MathPlotAI Logo MathPlotAI

Intersection of a Hyperboloid and a Cylinder

Intersection of a Hyperboloid and a Cylinder
Loading the math board and drawing, please wait…
2021-05-01 21:34
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates the intersection of a hyperboloid and a cylinder. Key mathematical concepts include solid geometry and coordinate geometry. Using commands like Surface, CurveCartesian, and Rotate in GeoGebra, the intersection can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Surface((a * cos(v)) - ((a * u) * sin(v)), (a * sin(v)) + ((a * u) * cos(v)), (c * u), u, (-b), t, v, 0°, β) → g
  2. CurveCartesian((a * sqrt(1 + u^(2))), ((a * sqrt(1 + u^(2))) * 0), (c * u), u, (-b), t) → i
  3. CurveCartesian(a - ((a * u) * 0), (a * 0) + (a * u), (c * u), u, (-b), t) → j
  4. OrthogonalLine(O, xOyPlane) → h
  5. Sequence(Rotate[j, (i * 30°), zAxis], i, 0, β / (30°), 1) → l2
  6. Sequence(Rotate[i, (k * 30°), zAxis], k, 0, α / (30°), 1) → l3
  7. CurveCartesian(((a * sqrt(1 + t^(2))) * cos(v)), ((a * sqrt(1 + t^(2))) * sin(v)), (c * t), v, 0°, α) → k
  8. CurveCartesian((a * cos(β)) - ((a * u) * sin(β)), (a * sin(β)) + ((a * u) * cos(β)), (c * u), u, (-b), t) → e
  9. Surface(((a * sqrt(1 + u^(2))) * cos(v)), ((a * sqrt(1 + u^(2))) * sin(v)), (c * u), u, (-b), t, v, 0°, α) → d
  10. CurveCartesian(((a * sqrt(1 + b^(2))) * cos(v)), ((a * sqrt(1 + b^(2))) * sin(v)), (c * (-b)), v, 0°, α) → l
  11. Sequence(CurveCartesian[(a * cos((i * 30°))) - ((a * u) * sin((i * 30°))), ((-a) * sin((i * 30°))) - ((a * u) * cos((i * 30°))), (c * u), u, (-b), t], i, 0, β / (30°), 1) → l1
  12. CurveCartesian(((a * sqrt(1 + u^(2))) * cos(α)), ((a * sqrt(1 + u^(2))) * sin(α)), (c * u), u, (-b), t) → f
  13. Circle(O, a, xOyPlane) → p
  14. CurveCartesian((((-a) * u) * 0), (a * u), (c * u), u, (-b), t) → a_1
  15. CurveCartesian((((-a) * u) * sin(β)), ((a * u) * cos(β)), (c * u), u, (-b), t) → w
  16. Rotate(a_1, 90°, h) → b_1
  17. Rotate(b_1, α, h) → c_1
  18. Surface((((-a) * u) * sin(v)), ((a * u) * cos(v)), (c * u), u, (-b), t, v, 0°, 360°) → g_1
  19. CurveCartesian((((-a) * t) * sin(v)), ((a * t) * cos(v)), (c * t), v, 0°, 360°) → h_1
  20. CurveCartesian((((-a) * (-b)) * sin(v)), ((a * (-b)) * cos(v)), (c * (-b)), v, 0°, 360°) → i_1
  21. Point(xAxis) → A
  22. OrthogonalPlane(A, xAxis) → p_1
  23. Intersect(yAxis, p, 2) → B
  24. Line(B, xAxis) → k_1
  25. Intersect(k_1, p_1) → C
  26. Line(C, A) → l_1
  27. Point(l_1) → D
  28. OrthogonalLine(D, p_1) → m_1
  29. Circle(m_1, C) → q_1
  30. Cylinder(q_1, -37.694964153002026) → n_1, r_1, o_1
  31. Point(k) → E
  32. Line(E, h) → s_1
  33. Intersect(s_1, xOyPlane) → F
  34. Vector(F, E) → u_1
  35. Translate(B, u_1) → B'
  36. Circle(h, B') → t_1
  37. Tangent(E, t_1) → f_2, g_2
  38. Circle(h, E) → c_2
  39. Intersect(f_2, c_2, 1) → G
  40. Vector(E, F) → v_1
  41. Translate(G, v_1) → G'
  42. Translate(G', v_1) → G''
  43. Segment(E, G'') → h_2
  44. Intersect(h_2, o_1) → H, I
  45. Locus(H, E) → ponthalmaz1
  46. Locus(I, E) → ponthalmaz2

AI Drawing Prompt Reference

Construct the intersection of a hyperboloid with a cylinder in 3D. Set a spatial reference with a chosen center and axes, then build a base parametric curve on the hyperboloid and obtain the hyperboloid surface through rotational symmetry. Construct the cylinder surface using a circular cross-section and the cylinder axis, then find and display the curve formed by intersecting the two surfaces. Select points on the intersection, connect them with a line, and use circle and tangent relationships on the relevant circular cross-sections. Finally, translate the key intersection-derived points along a determined direction and draw the resulting segment and two loci (point sets) defined with respect to the translated/anchor points.
The prompt above was compiled from the original drawing commands and is for reference only.