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圆锥截线

圆锥截线
几何画板及绘图内容加载中，请稍等……
2015-07-10 22:35
解析几何微积分变换最值优化几何作图
①本范例演示圆锥截线（椭圆、双曲线、抛物线）的形成与性质；②涉及核心数学概念包括解析几何、导数、几何变换等；③构造中使用了Point, Line, Intersect, Cone, Circle, Tangent, UnitOrthogonalVector, OrthogonalPlane, IntersectPath, Focus, Polygon, Mirror, Sphere, Segment, Ellipse, Hyperbola, Parabola, Angle等GeoGebra命令与对象；④适合高中及以上数学学习者和教师用于教学演示与探索。

GeoGebra 命令及参数

  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