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Gabriel Horn

Gabriel Horn
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2023-03-15 21:39
Coordinate GeometrySolid GeometryParametric EquationGeometric ConstructionGraphing
In collections geometriadinamica.es
This example demonstrates the Gabriel Horn, a fascinating three-dimensional geometric shape formed by the intersection of a paraboloid and a cylindrical surface. Key math concepts include solid geometry, coordinate geometry, rotational solids, parametric equations, curves, and surfaces. Construction highlights involve commands like Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence from GeoGebra, emphasizing parametric representation and 3D visualization. Suitable for high school and advanced students studying 3D geometry, coordinate geometry, and parametric equations, as well as mathematical modeling and visualization education.

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 / 4 * (((1 / v * cos(α)) * sin(zAngRot)) + ((1 / v * sin(α)) * cos(zAngRot)))), (r / 4 * ((((-1 / v * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + (((1 / v * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((v + 4) * cos(zAngInc)))), v, v1, v2) → curva2
  32. Sequence(CurveCartesian[(r / 4 * (((1 / v * cos(u)) * sin(zAngRot)) + ((1 / v * sin(u)) * cos(zAngRot)))), (r / 4 * ((((-1 / v * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((1 / v * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((v + 4) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list2
  33. CurveCartesian((r / 4 * (((1 / -1 * cos(u)) * sin(zAngRot)) + ((1 / -1 * sin(u)) * cos(zAngRot)))), (r / 4 * ((((-1 / -1 * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((1 / -1 * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((-1 + 4) * cos(zAngInc)))), u, u1, u2) → curva1
  34. CurveCartesian((r / 4 * (((1 / β * cos(u)) * sin(zAngRot)) + ((1 / β * sin(u)) * cos(zAngRot)))), (r / 4 * ((((-1 / β * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((1 / β * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((β + 4) * cos(zAngInc)))), u, u1, u2) → a1
  35. Sequence(CurveCartesian[(r / 4 * (((1 / v * cos(u)) * sin(zAngRot)) + ((1 / v * sin(u)) * cos(zAngRot)))), (r / 4 * ((((-1 / v * cos(u)) * cos(zAngRot)) * sin(zAngInc)) + (((1 / v * sin(u)) * sin(zAngRot)) * sin(zAngInc)) + ((v + 4) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1

AI Drawing Prompt Reference

Draw a three-dimensional Gabriel's Horn shape formed by the intersection of a paraboloid and a cylindrical surface. Start by constructing two perpendicular reference segments with points on them, then create auxiliary circles and ellipses. Use the intersection points of ellipses with perpendicular lines to determine rotation angles. Finally, generate a series of parametric curves arranged in a grid pattern to form the horn-like surface, showing a symmetric rotational solid with clear outlines and internal structure.
The prompt above was compiled from the original drawing commands and is for reference only.