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Animation Demonstrating Geometric Transformations and Integration

Animation Demonstrating Geometric Transformations and Integration
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2017-04-03 09:48
TriangleIntegralGeometric Construction
This example demonstrates geometric transformations like rotations and transformations of triangles and ellipses through animation, while also illustrating the calculation of definite integrals and Riemann sums. Core mathematical concepts include angles, triangles, statistical characteristics, ellipses, definite integrals, prisms, and upper and lower sums. The construction utilizes commands such as Rotate, Angle, Segment, Ray, Point, LineBisector, Line, OrthogonalLine, Intersect, Dilate, Midpoint, Min, Circle, CircleSector, and Polygon in GeoGebra. It is suitable for high school and above, aiding in understanding geometric transformations and calculus concepts.

GeoGebra Commands & Parameters

  1. Rotate(A, (-((a * π / 180))), B) → A'
  2. Angle(A', B, A) → α
  3. Segment(A, B) → f
  4. Ray(B, A') → g
  5. Point(g) → C
  6. Segment(A, C) → h
  7. Segment(B, C) → i
  8. LineBisector(i) → j
  9. LineBisector(f) → k
  10. Line(A, B) → m
  11. OrthogonalLine(C, m) → n
  12. Intersect(m, n) → E
  13. Angle(E, C, A) → γ
  14. Angle(E, A, C) → ε
  15. Dilate(C, If[3 < o ≤ 4, 4 - o, 0], E) → C'_1
  16. Dilate(B, If[3 < o ≤ 4, 4 - o, 0], E) → B'
  17. Midpoint(A, B) → J
  18. Segment(A, E) → g_1
  19. Segment(A, J) → h_1
  20. Min(f, i) → u
  21. Circle(B, (0.2 * u)) → r_1
  22. Intersect(r_1, f) → K,
  23. Intersect(r_1, i) → L,
  24. CircleSector(B, L, K) → s_1
  25. Point(f) → M
  26. Segment(C, M) → p
  27. Line(C, M) → q
  28. OrthogonalLine(A, q) → r
  29. Intersect(q, r) → N
  30. Line(C, B) → s
  31. OrthogonalLine(A, s) → c
  32. Intersect(s, c) → O
  33. Line(N, O) → e
  34. OrthogonalLine(C, e) → i_1
  35. Intersect(e, i_1) → P
  36. Angle(A, C, M) → θ
  37. Angle(O, C, P) → ι
  38. Segment(C, P) → j_1
  39. Min({h, p, i, j_1}) → v
  40. Circle(C, (0.4 * v)) → c_1
  41. Intersect(c_1, h) → Q,
  42. Intersect(c_1, p) → R,
  43. CircleSector(C, Q, R) → d_1
  44. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], M) → C'_2
  45. Segment(C, C'_2) → t
  46. Angle(M, N, A) → κ_1
  47. Dilate(M, If[1 < o ≤ 2, 2 - o, 0], N) → M'
  48. Dilate(A, If[1 < o ≤ 2, 2 - o, 0], N) → A'_1
  49. Segment(M, M') → f_1
  50. Segment(A, A'_1) → k_1
  51. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], O) → A'_2
  52. Segment(A, A'_2) → l_1
  53. Dilate(N, If[3 < o ≤ 4, 4 - o, 0], O) → N'
  54. Segment(N, N') → m_1
  55. Dilate(O, If[4 < o ≤ 5, 5 - o, 0], P) → O'
  56. Dilate(C, If[4 < o ≤ 5, 5 - o, 0], P) → C'_3
  57. Angle(C, P, O) → ζ
  58. Segment(O, O') → n_1
  59. Segment(C, C'_3) → p_1
  60. Rotate(Q, If[5 < o ≤ 6, ((o - 5) * θ), θ], C) → Q'
  61. CircleSector(C, Q, Q') → e_1
  62. Angle(A, C, B) → κ
  63. Polygon(A, B, C) → poly1, c_2, a_1, b
  64. Rotate(d_1, If[6.5 < o ≤ 7.5, ((o - 6.5) * κ), κ], C) → q_1
  65. Circle(B, 1) → d
  66. Circle(C, 1) → t_1
  67. Intersect(d, s) → D, F
  68. Intersect(t_1, s) → G, H
  69. Ray(B, F) → l
  70. Ray(C, G) → b_1
  71. Angle(C, O, A) → β