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Animation 135: Dynamic Geometry and Function Visualization

Animation 135: Dynamic Geometry and Function Visualization
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2016-11-06 11:41
AngleDerivativeTransformationGeometric 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 CCSS High School: Geometry (Congruence) Volume 2
①This example demonstrates a dynamic animation that integrates geometric figures with function graphs, showcasing complex interactions among lines, circles, angles, vectors, polynomial functions, and derivatives. ②Core mathematical concepts include angles, straight lines, complex numbers, polynomial functions, derivatives, and secant lines, with transformations such as rotation, translation, and dilation creating rich geometric relationships. ③Construction highlights: Based on the construction summary, GeoGebra commands such as Point, Line, Rotate, Angle, Ray, Circle, OrthogonalLine, Intersect, Segment, CircleSector, Dilate, Vector, and Translate are used, generating numerous objects (A, B, f, α, B', β, g, γ, B'', δ, h, a, c, i, C, d, D, e, E, j, k, l, F, G, H, m, p, I, J, n, K, L, M, N, q, b, A', r, F', s) to illustrate dynamic changes via sliders or animation. ④Suitable for high school and above, this example helps explore relationships between geometric transformations, function derivatives, and secant slopes, and serves as a visual teaching tool or enrichment material for mathematical competitions.

GeoGebra Commands & Parameters

  1. Point(y = 0) → A
  2. Line(A, B) → f
  3. Rotate(B, α, A) → B'
  4. Angle(B, A, B') → β
  5. Line(A, B') → g
  6. Rotate(B', γ, A) → B''
  7. Angle(B', A, B'') → δ
  8. Ray(A, B'') → h
  9. Circle(A, a) → c
  10. OrthogonalLine(A, g) → i
  11. Intersect(c, i, 2) → C
  12. Circle(C, a) → d
  13. Intersect(d, i, 2) → D
  14. Circle(D, a) → e
  15. Intersect(e, i, 2) → E
  16. Line(C, g) → j
  17. Line(D, g) → k
  18. Line(E, g) → l
  19. Intersect(h, j) → F
  20. Intersect(k, h) → G
  21. Intersect(l, h) → H
  22. Segment(A, F) → m
  23. Circle(A, (0.3 * m)) → p
  24. Point(l) → I
  25. Point(g) → J
  26. Segment(I, J) → n
  27. Intersect(k, n) → K
  28. Intersect(j, n) → L
  29. Intersect(p, g, 2) → M
  30. Intersect(p, h, 1) → N
  31. CircleSector(A, M, N) → q
  32. Dilate(A, If[0 < b ≤ 1, 1 - b, 0], F) → A'
  33. Segment(A, A') → r
  34. Dilate(F, If[1 < b ≤ 2, 2 - b, 0], G) → F'
  35. Segment(F, F') → s
  36. Dilate(G, If[2 < b ≤ 3, 3 - b, 0], H) → G'
  37. Segment(G, G') → t
  38. Dilate(A, If[3 < b ≤ 4, 4 - b, 0], F) → A'_1
  39. Vector(A, A'_1) → u
  40. Translate(q, u) → q'
  41. Dilate(F, If[4 < b ≤ 5, 5 - b, 0], G) → F'_1
  42. Vector(F, F'_1) → v
  43. Translate(q', v) → q''
  44. Dilate(G, If[5 < b ≤ 6, 6 - b, 0], H) → G'_1
  45. Vector(G, G'_1) → w
  46. Translate(q'', w) → q'''
  47. Rotate(q''', If[6 < b ≤ 7, ((-(b - 6)) * π), (-π)], H) → c_1
  48. Rotate(A, If[7 < b ≤ 8, ((b - 7) * π), π], F) → A'_2
  49. Rotate(A', If[7 < b ≤ 8, ((b - 7) * π), π], F) → A''
  50. Segment(A'_2, A'') → r'
  51. Rotate(A'_2, If[8 < b ≤ 9, ((b - 8) * π), π], G) → O
  52. Rotate(A'', If[8 < b ≤ 9, ((b - 8) * π), π], G) → A'''
  53. Segment(O, A''') → r''
  54. Dilate(L, If[12.5 < b ≤ 13.5, 13.5 - b, 0], K) → K'
  55. Dilate(J, If[11.5 < b ≤ 12.5, 12.5 - b, 0], L) → L'
  56. Rotate(I, If[14.5 < b ≤ 15.5, ((-(b - 15.5)) * π), (-π)], K) → I'_1
  57. Rotate(I, If[14.5 < b ≤ 15.5, ((-(b - 14.5)) * π), (-π)], K) → I'_2
  58. Rotate(I'_2, If[15.5 < b ≤ 16.5, ((-(b - 15.5)) * π), (-π)], L) → P
  59. Circle(E, a) → d_1
  60. Intersect(d_1, i, 2) → R
  61. Line(R, g) → k_1
  62. Intersect(k_1, h) → S
  63. Circle(R, a) → e_1
  64. Intersect(e_1, i, 2) → U
  65. Line(U, g) → m_1
  66. Intersect(m_1, h) → V
  67. Point(k_1) → T
  68. Point(m_1) → W
  69. Segment(J, T) → l_1
  70. Segment(J, L') → j_1
  71. Segment(L, K') → q_1
  72. Dilate(K, If[13.5 < b ≤ 14.5, 14.5 - b, 0], I) → K'_1
  73. Segment(K, K'_1) → f_1
  74. Rotate(J, If[14.5 < b ≤ 15.5, ((b - 14.5) * π), π], L) → J'
  75. Rotate(L', If[14.5 < b ≤ 15.5, ((b - 14.5) * π), π], L) → L''
  76. Segment(J', L'') → g_1
  77. Rotate(J', If[15.5 < b ≤ 16.5, ((b - 15.5) * π), π], K) → J''
  78. Rotate(L'', If[15.5 < b ≤ 16.5, ((b - 15.5) * π), π], K) → L'''
  79. Segment(J'', L''') → h_1
  80. Intersect(l, l_1) → Q
  81. Circle(T, Q) → r_1
  82. Circle(Q, Radius[r_1]) → s_1
  83. Intersect(s_1, l_1, 1) → Z
  84. Circle(Z, Radius[r_1]) → t_1
  85. Intersect(t_1, l_1, 1) → A_1
  86. Dilate(H, If[6 < b ≤ 7, 7 - b, 0], S) → H'
  87. Vector(H, H') → u_1
  88. Translate(q''', u_1) → c_2
  89. Rotate(c_2, If[7 < b ≤ 8, ((b - 7) * π), π], S) → d_2
  90. Segment(H, H') → i_1
  91. Rotate(O, If[9 < b ≤ 10, ((b - 9) * π), π], H) → O'
  92. Rotate(A''', If[9 < b ≤ 10, ((b - 9) * π), π], H) → B_1
  93. Segment(O', B_1) → r'''
  94. Dilate(J, If[10 < b ≤ 11, 11 - b, 0], A_1) → J'_1
  95. Segment(J, J'_1) → p_1
  96. Dilate(A_1, If[11 < b ≤ 12, 12 - b, 0], Z) → C_1
  97. Segment(A_1, C_1) → a_1
  98. Dilate(Z, If[12 < b ≤ 13, 13 - b, 0], Q) → Z'
  99. Segment(Z, Z') → b_1
  100. Dilate(Q, If[13 < b ≤ 14, 14 - b, 0], T) → Q'
  101. Segment(Q, Q') → g_2
  102. Rotate(J, If[14 < b ≤ 15, ((b - 14) * π), π], A_1) → J'_2
  103. Rotate(J'_1, If[14 < b ≤ 15, ((b - 14) * π), π], A_1) → D_1
  104. Segment(J'_2, D_1) → h_2
  105. Rotate(J'_2, If[15 < b ≤ 16, ((b - 15) * π), π], Z) → E_1
  106. Rotate(D_1, If[15 < b ≤ 16, ((b - 15) * π), π], Z) → F_1
  107. Segment(E_1, F_1) → i_2
  108. Rotate(E_1, If[16 < b ≤ 17, ((b - 16) * π), π], Q) → G_1
  109. Rotate(F_1, If[16 < b ≤ 17, ((b - 16) * π), π], Q) → H_1
  110. Segment(G_1, H_1) → j_2

AI Drawing Prompt Reference

Create a dynamic geometric animation combining shapes and function graphs. Start with a point on a horizontal line, then rotate it to form rays and angles. Construct a series of equal circles along a perpendicular line, intersecting with parallel lines to generate key points. Connect these points to form segments and a circular sector. Then, translate and rotate the sector along a path while using dilation to create moving segments between points, simulating function changes. The scene includes lines, circles, angles, sectors, vectors, and dynamic segments, controlled by a slider to show different transformation stages.
The prompt above was compiled from the original drawing commands and is for reference only.