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圆锥曲线的定义：椭圆、双曲线与抛物线的动态演示

圆锥曲线的定义：椭圆、双曲线与抛物线的动态演示
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2018-08-22 16:55
圆解析几何立体几何变换轨迹
所属专题 Conics
本范例通过GeoGebra动态展示圆锥曲线（椭圆、双曲线、抛物线）的统一定义，即到定点（焦点）与到定直线（准线）距离之比为常数的轨迹。核心数学概念包括圆锥曲线的焦点、准线、离心率，以及三维圆锥与平面的截交线。作图要点：利用Circle、Point、Focus、Line、Ray等命令构造焦点和准线，通过Distance和Angle度量距离与角度，使用Rotate、Mirror、Translate等变换调整图形，并借助Cone、Plane、OrthogonalPlane、IntersectPath生成三维圆锥截面，配合按钮（button1等）和滑块实现动态切换。适合高中或大学低年级解析几何与立体几何教学，帮助学生直观理解圆锥曲线的几何本质及离心率对曲线类型的影响。

GeoGebra 命令及参数

  1. Circle(H16, 4) → c16
  2. Point(c16) → M16
  3. LaTeX(S5, true, true) → text1
  4. Rotate(A6, ((n6 * 360))°, O6) → M6
  5. Segment(O6, M6) → s6
  6. Segment(F71, M7) → s71
  7. Segment(M7, F62) → s72
  8. Focus(e8) → F8, F'8
  9. Intersect(e8, j8) → I8, J8
  10. Intersect(m8_1, i8) → C
  11. Intersect(m8_1, i8) → D
  12. Intersect(m8_1, i8) → E
  13. Intersect(m8_1, i8) → F_1
  14. Intersect(e8, i8) → G8, H8
  15. Circle(F8, 11.5) → g8
  16. Ray(F8, H8) → h8
  17. Intersect(g8, h8) → A8,
  18. Segment(A8, H8) → k8
  19. Segment(F'8, H8) → m8
  20. Circle(F'8, 11.5) → p8
  21. Ray(F'8, I8) → n8
  22. Intersect(p8, n8) → C8,
  23. Segment(C8, I8) → q8
  24. Segment(I8, F8) → r8
  25. Segment(F8, A8) → f8
  26. Segment(F'8, C8) → f'8
  27. OrthogonalLine(D9, d9) → p9
  28. Segment(F9, D9) → s9
  29. LineBisector(D9, F9) → h9
  30. Intersect(p9, h9) → M9
  31. Intersect(h9, d9) → H9
  32. Segment(M9, F9) → f9
  33. Segment(M9, D9) → g9
  34. Intersect(s9, h9) → G9
  35. Polygon(E9, D9, N9) → t1, f, g, h
  36. If(a9 > 0, Angle[H9, D9, M9], Angle[M9, D9, H9]) → α
  37. Segment(N9, M9) → i
  38. Segment(N9, M9) → l91
  39. Segment(F9, M9) → l92
  40. Segment(A10, B10) → j
  41. Point(q_1) → C10
  42. Segment(A10, C10) → k
  43. Segment(C10, B10) → l
  44. Line(J, xAxis) → m
  45. Point(m) → G10_1
  46. OrthogonalLine(G10_1, m) → n
  47. Point(n) → H10_1
  48. Line(H10_1, l) → p
  49. Intersect(p, m) → I10_1
  50. Circle(G10_1, I10_1) → d10_1
  51. Intersect(d10_1, n) → G10_2, K10_1
  52. Line(K10_1, m) → q
  53. Distance(G10_1, H10_1) → r10
  54. Circle(F10, r10) → e10
  55. Intersect(e10, q) → H10_2, M10
  56. OrthogonalLine(M10, m) → s
  57. Intersect(m, s) → N10
  58. Segment(M10, N10) → t
  59. Polygon(B10, C10, A10) → t2, a10_1, b10, c10
  60. Polygon(I10_1, H10_1, G10_1) → t3, g10, i10, h10
  61. OrthogonalLine(H10, d10_2) → a10_2
  62. Intersect(d10_2, a10_2) → I10
  63. Point(d10_2) → K10
  64. Angle(K10, I10, H10) → γ
  65. Segment(G10, H10) → a10
  66. Segment(H10, I10) → d10
  67. Segment(B12, A12) → f12
  68. Segment(C12, B12) → h12
  69. Point(i12) → G12
  70. OrthogonalLine(G12, i12) → j12
  71. Line(H12, h12) → k12
  72. Intersect(i12, k12) → N12
  73. Circle(G12, N12) → c122
  74. Intersect(c122, j12) → J12, K12
  75. Line(K12, i12) → l12
  76. Distance(G12, H12) → r12
  77. Circle(F, r12) → c121
  78. Intersect(c121, l12) → L12, M12
  79. OrthogonalLine(M12, i12) → m122
  80. Intersect(m122, i12) → G
  81. Segment(M12, G) → n12
  82. Segment(F, M12) → p12
  83. Polygon(A12, C12, B12) → t4, b12, a12, c12
  84. Segment(G12, N12) → h121
  85. Segment(K12, G12) → r
  86. Segment(H12, G12) → i121
  87. Mirror(L12, i12) → L12'
  88. Mirror(M12, i12) → M12'
  89. Mirror(G, (x(F), y(G))) → P'
  90. Segment(L12, P') → a
  91. Segment(L12, F) → d
  92. Rotate(B13, t13, F13) → M13
  93. Segment(M13, F'12) → f13
  94. LineBisector(f13) → d131
  95. Line(F13, M13) → d132
  96. Intersect(d132, d131) → E13
  97. Segment(F13, M13) → f132
  98. Segment(E13, F'12) → f133
  99. Intersect(d131, f13) → A13
  100. Circle(F14, 5) → c14
  101. Point(c14) → B14
  102. Rotate(B14, t14, F14) → P14
  103. Segment(P14, F'14) → f_1
  104. LineBisector(f_1) → g14
  105. Line(F14, P14) → h14_1
  106. Intersect(g14, h14_1) → H14
  107. Segment(H14, F'14) → i_1
  108. Intersect(g14, f_1) → E14
  109. Angle(H14, E14, F'14) → ε
  110. Segment(H14, P14) → g_1
  111. Segment(P15, F15) → h15_2
  112. OrthogonalLine(P15, h15_1) → j15_1
  113. LineBisector(h15_2) → k_1
  114. Intersect(k_1, j15_1) → H15_1
  115. Segment(H15_1, F15) → h15
  116. Segment(H15_1, P15) → j15
  117. Point(h15_1) → H15
  118. Angle(H15, P15, H15_1) → ζ
  119. Segment(F14, P14) → h14
  120. Segment(M16, F16) → h_1
  121. LineBisector(h_1) → j_1
  122. Point(l14) → I
  123. Segment(I, H) → l_1
  124. LineBisector(l_1) → m_1
  125. Circle(A17, B17) → d_1
  126. LineBisector(F17, M17) → n_1
  127. Segment(F17, M17) → p_1
  128. Point(h15_1) → K
  129. Angle(K, P15, H15_1) → η
  130. Plane(C5, (0, 0, 0), (0, 1, 0)) → b5
  131. Circle(C5, 3, b5) → c5
  132. Point(c5) → M5
  133. Segment(C5, M5) → f5
  134. OrthogonalPlane(C5, f5) → d5
  135. Cone((0, 0, -10), (0, 0, 0), 7) → e5, c, b
  136. Cone((0, 0, -10), (0, 0, 0), 7) → h5, c_1, o
  137. Cone((0, 0, -10), (0, 0, 0), 7) → g5, e_1, u
  138. Mirror(e5, (0, 0, 0)) → e5'
  139. IntersectPath(d5, e5) → k5
  140. IntersectPath(d5, e5') → p5
  141. LaTeX(M5, true, true) → text30
  142. Circle(F13, 6) → e
  143. Distance(A, C) → AC
  144. Distance(A, B) → AB