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Ellipse from Conic Section

Ellipse from Conic Section
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2016-10-17 17:31
Coordinate GeometryGeometric Construction
This example demonstrates how an ellipse can be obtained by intersecting a cone. Key mathematical concepts include conic sections in coordinate geometry. Construction highlights the use of point, cone, intersection, line, orthogonal line, plane, intersect path, polygon, angular bisector, circle, sphere, segment, and other GeoGebra commands. Suitable for high school students or educators teaching coordinate geometry.

GeoGebra Commands & Parameters

  1. Point(zAxis) → A
  2. Cone(A_1, A, 4) → a, a_2, a_3
  3. Intersect(xAxis, a_3, 2) → B
  4. Intersect(yAxis, xAxis) → C_1
  5. Line(C_1, B) → f
  6. Intersect(xAxis, a_3, 1) → C
  7. Point(xAxis) → E_1
  8. Point(xAxis) → F
  9. OrthogonalLine(E_1, xOyPlane) → g
  10. Point(g) → G
  11. Plane(A, E_1, F) → b
  12. OrthogonalLine(F, b) → h
  13. Point(h) → H
  14. Plane(G, F, H) → c
  15. IntersectPath(c, a) → d
  16. Line(G, F) → i
  17. Intersect(i, d) → D, E
  18. Line(A, C) → j
  19. Line(A, B) → k
  20. Polygon(A, E, D) → Polígon1, k_1, a_1, l
  21. AngularBisector(i, j) → m, n
  22. AngularBisector(k_1, l) → p, q
  23. Intersect(n, q) → I_1
  24. OrthogonalLine(I_1, a_1, Polígon1) → r
  25. Intersect(i, r) → F_1
  26. OrthogonalLine(I_1, k_1, Polígon1) → s
  27. Intersect(j, s) → P_1
  28. Circle(I_1, F_1, Polígon1) → e
  29. Sphere(I_1, P_1) → o
  30. Circle(q, P_1) → t
  31. AngularBisector(i, k) → f_1, g_1
  32. Intersect(q, g_1) → I_2
  33. OrthogonalLine(I_2, a_1, Polígon1) → h_1
  34. Intersect(h_1, i) → F_2
  35. OrthogonalLine(I_2, k, Polígon1) → i_1
  36. Intersect(k, i_1) → P_2
  37. Circle(I_2, F_2, Polígon1) → d_1
  38. Sphere(I_2, F_2) → u
  39. Circle(q, P_2) → c_1
  40. Point(d) → L
  41. Segment(F_1, L) → j_1
  42. Line(L, F_1) → n_1
  43. Line(A, L) → m_1
  44. Intersect(m_1, t) → T,
  45. Segment(T, L) → p_1
  46. Segment(L, F_2) → l_1
  47. Line(L, F_2) → r_1
  48. Intersect(m_1, c_1) → U,
  49. Segment(L, U) → q_1

AI Drawing Prompt Reference

Draw an ellipse obtained by intersecting a cone, demonstrating how a cone and a plane intersect to form an ellipse. Start by sketching the cone and the intersecting plane, then locate points on the ellipse and connect them. Add auxiliary lines like angle bisectors and perpendiculars, and use them to construct circles and spheres, completing the ellipse with these elements.
The prompt above was compiled from the original drawing commands and is for reference only.