin(zAngRot)) * sin(zAngInc)) + ((u - tanh(u)) * cos(zAngInc)))), u, u1, u2) → curva2
  32. Sequence(CurveCartesian[(r * ((cos(v) / cosh(u) * sin(zAngRot)) + (sin(v) / cosh(u) * cos(zAngRot)))), (r * (((((-cos(v))) / cosh(u) * cos(zAngRot)) * sin(zAngInc)) + ((sin(v) / cosh(u) * sin(zAngRot)) * sin(zAngInc)) + ((u - tanh(u)) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1A
  33. CurveCartesian((r * ((cos(v) / cosh(β) * sin(zAngRot)) + (sin(v) / cosh(β) * cos(zAngRot)))), (r * (((((-cos(v))) / cosh(β) * cos(zAngRot)) * sin(zAngInc)) + ((sin(v) / cosh(β) * sin(zAngRot)) * sin(zAngInc)) + ((β - tanh(β)) * cos(zAngInc)))), v, v1, v2) → a1
  34. CurveCartesian((r * ((cos(v) * sin(zAngRot)) + (sin(v) * cos(zAngRot)))), (r * ((((-cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((sin(v) * sin(zAngRot)) * sin(zAngInc)))), v, v1, v2) → curva1A
  35. Sequence(CurveCartesian[(r * ((cos(v) / cosh(u) * sin(zAngRot)) + (sin(v) / cosh(u) * cos(zAngRot)))), (r * (((((-cos(v))) / cosh(u) * cos(zAngRot)) * sin(zAngInc)) + ((sin(v) / cosh(u) * sin(zAngRot)) * sin(zAngInc)) + ((u - tanh(u)) * cos(zAngInc)))), u, u1, u2], v, v1, v2, (2 * paso)) → list1

AI Drawing Prompt Reference

Draw a pseudosphere, a surface with negative curvature. Start with a center point and a small circle around it. Construct two perpendicular diameters through the center. On each diameter, mark two endpoints)Skip, and use each pair of endpoints as foci to draw two ellipses that intersect. Place a moving point on each ellipse and measure their angles relative to the center. Use these angles as rotation and tilt parameters to generate a family of parametric curves that weave together in space, forming a grid-like surface. Finally, add a boundary curve and a central curve on the surface to complete the shape.
The prompt above was compiled from the original drawing commands and is for reference only.