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Animation Demonstrating Rhombus and Parametric Curves

Animation Demonstrating Rhombus and Parametric Curves
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2016-06-15 01:32
PolygonAngleOptimizationParametric EquationGeometric Construction
This example demonstrates the angle variations of a rhombus and the drawing of parametric curves through animation. Core mathematical concepts include angles, rhombus, parametric curves, tangent line equations, and optimization problems. Key construction elements involve GeoGebra commands and objects such as Circle, Line, Intersect, CircleArc, Point, Segment, Min, Dilate, Angle, Rotate, CircleSector, Polygon, Vector, and Translate. Suitable for high school and above, this example helps understand geometric transformations and the application of parametric equations.

GeoGebra Commands & Parameters

  1. Circle(B, A) → f
  2. Circle(A, B) → c
  3. Line(A, B) → b
  4. Intersect(c, b, 2) → C
  5. CircleArc(A, B, C) → d
  6. Point(d) → D
  7. Segment(D, B) → s
  8. Circle(D, A) → e
  9. Intersect(e, f, 2) → E
  10. Segment(A, E) → n
  11. Intersect(n, s) → L
  12. Segment(A, B) → a
  13. Segment(A, D) → g
  14. Segment(D, E) → h
  15. Segment(E, B) → i
  16. Min(g, h) → j
  17. Dilate(B, If[0 < l ≤ 1, 1 - l, 0], A) → B'
  18. Segment(B, B') → m
  19. Angle(E, B, A) → α
  20. Angle(D, E, B) → β
  21. Rotate(B, If[1 < l ≤ 2, ((-(l - 1)) * α), α], B) → B'_1
  22. Rotate(B', If[1 < l ≤ 2, ((-(l - 1)) * α), (-α)], B) → B''
  23. Segment(B'_1, B'') → m'
  24. Rotate(B'_1, If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → J
  25. Rotate(B'', If[2 < l ≤ 3, ((-(l - 2)) * β), (-β)], E) → B'''
  26. Segment(J, B''') → m''
  27. Rotate(J, If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → J'
  28. Rotate(B''', If[3 < l ≤ 4, ((-(l - 3)) * α), (-α)], D) → K
  29. Segment(J', K) → m'''
  30. Circle(B, (0.2 * j)) → k
  31. Circle(E, (0.2 * j)) → p
  32. Intersect(p, m'', 1) → F
  33. Intersect(p, m', 1) → G
  34. Intersect(k, m', 1) → H
  35. Intersect(k, m, 1) → I
  36. CircleSector(B, H, I) → q
  37. CircleSector(E, F, G) → r
  38. Polygon(A, B, E, D) → poly1, a_1, b_1, e_1, d_1
  39. Dilate(B, If[5 < l ≤ 6, 6 - l, 0], D) → B'_2
  40. Vector(B, B'_2) → u
  41. Translate(q, u) → q'
  42. Rotate(q', If[6 < l ≤ 7, ((l - 6) * π), 3.141592653589793], D) → q''
  43. Dilate(E, If[7 < l ≤ 8, 8 - l, 0], A) → E'
  44. Vector(E, E') → v
  45. Translate(r, v) → r'
  46. Rotate(r', If[8 < l ≤ 9, ((-(l - 8)) * π), -3.141592653589793], A) → r''
  47. Dilate(D, If[9 < l ≤ 10, 10 - l, 0], L) → D'
  48. Vector(D, D') → w
  49. Min(n / 2, s / 2) → o
  50. Circle(L, (0.2 * o)) → t
  51. Intersect(t, s, 2) → M
  52. Intersect(t, n, 2) → N
  53. CircleSector(L, M, N) → c_1
  54. Segment(D, D') → f_1
  55. Dilate(L, If[10 < l ≤ 11, 11 - l, 0], B) → L'
  56. Circle(D, (0.2 * j)) → h_1
  57. Intersect(k, s) → P,
  58. Intersect(s, f_1) → Q
  59. Intersect(h_1, s, 1) → O
  60. Circle(A, (0.2 * j)) → k_1
  61. Intersect(k_1, m, 1) → R
  62. Intersect(k_1, n, 1) → S
  63. Intersect(k_1, m''', 1) → T
  64. Intersect(h_1, m'', 1) → U
  65. CircleSector(D, O, U) → q_1
  66. Dilate(A, If[11 < l ≤ 12, 12 - l, 0], L) → A'
  67. Segment(A, A') → i_1
  68. Dilate(L, If[12 < l ≤ 13, 13 - l, 0], E) → L'_1
  69. Segment(L, L'_1) → j_1
  70. Intersect(p, m'', 1) → V
  71. Intersect(p, n, 1) → W
  72. CircleSector(A, S, T) → s_1
  73. CircleSector(B, P, I) → p_1
  74. Intersect(p, m', 1) → Z
  75. CircleSector(E, W, Z) → r_1
  76. Segment(L, L') → g_1
  77. Rotate(L, If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L'_2
  78. Rotate(L', If[13 < l ≤ 14, ((-(l - 13)) * π), -3.141592653589793], L) → L''
  79. Segment(L'_2, L'') → l_1
  80. Rotate(L, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → L'_3
  81. Rotate(L'_1, If[14 < l ≤ 15, ((-(l - 14)) * π), 3.141592653589793], L) → A_1
  82. Segment(L'_3, A_1) → m_1
  83. Rotate(q_1, If[15 < l ≤ 16, ((-(l - 15)) * α / 2), ((-α)) / 2], D) → t_1
  84. Rotate(p_1, If[15 < l ≤ 16, ((-(l - 15)) * α / 2), ((-α)) / 2], B) → c_2
  85. Rotate(s_1, If[16 < l ≤ 17, ((-(l - 16)) * β / 2), ((-β)) / 2], A) → d_2
  86. Rotate(r_1, If[16 < l ≤ 17, ((-(l - 16)) * β / 2), ((-β)) / 2], E) → e_2
  87. Rotate(c_1, If[17 < l ≤ 18, ((l - 17) * π / 2), 1.5707963267948966], L) → f_2

AI Drawing Prompt Reference

Draw a rhombus and demonstrate its angle variations and the drawing of parametric curves through animation. Start by drawing two intersecting circles, finding their intersection points and arcs, then connect these points to form the rhombus. Next, draw parametric curves and tangent line equations, showing the solution to optimization problems. Finally, use rotations and scalings to illustrate geometric transformations and the application of parametric equations.
The prompt above was compiled from the original drawing commands and is for reference only.