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Conic Sections by Sérgio Dantas

Conic Sections by Sérgio Dantas
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2015-07-10 22:35
Coordinate GeometryCalculusTransformationOptimizationGeometric Construction
In collections intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve intersection: surface-surface, curve-curve
This example demonstrates the formation and properties of conic sections including ellipses, hyperbolas, and parabolas. It involves core mathematical concepts such as coordinate geometry, derivatives, and geometric transformations. GeoGebra commands and objects used include Point, Line, Intersect, Cone, Circle, Tangent, UnitOrthogonalVector, OrthogonalPlane, IntersectPath, Focus, Polygon, Mirror, Sphere, Segment, Ellipse, Hyperbola, Parabola, Angle. Suitable for high school mathematics students and teachers for teaching demonstrations and explorations.

GeoGebra Commands & Parameters

  1. Point(yAxis) → H
  2. Line((0, 0), (cos(α), sin(α))) → r_1
  3. Intersect(OrthogonalLine[H, yAxis], r_1) → D
  4. Cone((0, 0, y(H)), (0, 0, 0), Distance[H, D]) → superior, f, g
  5. Point(yAxis) → C_1
  6. Intersect(OrthogonalLine[C_1, r_1], r_1) → B
  7. Circle(C_1, B) → circ_1
  8. Point(circ_1) → G_1
  9. Tangent(G_1, circ_1) → s,
  10. UnitOrthogonalVector(s) → u
  11. OrthogonalPlane((0, x(G_1), y(G_1)), Vector[(0, x(u), y(u))]) → h
  12. IntersectPath(h, superior) → c_1
  13. Focus(c_1) → F_1, F_2
  14. Circle(G_1, Distance[F_1, F_2]) → c
  15. Intersect(c, s) → I_2, I_1
  16. If(y(I_1) < y(G_1), I_1, I_2) → G_2
  17. Polygon((0, 0), D, Mirror[D, yAxis]) → pol1, a, b, d
  18. Mirror(pol1, (0, 0)) → pol1'
  19. Mirror(superior, (0, 0, 0)) → inferior
  20. Sphere((0, 0, y(C_1)), Radius[circ_1]) → esf_1
  21. Corner(c_1) → F, G
  22. IntersectPath(h, inferior) → c_2
  23. Corner(c_2) → H_1, I
  24. Vertex(c_1) → V_1, V_2, M, N
  25. Mirror(r_1, yAxis) → r_2
  26. Intersect(s, r_2) → X_1
  27. Intersect(r_1, s) → X_2
  28. Segment(Intersect[Circle[(0, 0), D], s, 1], Intersect[Circle[(0, 0), D], s, 2]) → j
  29. Mirror(A, (0, 0)) → A'
  30. Mirror(C, (0, 0)) → C'
  31. Ellipse(A, A', C) → e
  32. Mirror(E, (0, 0)) → E'
  33. Hyperbola(A, A', E) → k
  34. If(α ≟ β, (0, Distance[V_1, F_1]), (0, 0)) → J
  35. Parabola(J, OrthogonalLine[(0, (-y(J))), yAxis]) → p
  36. Point(k) → O
  37. Intersect
  38. Intersect(c_1, Circle[V_2, Distance[E, O], Vector[OrthogonalVector[h]]]) → L_1, L_2, L, P
  39. Segment(A, O) → i
  40. If(y(O) > 0, Segment[F_2, L_2], Segment[F_2, L_1]) → m
  41. If(y(O) > 0, Segment[F_1, L_2], Segment[F_1, L_1]) → n
  42. If(α ≟ β, 1, If[(-α) < β < α, 2, 3]) → q
  43. Point(e) → K_1
  44. Segment(A, K_1) → r
  45. Segment(K_1, A') → t
  46. Angle(C, (0, 0), K_1) → γ
  47. Intersect(Ray[Center[c_1], Rotate[V_2, γ, Center[c_1], h]], c_1) → K_2, Q
  48. Segment(F_2, K_2) → a_1
  49. Segment(K_2, F_1) → b_1
  50. Point(p) → K
  51. Segment(J, K) → e_1
  52. Segment(K, (x(K), (-y(J)))) → e_2
  53. If(α ≟ β, Line[Mirror[F_1, V_1], Tangent[V_1, c_1]]) → l
  54. Segment(O, A') → f_1
  55. Intersect
  56. Intersect(c_1, Circle[V_1, Distance[K, (0, 0)], Vector[OrthogonalVector[h]]]) → R, S, T, U
  57. If(x(K) > 0, Intersect[l, OrthogonalLine[S, l, h]], Intersect[l, OrthogonalLine[R, l, h]]) → V