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Geometric Construction of Conics using Eccentricity

Geometric Construction of Conics using Eccentricity
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2014-02-24 04:40
Coordinate GeometryGeometric ConstructionSlider Animation
In collections Conic Sections
This example demonstrates the geometric construction of conic sections based on their eccentricity. It involves core mathematical concepts such as eccentricity, ellipse, hyperbola, and geometric constructions. The construction utilizes GeoGebra commands and objects like Circle, OrthogonalLine, Point, Ray, Intersect, Segment, Line, Locus, Distance, Midpoint, Mirror, and Polygon. It is suitable for high school and above students to understand the geometric definitions and properties of conic sections.

GeoGebra Commands & Parameters

  1. Circle(B, e) → f
  2. OrthogonalLine(A, a) → b
  3. Point(b) → F_1
  4. Ray(B, C_2) → c
  5. Ray(B, D_2) → d
  6. Intersect(f, d) → D, N
  7. Point(B, Vector[20 * Vector[B, D]]) → E_1
  8. Circle(B, 1) → e_1
  9. Intersect(e_1, c) → C,
  10. Segment(D, C) → g
  11. Point(d) → G
  12. Line(G, g) → h
  13. Intersect(h, c) → H
  14. Segment(G, H) → i
  15. Segment(G, B) → j
  16. Segment(H, B) → k
  17. Intersect(a, b) → I
  18. Circle(I, k) → p
  19. Intersect(p, b) → J, K
  20. Line(J, a) → l
  21. Circle(F_1, j) → q
  22. Intersect(q, l) → L, M
  23. Locus(M, G) → loc1
  24. Locus(L, G) → loc2
  25. Segment(L, F_1) → n
  26. OrthogonalLine(L, a) → r
  27. Intersect(a, r) → Q_1
  28. Segment(L, Q_1) → s
  29. Distance(F_1, a) → df
  30. Circle(F_1, dv) → t
  31. Intersect(t, b) → R_1, S
  32. Circle(S, AB) → c_1
  33. Intersect(c_1, b) → O, P_1
  34. Midpoint(S, P_1) → T
  35. Mirror(a, T) → a'
  36. Circle(P_1, dv) → d_1
  37. Intersect(d_1, b) → U, F_2
  38. Circle(F_2, n) → f_1
  39. Intersect(a', b) → V
  40. Circle(V, s) → g_1
  41. Intersect(g_1, b) → W, Z
  42. Line(Z, a') → a_1
  43. Intersect(f_1, a_1) → Q, P
  44. Locus(P, G) → loc3
  45. Polygon(Corner[1], Corner[2], (x(Corner[2]), y(Corner[2]) + 0.28 * Distance[Corner[3], Corner[2]]), (x(Corner[4]), y(Corner[1]) + 0.3 * Distance[Corner[3], Corner[2]])) → Bottom, m, b_1, h_1, i_1
  46. Locus(Q, G) → loc4
  47. Point(B, Vector[20 * Vector[B, C]]) → F_3
  48. Segment(P, F_2) → m_1
  49. OrthogonalLine(P, a') → n_1
  50. Intersect(n_1, a') → E
  51. Point(m_1) → F
  52. Segment(F, E) → p_1
  53. Segment(F_1, M) → q_1
  54. Mirror(Q_1, b) → R
  55. Segment(M, R) → r_1
  56. Mirror(E, b) → E'
  57. Segment(Q, E') → s_1
  58. Segment(Q, F_2) → t_1

AI Drawing Prompt Reference

Draw an ellipse based on a given focus and directrix. Start by creating a circle and a line perpendicular to it, then determine the focus F1. Draw a ray from the focus to the circle's center, and find the intersection point C. Connect C and D to form part of the ellipse. Use mirroring and rotation to construct the rest of the ellipse, ultimately forming a complete ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.