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Definitions of Conics: Dynamic Demonstration of Ellipse, Hyperbola, and Parabola

Definitions of Conics: Dynamic Demonstration of Ellipse, Hyperbola, and Parabola
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2018-08-22 16:55
CircleCoordinate GeometrySolid GeometryTransformationLocus
In collections Conics
This GeoGebra example dynamically illustrates the unified definition of conic sections (ellipse, hyperbola, parabola) as the locus of points whose ratio of distances to a fixed point (focus) and a fixed line (directrix) is constant. It covers core mathematical concepts such as focus, directrix, eccentricity, and the intersection of a plane with a cone. The construction uses commands like Circle, Point, Focus, Line, Ray, Distance, Angle, Rotate, Mirror, and Translate to build the 2D locus, while Cone, Plane, OrthogonalPlane, and IntersectPath create the 3D cross-section. Interactive buttons and sliders allow users to switch between conic types and adjust parameters. Suitable for high school or early college courses in analytic geometry and solid geometry, this example helps students grasp the geometric essence of conics and how eccentricity determines the curve type.

GeoGebra Commands & Parameters

  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