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Hyperboloid of Two Sheets

Hyperboloid of Two Sheets
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2023-03-15 20:36
Solid GeometryOptimizationParametric Equation
In collections geometriadinamica.es
This example demonstrates the 3D geometric shape of a hyperboloid of two sheets and its properties. The core mathematical concepts involved include solid geometry, parametric equations, inequalities, and 3D curve plotting. The construction highlights the use of GeoGebra commands and objects such as Segment, Point, OrthogonalLine, Circle, Intersect, Ellipse, Angle, CurveCartesian, and Sequence as outlined in the construction summary. It is suitable for high school and above students studying solid geometry, calculus, and parametric equations, as well as for teaching demonstrations and geometric construction practice.

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, yAxis) → zz1
  12. Circle(C, 0.62) → zz0
  13. Intersect(zz0, zz1, 1) → ZZ1
  14. Intersect(zz0, zz1, 2) → ZZ2
  15. Ellipse(ZZ1, ZZ2, 0.65) → zElipse1
  16. Intersect(zElipse1, zz1) → ZZ5, ZZ6
  17. Point(zElipse1) → XY
  18. Angle(ZZ6, C, XY) → zAngRot
  19. OrthogonalLine(C, xAxis) → zz2
  20. Intersect(zz0, zz2, 2) → ZZ3
  21. Intersect(zz0, zz2, 1) → ZZ4
  22. Ellipse(ZZ3, ZZ4, 0.65) → zElipse2
  23. Intersect(zElipse2, zz2) → ZZ7, ZZ8
  24. Point(zElipse2) → Z
  25. Angle(ZZ8, C, Z) → zAngInc
  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 * ((((a * sqrt(u^(2) - 1)) * cos(α)) * sin(zAngRot)) + (((b * sqrt(u^(2) - 1)) * sin(α)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(u^(2) - 1)) * cos(α)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(u^(2) - 1)) * sin(α)) * sin(zAngRot)) * sin(zAngInc)) + ((c * u) * cos(zAngInc)))), u, (-u2), u2) → curva2
  32. Sequence(CurveCartesian[(r * ((((a * sqrt(u^(2) - 1)) * cos(v)) * sin(zAngRot)) + (((b * sqrt(u^(2) - 1)) * sin(v)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(u^(2) - 1)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(u^(2) - 1)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((c * u) * cos(zAngInc)))), v, v1, v2], u, u1, u2, paso) → list1A
  33. Sequence(CurveCartesian[(r * ((((a * sqrt(u^(2) - 1)) * cos(v)) * sin(zAngRot)) + (((b * sqrt(u^(2) - 1)) * sin(v)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(u^(2) - 1)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(u^(2) - 1)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((c * u) * cos(zAngInc)))), v, v1, v2], u, (-u2), (-u1), paso) → list2A
  34. CurveCartesian((r * ((((a * sqrt(1.5^(2) - 1)) * cos(v)) * sin(zAngRot)) + (((b * sqrt(1.5^(2) - 1)) * sin(v)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(1.5^(2) - 1)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(1.5^(2) - 1)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((c * 1.5) * cos(zAngInc)))), v, v1, v2) → curva1
  35. CurveCartesian((r * ((((a * sqrt(β^(2) - 1)) * cos(v)) * sin(zAngRot)) + (((b * sqrt(β^(2) - 1)) * sin(v)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(β^(2) - 1)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(β^(2) - 1)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((c * β) * cos(zAngInc)))), v, v1, v2) → a1
  36. Sequence(CurveCartesian[(r * ((((a * sqrt(u^(2) - 1)) * cos(v)) * sin(zAngRot)) + (((b * sqrt(u^(2) - 1)) * sin(v)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(u^(2) - 1)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(u^(2) - 1)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((c * u) * cos(zAngInc)))), u, (-u2), (-u1)], v, v1, v2, paso) → list2
  37. Sequence(CurveCartesian[(r * ((((a * sqrt(u^(2) - 1)) * cos(v)) * sin(zAngRot)) + (((b * sqrt(u^(2) - 1)) * sin(v)) * cos(zAngRot)))), (r * ((((((-a) * sqrt(u^(2) - 1)) * cos(v)) * cos(zAngRot)) * sin(zAngInc)) + ((((b * sqrt(u^(2) - 1)) * sin(v)) * sin(zAngRot)) * sin(zAngInc)) + ((c * u) * cos(zAngInc)))), u, u1, u2], v, v1, v2, paso) → list1

AI Drawing Prompt Reference

Draw a hyperboloid of two sheets, including two elliptical rings, two cone bodies, two cone bodies' intersection points, two cone bodies' intersection points' line segments, two cone bodies' intersection points' line segments' intersection points, and a hyperbola. The hyperbola's parametric equation is determined by parameters r, a, b, c, u, v, α, β, zAngRot, zAngInc, etc.
The prompt above was compiled from the original drawing commands and is for reference only.