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Animation 175: Dynamic Construction of Multiple Geometric Elements

Animation 175: Dynamic Construction of Multiple Geometric Elements
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2017-01-23 11:02
PolygonVectorTransformationGeometric ConstructionSlider Animation
In collections Congruence (CO) Cluster Congruence (Vol 2) CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Gail LeBlancCCSS High School: Geometry (Congruence) Volume 2 Area, Surface Area, Volume, Cross Section, 3D
① This example demonstrates a composite geometric animation involving rotations, translations, and dilations, where sliders control parameters such as angles to dynamically show the changing positional relationships among points, lines, circles, polygons, and vectors. ② Core mathematical concepts include planar geometry transformations (rotation and translation), vector operations, properties of parallelograms, intersections of lines and circles, midpoints and perpendicular lines, with additional elements of complex numbers and solid geometry (cylinder), as well as an application context of interest calculation. ③ Construction highlights: Based on the construction summary, the GeoGebra commands used include Rotate, Angle, Ray, Segment, Intersect, Midpoint, Line, OrthogonalLine, Polygon, Point, Circle, Dilate, Min, CircleSector, Vector, and Translate. Numerous objects such as A, B, α, β, A', γ, f, B', δ, g, h, C, i, j, D, E, l, m, F, n, poly1, c, d, e, G, p, poly2, b, d_1, e_1, a, q, r, H, t, I, k, s, c_1, f_1 are created, with sliders driving angle parameters for dynamic effects. ④ Suitable for high school and above, this example is useful for teaching geometric transformations, vectors, and complex numbers, and can serve as a reference for mathematical modeling or dynamic demonstrations.

GeoGebra Commands & Parameters

  1. Rotate(A, α, B) → A'
  2. Angle(A, B, A') → γ
  3. Ray(B, A') → f
  4. Rotate(B, (-β), A) → B'
  5. Angle(B', A, B) → δ
  6. Ray(A, B') → g
  7. Segment(B, A) → h
  8. Intersect(f, g) → C
  9. Segment(B, C) → i
  10. Segment(A, C) → j
  11. Midpoint(i) → D
  12. Midpoint(j) → E
  13. Line(B, A) → l
  14. OrthogonalLine(C, l) → m
  15. Intersect(l, m) → F
  16. Segment(C, F) → n
  17. Polygon(D, E, C) → poly1, c, d, e
  18. Midpoint(n) → G
  19. Line(C, l) → p
  20. Polygon(B, D, E, A) → poly2, b, d_1, e_1, a
  21. Segment(D, C) → q
  22. Segment(B, A) → r
  23. Point(p) → H
  24. OrthogonalLine(H, p) → t
  25. Intersect(l, t) → I
  26. Segment(H, I) → s
  27. Circle(I, 3) → c_1
  28. Circle(H, 3) → f_1
  29. Intersect(f_1, p, 1) → J
  30. Intersect(f_1, p, 2) → K
  31. Intersect(c_1, l, 2) → L
  32. Intersect(c_1, s, 1) → M
  33. Intersect(c_1, l, 1) → N
  34. Angle(J, H, I) → ε
  35. Angle(M, I, N) → ζ
  36. Dilate(H, If[0 < k ≤ 1, 1 - k, 0], I) → H'
  37. Segment(H, H') → g_1
  38. Angle(C, F, B) → η
  39. Dilate(C, If[1 < k ≤ 2, 2 - k, 0], G) → C'
  40. Segment(C, C') → h_1
  41. Dilate(G, If[2 < k ≤ 3, 3 - k, 0], F) → G'
  42. Segment(G, G') → i_1
  43. Rotate(C, If[3 < k ≤ 4, ((-(k - 3)) * π), (-π)], G) → C'_1
  44. Rotate(C', If[3 < k ≤ 4, ((-(k - 3)) * π), (-π)], G) → C''
  45. Segment(C'_1, C'') → j_1
  46. Dilate(G, If[4 < k ≤ 5, 5 - k, 0], D) → G'_1
  47. Dilate(G, If[4 < k ≤ 5, 5 - k, 0], E) → G'_2
  48. Segment(G, G'_1) → k_1
  49. Segment(G, G'_2) → l_1
  50. Segment(B, F) → m_1
  51. Segment(B, D) → n_1
  52. Segment(D, G) → p_1
  53. Segment(D, C) → q_1
  54. Min({m_1, n_1, p_1, q_1}) → o
  55. Circle(B, (0.6 * o)) → r_1
  56. Intersect(r_1, l) → O, P
  57. Segment(B, C) → s_1
  58. Intersect(r_1, s_1, 1) → Q
  59. CircleSector(B, P, Q) → t_1
  60. Rotate(D, If[6 < k ≤ 7, ((-(k - 6)) * π), (-π)], E) → D'
  61. Rotate(E, If[6 < k ≤ 7, ((-(k - 6)) * π), (-π)], E) → E'
  62. Rotate(C, If[6 < k ≤ 7, ((-(k - 6)) * π), (-π)], E) → C'_2
  63. Polygon(D', E', C'_2) → poly1', c'_2, d', e'
  64. Segment(D, C) → a_1
  65. Segment(D, B) → b_1
  66. Segment(C, E) → f_2
  67. Segment(E, A) → g_2
  68. Segment(E, C'_2) → h_2
  69. Segment(C'_2, D') → i_2
  70. Segment(D', E) → j_2
  71. Dilate(D, If[7 < k ≤ 8, 8 - k, 0], C) → D'_1
  72. Segment(D, D'_1) → k_2
  73. Dilate(B, If[8 < k ≤ 9, 9 - k, 0], D) → B'_1
  74. Vector(B, B'_1) → u
  75. Translate(t_1, u) → c_2
  76. Dilate(B, If[9 < k ≤ 10, 10 - k, 0], A) → B'_2
  77. Vector(B, B'_2) → v
  78. Translate(t_1, v) → d_2
  79. Circle(A, Radius[r_1]) → e_2
  80. Intersect(e_2, l, 1) → R
  81. Intersect(e_2, g_2, 1) → S
  82. Segment(B, A) → l_2
  83. Segment(A, C) → m_2
  84. Min({l_2, m_2}) → w
  85. Circle(A, (0.25 * w)) → p_2
  86. Intersect(g_2, m_2) → T
  87. Intersect(g_2, m_2) → U
  88. Intersect(p_2, m_2) → V,
  89. Intersect(p_2, l_2) → W,
  90. CircleSector(A, V, W) → q_2