MathPlotAI Logo MathPlotAI

Animation 97: Composite Similarity Transformations with Dynamic Geometry

Animation 97: Composite Similarity Transformations with Dynamic Geometry
Loading the math board and drawing, please wait…
2016-07-09 11:39
FunctionTransformationGeometric ConstructionSlider Animation
①This GeoGebra example demonstrates a dynamic geometric figure formed by composite transformations including translation, rotation, and dilation (homothety), featuring similar triangles, function graphs, and logarithmic curves in continuous animation. ②Core mathematical concepts include similarity transformations (dilation, rotation, translation), similar triangles, function graphs (logarithmic), vectors and polygons, with an introductory touch of probability. ③Construction highlights: from the summary, commands such as Segment, Circle, Point, Angle, Rotate, Ray, Intersect, Line, Dilate, Vector, Polygon, Translate, and If are used, creating multiple points (A, B, C, D, E, F, G, etc.), segments, circles, angles, vectors, polygons (poly1, poly2), and transformed copies (D', E', D'_1) via rotation, scaling, and translation, likely controlled by sliders or parameters. ④Suitable for upper secondary or early university students to understand composite geometric transformations, similarity criteria and properties, and the integration of functions with transformations in dynamic geometry; also useful for teacher demonstrations or student exploration.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b
  3. Segment(C, A) → c
  4. Circle(D, (k * a)) → d
  5. Point(d) → E
  6. Segment(D, E) → e
  7. Circle(E, (k * b)) → f
  8. Angle(A, B, C) → α
  9. Rotate(D, α, E) → D'
  10. Angle(D, E, D') → β
  11. Ray(E, D') → g
  12. Intersect(f, g, 1) → F
  13. Angle(C, A, B) → γ
  14. Rotate(E, (-γ), D) → E'
  15. Angle(E', D, E) → δ
  16. Ray(D, E') → h
  17. Segment(D, E') → i
  18. Segment(E', E) → j
  19. Line(E, D') → l
  20. Line(E, D') → n
  21. Circle(D, (k * c)) → p
  22. Ray(D, E') → m
  23. Intersect(p, m, 1) → G
  24. Segment(E, G) → q
  25. Segment(D, G) → r
  26. Dilate(D, If[0 < s ≤ 1, 1 - s], A) → D'_1
  27. Vector(D, D'_1) → u
  28. Polygon(A, C, B) → poly1, b_1, a_1, c_1
  29. Polygon(D, E, G) → poly2, g_1, d_1, e_1
  30. Translate(D, u) → D'_2
  31. Translate(E, u) → E'_1
  32. Translate(G, u) → G'
  33. Polygon(D'_2, E'_1, G') → poly2', g', d'_2, e'_1
  34. Vector(D, A) → v
  35. Translate(D, v) → D'_3
  36. Translate(E, v) → E'_2
  37. Translate(G, v) → G'_1
  38. Polygon(D'_3, E'_2, G'_1) → poly2'_1, g'_1, d'_3, e'_2
  39. Segment(D'_1, G') → t
  40. Segment(E'_1, D'_1) → h_1
  41. Segment(A, G'_1) → i_1
  42. Segment(G'_1, E'_2) → j_1
  43. Segment(E'_2, A) → k_1
  44. Angle(C, A, E'_2) → ε
  45. If(ε > γ, ε - γ, γ - ε) → ζ
  46. Rotate(D'_3, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → H
  47. Rotate(E'_2, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → I
  48. Rotate(G'_1, If[1 < s ≤ 2, ((-(s - 1)) * ζ), (-ζ)], A) → J
  49. Polygon(H, I, J) → poly3, j_2, h_2, i_2
  50. Segment(A, J) → l_1
  51. Segment(A, I) → m_1
  52. Segment(I, J) → n_1
  53. Dilate(J, If[2 < s ≤ 3, 3 - s, 0], C) → J'
  54. Dilate(I, If[2 < s ≤ 3, 3 - s, 0], B) → I'
  55. Segment(I', J') → p_1
  56. Angle(A, I', J') → η
  57. Angle(A, I, J) → θ
  58. Angle(D'_2, E'_1, G') → ι
  59. Angle(J, A, I) → λ
  60. Angle(G', D'_1, E'_1) → μ
  61. Polygon(A, J', I') → poly4, i', a_2, j'
  62. Angle(A, I, J) → ν
  63. Segment(G'_1, E'_2) → q_1
  64. Segment(G', E'_1) → f_1
  65. Angle(G'_1, A, E'_2) → κ
  66. Angle(A, E'_2, G'_1) → ξ
  67. Rotate(D'_3, ((s - 1) * ζ), A) → K
  68. Rotate(D'_3, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → L
  69. Rotate(E'_2, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → M
  70. Rotate(G'_1, If[1 < s ≤ 2, ((s - 1) * ζ), ζ], A) → N
  71. Polygon(L, M, N) → poly5, n_2, l_2, m_2
  72. Dilate(M, If[2 < s ≤ 3, 3 - s, 0], B) → M'
  73. Dilate(N, If[2 < s ≤ 3, 3 - s, 0], C) → N'
  74. Segment(M', N') → r_1
  75. Polygon(A, M', N') → poly6, n', a_3, m'
  76. Angle(A, M', N') → ο
  77. Angle(A, M, N) → ρ
  78. Segment(A, M) → t_1
  79. Segment(M, N) → b_2
  80. Segment(A, N) → c_2
  81. Angle(B, A, C) → σ
  82. Angle(G, E, D) → τ
  83. Angle(N, M, A) → υ
  84. Angle(N', M', A) → ϕ

AI Drawing Prompt Reference

Draw a dynamic geometric figure: start with a triangle, then construct a circle centered at an external point with a scaled radius, pick a point on the circle, and connect to form a second triangle. Apply rotations, dilations, and translations to create multiple similar triangle copies, linking corresponding vertices with segments. Add vectors to show translation directions, and use conditional angles to control rotation, making the figure animate smoothly. The composition includes triangles, circles, segments, vectors, and angle labels, illustrating composite similarity transformations.
The prompt above was compiled from the original drawing commands and is for reference only.