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Hyperboloid as a Ruled Surface

Hyperboloid as a Ruled Surface
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2020-02-09 14:46
CirclePolygonSolid GeometryTransformationGeometric Construction
① This applet demonstrates a one-sheet hyperboloid generated by a family of straight lines, visually illustrating its nature as a ruled surface. ② Core mathematical concepts include ruled surfaces, hyperboloids, spatial lines and circles, rotation and mirror transformations, and the construction of polygons and segments in 3D space. ③ Construction highlights: from the summary, the applet uses GeoGebra commands such as Intersect, Circle, Rotate, Polygon, Point, OrthogonalPlane, Mirror, and Segment, with multiple auxiliary objects (A, c, B, d, n, C, D, α, etc.) to create and rotate line segments, then mirror them to form the ruled surface. ④ Suitable for high school to undergraduate courses in solid geometry, analytic geometry, or mathematical modeling, this applet helps students grasp the geometric essence of ruled surfaces and can serve as a visual aid for 3D printing or interactive teaching.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Circle(A, 2) → c
  3. Intersect(xAxis, yAxis) → B
  4. Circle(B, 2) → d
  5. Intersect(c, xAxis, 2) → C
  6. Intersect(d, xAxis, 2) → D
  7. Rotate(D, (360°) / n, B) → D'
  8. Rotate(C, α, A) → C'
  9. Polygon(D, D', n) → poly2, j_1, t_1, a_1, b_1, c_1, d_1, e_1, f_2, g_2, h_2, i_2, k_1, j_2, m_3, p_3, r_3, s_3, t_3, d_3, e_3, f_4, g_4, l_1, l_4, W, Z, A_1, B_1, C_1, D_1, E_1, F_1, G_1, m_1, H_1, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, U_1, n_1, W_1, A_2, B_2, C_2, G_2, H_2, I_2, J_2, O_2, p_1, q_1, r_1, s_1
  10. Rotate(C', (360°) / n, A) → C''
  11. Polygon(C', C'', n) → poly1, f, q, r, s, t, a, b, e, f_1, g_1, h_1, g, i_1, n_3, q_3, a_3, b_3, c_3, h_4, i_4, j_4, k_4, h, m_4, E, F, G, H, I, J, K, L, M, i, N, O, R, S, T, U, V, Q_1, R_1, V_1, j, Z_1, D_2, E_2, F_2, K_2, L_2, M_2, N_2, P_2, k, l, m, p
  12. Point(zAxis) → P
  13. OrthogonalPlane(P, zAxis) → p_2
  14. Mirror(poly1, p_2) → poly1'
  15. Mirror(E, p_2) → E'
  16. Mirror(c, p_2) → c'
  17. Mirror(C', p_2) → C''_1
  18. Mirror(C'', p_2) → C'''
  19. Mirror(F, p_2) → F'
  20. Mirror(G, p_2) → G'
  21. Mirror(H, p_2) → H'
  22. Mirror(I, p_2) → I'
  23. Mirror(J, p_2) → J'
  24. Mirror(K, p_2) → K'
  25. Mirror(L, p_2) → L'
  26. Mirror(M, p_2) → M'
  27. Mirror(N, p_2) → N'
  28. Mirror(O, p_2) → O'
  29. Mirror(R, p_2) → R'
  30. Mirror(S, p_2) → S'
  31. Mirror(T, p_2) → T'
  32. Mirror(U, p_2) → U'
  33. Mirror(V, p_2) → V'
  34. Mirror(Q_1, p_2) → S_1
  35. Mirror(R_1, p_2) → T_1
  36. Segment(C''', D') → k_2
  37. Segment(C''_1, D) → l_2
  38. Segment(E', W) → m_2
  39. Segment(F', Z) → n_2
  40. Segment(G', A_1) → r_2
  41. Segment(H', B_1) → s_2
  42. Segment(I', C_1) → t_2
  43. Segment(J', D_1) → a_2
  44. Segment(K', E_1) → b_2
  45. Segment(L', F_1) → c_2
  46. Segment(M', G_1) → q_2
  47. Segment(N', H_1) → d_2
  48. Segment(O', I_1) → e_2
  49. Segment(R', J_1) → f_3
  50. Segment(S', K_1) → g_3
  51. Segment(T', L_1) → h_3
  52. Segment(U', M_1) → i_3
  53. Segment(V', N_1) → j_3
  54. Segment(S_1, O_1) → k_3
  55. Segment(T_1, P_1) → l_3

AI Drawing Prompt Reference

Draw a one-sheet hyperboloid, demonstrating it as a ruled surface generated by a family of straight lines. Start by creating two circles on coordinate planes, then rotate and mirror points on these circles to form lines. Connect these lines into polygons and use mirror operations to construct the complete hyperboloid.
The prompt above was compiled from the original drawing commands and is for reference only.