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Animation 110: Quadrilateral Transformation

Animation 110: Quadrilateral Transformation
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2016-09-07 03:40
PolygonTransformationGeometric Construction
This example demonstrates the transformation process of a quadrilateral, including translation, rotation, and dilation. Key mathematical concepts include geometric transformations, symmetry, and continuity. Using GeoGebra commands such as Circle, Point, Polygon, and Dilate, the example dynamically illustrates the properties and changes of the quadrilateral. Suitable for students at the junior high school level and above to learn about geometric transformations and symmetry.

GeoGebra Commands & Parameters

  1. Circle(A, b) → d
  2. Point(d) → B
  3. Circle(B, c) → e
  4. Point(e) → C
  5. Polygon(A, B, 4) → poly1, f, g, h, i, D, E
  6. Polygon(C, B, 4) → poly2, j, k, l, m, F, G
  7. Midpoint(D, A) → H
  8. Midpoint(F, C) → I
  9. Midpoint(D, C) → J
  10. Midpoint(A, F) → K
  11. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], A) → B'
  12. Dilate(B, If[0 < a ≤ 1, 1 - a, 0], F) → B'_1
  13. Segment(B, B') → n
  14. Segment(B, B'_1) → p
  15. Rotate(B, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → B'_2
  16. Rotate(B'_1, If[1 < a ≤ 2, ((-(a - 1)) * π / 2), ((-π)) / 2], F) → L
  17. Segment(B'_2, L) → p'
  18. Rotate(B, If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B'_3
  19. Rotate(B', If[1 < a ≤ 2, ((a - 1) * π / 2), π / 2], A) → B''
  20. Segment(B'_3, B'') → n'
  21. Rotate(B'_2, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → M
  22. Rotate(L, If[2 < a ≤ 3, ((-(a - 2)) * π / 2), ((-π)) / 2], G) → L'
  23. Segment(M, L') → p''
  24. Rotate(B'_3, If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → N
  25. Rotate(B'', If[2 < a ≤ 3, ((a - 2) * π / 2), π / 2], E) → B'''
  26. Segment(N, B''') → n''
  27. Rotate(N, If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → N'
  28. Rotate(B''', If[3 < a ≤ 4, ((a - 3) * π / 2), π / 2], D) → O
  29. Segment(N', O) → n'''
  30. Rotate(M, If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → M'
  31. Rotate(L', If[3 < a ≤ 4, ((-(a - 3)) * π / 2), ((-π)) / 2], C) → L''
  32. Segment(M', L'') → p'''
  33. Segment(F, G) → q
  34. Segment(F, B) → r
  35. Segment(A, B) → s
  36. Min(r, s) → o
  37. Circle(F, (0.15 * o)) → t
  38. Intersect(l, q) → P
  39. Intersect(t, q) → Q,
  40. Intersect(t, r) → R,
  41. CircleSector(F, Q, R) → c_1
  42. Dilate(F, If[4 < a ≤ 5, 5 - a, 0], C) → F'
  43. Vector(F, F') → u
  44. Translate(c_1, u) → d_1
  45. Rotate(d_1, If[5 < a ≤ 6, ((a - 5) * π), π], C) → e_1
  46. Dilate(F, If[6 < a ≤ 7, 7 - a, 0], A) → F'_1
  47. Dilate(C, If[6 < a ≤ 7, 7 - a, 0], E) → C'
  48. Vector(F, F'_1) → v
  49. Vector(C, C') → w
  50. Translate(e_1, w) → f_1
  51. Translate(c_1, v) → g_1
  52. Line(A, p) → h_1
  53. Line(E, p''') → i_1
  54. Point(h_1) → S
  55. Angle(S, A, E) → α
  56. Rotate(g_1, If[7 < a ≤ 8, ((-(a - 7)) * ((2 * π) - α)), (-((2 * π) - α))], A) → k_1
  57. Line(E, h_1) → j_1
  58. Point(j_1) → T
  59. Angle(D, E, T) → β
  60. Rotate(f_1, If[7 < a ≤ 8, ((-(a - 7)) * β), (-β)], E) → p_1
  61. Dilate(E, If[8 < a ≤ 9, 9 - a, 0], G) → E'
  62. Dilate(F, If[8 < a ≤ 9, 9 - a, 0], D) → F'_2
  63. Vector(E, E') → u_1
  64. Vector(F, F'_2) → v_1
  65. Translate(p_1, u_1) → q_1
  66. Translate(c_1, v_1) → r_1
  67. Line(G, n'') → l_1
  68. Point(l_1) → U
  69. Line(G, n') → m_1
  70. Point(m_1) → V
  71. Angle(C, G, V) → γ
  72. Rotate(q_1, If[8 < a ≤ 9, ((-(a - 8)) * γ), (-γ)], E') → s_1
  73. Line(D, p) → n_1
  74. Point(n_1) → W
  75. Angle(W, D, B) → δ
  76. Rotate(r_1, If[8 < a ≤ 9, ((a - 8) * δ), δ], F'_2) → t_1
  77. Midpoint(A, B) → Z
  78. Midpoint(B, C) → A_1
  79. Midpoint(Z, B) → B_1
  80. Midpoint(A_1, B) → C_1
  81. Dilate(C, If[9 < a ≤ 10, 10 - a, 0], C_1) → C'_1
  82. Dilate(A, If[9 < a ≤ 10, 10 - a, 0], B_1) → A'
  83. Vector(A, A') → w_1
  84. Vector(C, C'_1) → a_1
  85. Translate(e_1, a_1) → c_2
  86. Translate(k_1, w_1) → d_2
  87. Rotate(d_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], B_1) → e_2
  88. Rotate(c_2, If[10 < a ≤ 11, ((a - 10) * π / 2), π / 2], C_1) → f_2
  89. Dilate(C_1, If[11 < a ≤ 12, 12 - a, 0], B) → D_1
  90. Dilate(B_1, If[11 < a ≤ 12, 12 - a, 0], B) → E_1
  91. Vector(B_1, E_1) → b_1
  92. Vector(C_1, D_1) → u_2
  93. Translate(e_2, b_1) → g_2
  94. Translate(f_2, u_2) → h_2
  95. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], I) → F'_3
  96. Segment(F, F'_3) → i_2
  97. Dilate(D, If[12 < a ≤ 13, 13 - a, 0], H) → D'
  98. Segment(D, D') → j_2
  99. Rotate(F, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F'_4
  100. Rotate(F'_3, If[13 < a ≤ 14, ((a - 13) * π), π], I) → F_1
  101. Segment(F'_4, F_1) → k_2
  102. Rotate(D, If[13 < a ≤ 14, ((a - 13) * π), π], H) → D'_1
  103. Rotate(D', If[13 < a ≤ 14, ((a - 13) * π), π], H) → D''
  104. Segment(D'_1, D'') → l_2
  105. Dilate(A, If[14 < a ≤ 15, 15 - a, 0], K) → A'_1
  106. Dilate(C, If[14 < a ≤ 15, 15 - a, 0], J) → C'_2
  107. Segment(A, A'_1) → m_2
  108. Segment(C, C'_2) → n_2
  109. Dilate(K, If[15 < a ≤ 16, 16 - a, 0], F) → K'
  110. Dilate(J, If[15 < a ≤ 16, 16 - a, 0], D) → J'
  111. Segment(K, K') → p_2
  112. Segment(J, J') → q_2
  113. Rotate(A, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], K) → A'_2
  114. Rotate(A'_1, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], K) → G_1
  115. Segment(A'_2, G_1) → r_2
  116. Rotate(C, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], J) → C'_3
  117. Rotate(C'_2, If[16 < a ≤ 17, ((-(a - 16)) * π), (-π)], J) → H_1
  118. Segment(C'_3, H_1) → s_2
  119. Dilate(K, If[17 < a ≤ 18, 18 - a, 0], I) → K'_1
  120. Polygon(K, I, J, H) → poly3, k_3, i_3, j_3, h_3
  121. Segment(K, K'_1) → t_2
  122. Rotate(K, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], K) → K'_2
  123. Rotate(K'_1, If[18 < a ≤ 19, ((a - 18) * π / 2), π / 2], K) → I_1
  124. Segment(K'_2, I_1) → a_2
  125. Rotate(K'_2, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], H) → J_1
  126. Rotate(I_1, If[19 < a ≤ 20, ((a - 19) * π / 2), π / 2], H) → K_1
  127. Segment(J_1, K_1) → b_2
  128. Rotate(J_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], J) → L_1
  129. Rotate(K_1, If[20 < a ≤ 21, ((a - 20) * π / 2), π / 2], J) → M_1
  130. Segment(L_1, M_1) → f_3
  131. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], I) → B'_4
  132. Dilate(B, If[21 < a ≤ 22, 22 - a, 0], H) → B'_5
  133. Vector(B, B'_4) → v_2
  134. Vector(B, B'_5) → w_2
  135. Translate(h_2, v_2) → c_3
  136. Translate(g_2, w_2) → d_3
  137. Line(I, p) → g_3
  138. Point(g_3) → N_1
  139. Angle(N_1, I, J) → ε
  140. Rotate(c_3, If[22 < a ≤ 23, ((a - 22) * ε), ε], I) → e_3
  141. Line(H, n''') → l_3
  142. Point(l_3) → O_1
  143. Angle(K, H, O_1) → ζ
  144. Rotate(d_3, If[22 < a ≤ 23, ((-(a - 22)) * ζ), (-ζ)], H) → p_3
  145. Dilate(F, If[23 < a ≤ 24, 24 - a, 0], K) → F'_5
  146. Dilate(D, If[23 < a ≤ 24, 24 - a, 0], J) → D'_2
  147. Vector(D, D'_2) → u_3
  148. Vector(F, F'_5) → v_3
  149. Translate(c_1, v_3) → q_3
  150. Translate(t_1, u_3) → r_3
  151. Line(D, f_3) → m_3
  152. Point(m_3) → P_1
  153. Angle(B, D, P_1) → η
  154. Rotate(r_3, If[23 < a ≤ 24, ((a - 23) * η), η], D'_2) → s_3
  155. Line(K, p) → n_3
  156. Point(n_3) → Q_1
  157. Angle(Q_1, K, I) → θ
  158. Rotate(q_3, If[24 < a ≤ 25, ((a - 24) * (θ + π / 2)), θ + π / 2], K) → t_3

AI Drawing Prompt Reference

Draw a quadrilateral and then perform transformations such as translation, rotation, and dilation around a point. Start by drawing a circle and taking a point from it as one vertex of the quadrilateral, then draw another circle and take a point from it as another vertex. Next, draw two quadrilaterals with these points as vertices, then find their midpoints. After that, perform scaling and rotation transformations around a point to generate a series of new points and segments. Finally, draw some segments and arcs to complete the dynamic transformation process of the quadrilateral.
The prompt above was compiled from the original drawing commands and is for reference only.