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Conic Section of a Cone

Conic Section of a Cone
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2017-07-18 02:12
Coordinate GeometrySolid GeometryGeometric Construction
In collections Conic Sections' study using Geogebra applet
This example demonstrates the geometric construction of conic sections from a cone. Key mathematical concepts include cones, planes, sections, and the resulting ellipses, parabolas, and hyperbolas. Using GeoGebra commands such as Circle, Intersect, Rotate, etc., the formation process of these curves can be dynamically shown. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Circle(V', A, xOyPlane) → e
  2. Intersect(xAxis, e, 2) → B
  3. Rotate(B, α, V') → B'
  4. Angle(B, V', B', xOyPlane) → β
  5. OrthogonalLine(B', xOyPlane) → f
  6. Line(V', B') → g
  7. Line(D, g) → h_1
  8. Plane(h_1, f) → i
  9. OrthogonalLine(V', g, i) → j
  10. Intersect(h_1, j) → C_1
  11. Rotate(C_1, γ, D, i) → C'
  12. Angle(C_1, D, C', i) → δ
  13. Line(D, C') → l
  14. Intersect(g, l) → E
  15. OrthogonalLine(E, i) → m
  16. OrthogonalLine(C', i) → n
  17. Plane(l, n) → p
  18. OrthogonalLine(V, p) → q
  19. Intersect(q, p) → F
  20. Plane(E, F, V) → r
  21. Line(E, F) → s
  22. IntersectPath(p, d) → t
  23. Intersect(g, c_1, 2) → V_2
  24. Intersect(g, c_1, 1) → V_1
  25. Plane(xAxis, zAxis) → o
  26. OrthogonalLine(V, o) → f_1
  27. PointIn(o) → C_2
  28. Circle(f_1, C_2) → e_1
  29. Intersect(c_1, yAxis, 1) → G
  30. Line(V, G) → g_1
  31. IntersectPath(o, d) → k_1
  32. Intersect(k_1, e_1, 3) → H
  33. Line(V, H) → i_1
  34. Midpoint(V_2, G) → C
  35. If(u ≟ 0, v ≟ 0) → w
  36. If(u ≟ 0, v ≟ 0) → a_1
  37. If(u ≟ 0, v ≟ 0) → b_1
  38. If(u ≟ 0, v ≟ 0) → d_1
  39. If(u ≟ false, v ≟ false) → j_1
  40. If(u ≟ false, v ≟ false) → l_1
  41. If(u ≟ false, v ≟ false) → m_1
  42. If(u ≟ false, v ≟ false) → n_1
  43. If(u ≟ false, v ≟ false) → o_1
  44. If(u ≟ false, v ≟ false) → p_1
  45. If(u ≟ false, v ≟ false) → q_1
  46. If(u ≟ false, v ≟ false) → r_1
  47. If(u ≟ false, v ≟ false) → s_1
  48. If(u ≟ false, v ≟ false) → t_1
  49. If(u ≟ false, v ≟ false) → u_1
  50. If(u ≟ false, v ≟ false) → v_1
  51. If(u ≟ false, v ≟ false) → w_1
  52. If(u ≟ false, v ≟ false) → a_2
  53. If(u ≟ false, v ≟ false) → b_2
  54. If(u ≟ false, v ≟ false) → c_2
  55. Midpoint(G, V_2) → C_t
  56. Line(V, V_2) → f_2

AI Drawing Prompt Reference

Draw a 3D cone with its apex at a point in space and a circular base on a horizontal plane. Inside the cone, construct an inclined plane that intersects the cone, producing a section curve. By rotating and projecting, show how the curve changes from an ellipse to a parabola or hyperbola as the plane's angle varies. Also include auxiliary circles and lines in the coordinate planes, mark key intersection points, and display the projection of the section curve onto the base. The overall figure includes the cone, the cutting plane, the section curve, auxiliary geometric elements, and dynamic angle controls to visually demonstrate the generation of conic sections.
The prompt above was compiled from the original drawing commands and is for reference only.