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Animation 287: Transformation Between Circle and Square

Animation 287: Transformation Between Circle and Square
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2017-06-11 09:15
CirclePolygonVectorTransformationGeometric Construction
This example demonstrates the transformation relationship between a circle and a square. Key mathematical concepts include circles, squares, vectors, and transformations. Using GeoGebra commands such as Polygon, LineBisector, Intersect, etc., it shows how to achieve the transformation of shapes through various operations. Suitable for high school mathematics education, particularly in coordinate geometry and transformation geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. LineBisector(A, B) → j
  3. LineBisector(C, B) → k
  4. Intersect(k, j) → E
  5. Intersect(j, h) → F
  6. Circle(E, F) → c
  7. Intersect(g, k) → G
  8. Segment(B, F) → l
  9. Segment(A, G) → m
  10. Intersect(c, l) → H, I
  11. Segment(G, I) → n
  12. Intersect(l, m) → J
  13. Segment(G, J) → p
  14. Segment(J, I) → q
  15. Segment(D, A) → r
  16. Segment(C, B) → s
  17. Segment(B, A) → t
  18. Segment(I, B) → d
  19. Intersect(k, l) → K
  20. Segment(J, K) → e
  21. Line(E, l) → f_1
  22. Intersect(f_1, t) → L
  23. Intersect(f_1, m) → M
  24. Segment(K, G) → h_1
  25. Segment(E, K) → i_1
  26. Segment(D, C) → j_1
  27. Intersect(j, m) → N
  28. Intersect(c, r) → O, S
  29. Dilate(A, If[0 < u ≤ 1, 1 - u, 0], G) → A'
  30. Segment(A, A') → g_1
  31. Dilate(O, If[0 < u ≤ 1, 1 - u, 0], G) → O'
  32. Segment(O, O') → k_1
  33. Dilate(F, If[1 < u ≤ 2, 2 - u, 0], K) → F'
  34. Segment(F, F') → l_1
  35. Dilate(K, If[2 < u ≤ 3, 3 - u, 0], J) → K'
  36. Segment(K, K') → m_1
  37. Midpoint(J, I) → P
  38. Intersect(c, t) → Q, T
  39. Dilate(F, If[0 < u ≤ 1, 1 - u, 0], Q) → F'_1
  40. Segment(F, F'_1) → n_1
  41. Dilate(J, If[3 < u ≤ 4, 4 - u, 0], P) → J'
  42. Segment(J, J') → p_1
  43. Dilate(P, If[4 < u ≤ 5, 5 - u, 0], I) → P'
  44. Segment(P, P') → q_1
  45. Dilate(G, If[4 < u ≤ 5, 5 - u, 0], I) → G'
  46. Segment(G, G') → r_1
  47. Dilate(I, If[5 < u ≤ 6, 6 - u, 0], B) → I'
  48. Polygon(I, J, G) → poly2, g_2, i_2, j_2
  49. Polygon(K, J, G) → poly3, g_3, k_2, j_3
  50. Rotate(K, If[7 < u ≤ 8, ((u - 7) * π), π], J) → K'_1
  51. Rotate(K', If[7 < u ≤ 8, ((u - 7) * π), π], J) → K''
  52. Segment(K'_1, K'') → t_1
  53. Rotate(K'_1, If[8 < u ≤ 9, ((u - 8) * π), π], P) → R
  54. Rotate(K'', If[8 < u ≤ 9, ((u - 8) * π), π], P) → K'''
  55. Segment(R, K''') → a_1
  56. Dilate(J, If[10 < u ≤ 11, 11 - u, 0], I) → J'_1
  57. Midpoint(I, B) → U
  58. Dilate(I, If[5 < u ≤ 6, 6 - u, 0], U) → I'_1
  59. Segment(I, I'_1) → s_1
  60. Dilate(U, If[6 < u ≤ 7, 7 - u, 0], B) → U'
  61. Segment(U, U') → b_1
  62. Polygon(I, B, G) → poly4, g_4, i_3, b_2
  63. Rotate(R, If[9 < u ≤ 10, ((u - 9) * π), π], I) → R'
  64. Rotate(K''', If[9 < u ≤ 10, ((u - 9) * π), π], I) → V
  65. Segment(R', V) → c_1
  66. Rotate(R', If[10 < u ≤ 11, ((u - 10) * π), π], U) → R''
  67. Rotate(V, If[10 < u ≤ 11, ((u - 10) * π), π], U) → V'
  68. Segment(R'', V') → d_1
  69. Midpoint(K, B) → W
  70. Rotate(K, If[12 < u ≤ 13, ((u - 12) * π), π], W) → K'_2
  71. Rotate(J, If[12 < u ≤ 13, ((u - 12) * π), π], W) → J'_2
  72. Rotate(G, If[12 < u ≤ 13, ((u - 12) * π), π], W) → G'_1
  73. Polygon(K'_2, J'_2, G'_1) → poly3', g'_1, k'_2, j'_2
  74. Rotate(I, If[12 < u ≤ 13, ((u - 12) * π), π], W) → I'_2
  75. Rotate(J, If[12 < u ≤ 13, ((u - 12) * π), π], W) → J'_3
  76. Rotate(G, If[12 < u ≤ 13, ((u - 12) * π), π], W) → G'_2
  77. Polygon(I'_2, J'_3, G'_2) → poly2', g'_2, i'_2, j'_3
  78. Rotate(I, If[12 < u ≤ 13, ((u - 12) * π), π], W) → I'_3
  79. Rotate(B, If[12 < u ≤ 13, ((u - 12) * π), π], W) → B'
  80. Rotate(G, If[12 < u ≤ 13, ((u - 12) * π), π], W) → G'_3
  81. Polygon(I'_3, B', G'_3) → poly4', g'_3, i'_3, b'
  82. Dilate(E, If[16 < u ≤ 17, 17 - u, 0], D) → K'_3
  83. Vector(E, K'_3) → v
  84. Dilate(L, If[13 < u ≤ 14, 14 - u, 0], Q) → L'
  85. Vector(L, L') → w
  86. Translate(I'_3, w) → Z
  87. Translate(B', w) → B''
  88. Translate(G'_3, w) → A_1
  89. Polygon(Z, B'', A_1) → poly4'', a_2, z_1, b''
  90. Translate(I'_2, w) → B_1
  91. Translate(J'_3, w) → C_1
  92. Translate(G'_2, w) → D_1
  93. Polygon(B_1, C_1, D_1) → poly2'', d_2, b_3, c_2
  94. Translate(K'_2, w) → E_1
  95. Translate(J'_2, w) → F_1
  96. Translate(G'_1, w) → G_1
  97. Polygon(E_1, F_1, G_1) → poly3'', g_5, e_1, f_2
  98. Midpoint(E, L) → H_1
  99. Rotate(B_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → I_1
  100. Rotate(C_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → J_1
  101. Rotate(D_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → K_1
  102. Polygon(I_1, J_1, K_1) → poly2''', k_3, i_4, j_4
  103. Rotate(Z, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → Z'
  104. Rotate(B'', If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → B'''
  105. Rotate(A_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → L_1
  106. Polygon(Z', B''', L_1) → poly4''', l_2, z', b'''
  107. Rotate(E_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → M_1
  108. Rotate(F_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → N_1
  109. Rotate(G_1, If[14 < u ≤ 15, ((u - 14) * π), π], H_1) → O_1
  110. Polygon(M_1, N_1, O_1) → poly3''', o_1, m_2, n_2
  111. Dilate(Q, If[16 < u ≤ 17, 17 - u, 0], A) → Q'
  112. Vector(Q, Q') → u_1
  113. Vector(P_1, Q_1) → v_1
  114. Dilate(E, If[16 < u ≤ 17, 17 - u, 0], F) → E'
  115. Vector(E, E') → w_1
  116. Translate(Z, u_1) → Z'_1
  117. Translate(B'', u_1) → B'''_1
  118. Translate(A_1, u_1) → R_1
  119. Polygon(Z'_1, B'''_1, R_1) → poly4'''_1, r_2, z'_1, b'''_1
  120. Translate(Z, w_1) → Z'_2
  121. Translate(B'', w_1) → B'''_2
  122. Translate(A_1, w_1) → S_1
  123. Polygon(Z'_2, B'''_2, S_1) → poly4'''_2, s_2, z'_2, b'''_2
  124. Translate(Z, v) → Z'_3
  125. Translate(B'', v) → B'''_3
  126. Translate(A_1, v) → T_1
  127. Polygon(Z'_3, B'''_3, T_1) → poly4'''_3, t_2, z'_3, b'''_3
  128. Translate(B_1, v) → U_1
  129. Translate(C_1, v) → V_1
  130. Translate(D_1, v) → W_1
  131. Polygon(U_1, V_1, W_1) → poly2'''_1, w_2, u_2, v_2
  132. Translate(B_1, u_1) → Z_1
  133. Translate(C_1, u_1) → A_2
  134. Translate(D_1, u_1) → B_2
  135. Polygon(Z_1, A_2, B_2) → poly2'''_2, b_4, z_2, a_3
  136. Translate(B_1, w_1) → C_2
  137. Translate(C_1, w_1) → D_2
  138. Translate(D_1, w_1) → E_2
  139. Polygon(C_2, D_2, E_2) → poly2'''_3, e_2, c_3, d_3
  140. Translate(E_1, u_1) → F_2
  141. Translate(F_1, u_1) → G_2
  142. Translate(G_1, u_1) → H_2
  143. Polygon(F_2, G_2, H_2) → poly3'''_1, h_2, f_3, g_6
  144. Translate(E_1, v) → I_2
  145. Translate(F_1, v) → J_2
  146. Translate(G_1, v) → K_2
  147. Polygon(I_2, J_2, K_2) → poly3'''_2, k_4, i_5, j_5
  148. Translate(E_1, w_1) → L_2
  149. Translate(F_1, w_1) → M_2
  150. Translate(G_1, w_1) → N_2
  151. Polygon(L_2, M_2, N_2) → poly3'''_3, n_3, l_3, m_3
  152. Translate(Z', u_1) → Z''
  153. Translate(B''', u_1) → O_2
  154. Translate(L_1, u_1) → P_2
  155. Polygon(Z'', O_2, P_2) → poly5, p_2, z'', o_2
  156. Translate(I_1, u_1) → Q_2
  157. Translate(J_1, u_1) → R_2
  158. Translate(K_1, u_1) → S_2
  159. Polygon(Q_2, R_2, S_2) → poly6, s_3, q_2, r_3
  160. Translate(M_1, u_1) → T_2
  161. Translate(N_1, u_1) → U_2
  162. Translate(O_1, u_1) → V_2
  163. Polygon(T_2, U_2, V_2) → poly7, v_3, t_3, u_3
  164. Translate(I'_3, u_1) → W_2
  165. Translate(B', u_1) → B''_1
  166. Translate(G'_3, u_1) → Z_2
  167. Polygon(W_2, B''_1, Z_2) → poly4''_1, z_3, w_3, b''_1
  168. Translate(K, u_1) → K'_4
  169. Translate(J, u_1) → J'_4
  170. Translate(G, u_1) → G'_4
  171. Polygon(K'_4, J'_4, G'_4) → poly3'_1, g'_4, k'_4, j'_4
  172. Translate(I, u_1) → I'_4
  173. Translate(J, u_1) → J'_5
  174. Translate(G, u_1) → G'_5
  175. Polygon(I'_4, J'_5, G'_5) → poly2'_1, g'_5, i'_4, j'_5
  176. Translate(I'_2, u_1) → A_3
  177. Translate(J'_3, u_1) → B_3
  178. Translate(G'_2, u_1) → C_3
  179. Polygon(A_3, B_3, C_3) → poly2''_1, c_4, a_4, b_5
  180. Translate(K'_2, u_1) → D_3
  181. Translate(J'_2, u_1) → E_3
  182. Translate(G'_1, u_1) → F_3
  183. Polygon(D_3, E_3, F_3) → poly3''_1, f_4, d_4, e_3
  184. Translate(I, u_1) → I'_5
  185. Translate(B, u_1) → B'_1
  186. Translate(G, u_1) → G'_6
  187. Polygon(I'_5, B'_1, G'_6) → poly4'_1, g'_6, i'_5, b'_1
  188. Translate(M_1, v) → G_3
  189. Translate(N_1, v) → H_3
  190. Translate(O_1, v) → I_3
  191. Polygon(G_3, H_3, I_3) → poly8, i_6, g_7, h_3
  192. Translate(I_1, v) → J_3
  193. Translate(J_1, v) → K_3
  194. Translate(K_1, v) → L_3
  195. Polygon(J_3, K_3, L_3) → poly9, l_4, j_6, k_5
  196. Translate(Z', v) → Z''_1
  197. Translate(B''', v) → M_3
  198. Translate(L_1, v) → N_3
  199. Polygon(Z''_1, M_3, N_3) → poly10, n_4, z''_1, m_4
  200. Translate(K, v) → K'_5
  201. Translate(J, v) → J'_6
  202. Translate(G, v) → G'_7
  203. Polygon(K'_5, J'_6, G'_7) → poly3'_2, g'_7, k'_5, j'_6
  204. Translate(I, v) → I'_6
  205. Translate(J, v) → J'_7
  206. Translate(G, v) → G'_8
  207. Polygon(I'_6, J'_7, G'_8) → poly2'_2, g'_8, i'_6, j'_7
  208. Translate(I, v) → I'_7
  209. Translate(B, v) → B'_2
  210. Translate(G, v) → G'_9
  211. Polygon(I'_7, B'_2, G'_9) → poly4'_2, g'_9, i'_7, b'_2
  212. Translate(I'_3, v) → O_3
  213. Translate(B', v) → B''_2
  214. Translate(G'_3, v) → P_3
  215. Polygon(O_3, B''_2, P_3) → poly4''_2, p_3, o_3, b''_2
  216. Translate(I'_2, v) → Q_3
  217. Translate(J'_3, v) → R_3
  218. Translate(G'_2, v) → S_3
  219. Polygon(Q_3, R_3, S_3) → poly2''_2, s_4, q_3, r_4
  220. Translate(K'_2, v) → T_3
  221. Translate(J'_2, v) → U_3
  222. Translate(G'_1, v) → V_3
  223. Polygon(T_3, U_3, V_3) → poly3''_2, v_4, t_4, u_4
  224. Translate(M_1, w_1) → W_3
  225. Translate(N_1, w_1) → Z_3
  226. Translate(O_1, w_1) → A_4
  227. Polygon(W_3, Z_3, A_4) → poly11, a_5, w_4, z_4
  228. Translate(I_1, w_1) → B_4
  229. Translate(J_1, w_1) → C_4
  230. Translate(K_1, w_1) → D_4
  231. Polygon(B_4, C_4, D_4) → poly12, d_5, b_6, c_5
  232. Translate(Z', w_1) → Z''_2
  233. Translate(B''', w_1) → E_4
  234. Translate(L_1, w_1) → F_4
  235. Polygon(Z''_2, E_4, F_4) → poly13, f_5, z''_2, e_4
  236. Translate(I'_3, w_1) → G_4
  237. Translate(B', w_1) → B''_3
  238. Translate(G'_3, w_1) → H_4
  239. Polygon(G_4, B''_3, H_4) → poly4''_3, h_4, g_8, b''_3
  240. Translate(K, w_1) → K'_6
  241. Translate(J, w_1) → J'_8
  242. Translate(G, w_1) → G'_{10}
  243. Polygon(K'_6, J'_8, G'_{10}) → poly3'_3, g'_{10}, k'_6, j'_8
  244. Translate(I, w_1) → I'_8
  245. Translate(J, w_1) → J'_9
  246. Translate(G, w_1) → G'_{11}
  247. Polygon(I'_8, J'_9, G'_{11}) → poly2'_3, g'_{11}, i'_8, j'_9
  248. Translate(I'_2, w_1) → I_4
  249. Translate(J'_3, w_1) → J_4
  250. Translate(G'_2, w_1) → K_4
  251. Polygon(I_4, J_4, K_4) → poly2''_3, k_6, i_7, j_7
  252. Translate(K'_2, w_1) → L_4
  253. Translate(J'_2, w_1) → M_4
  254. Translate(G'_1, w_1) → N_4
  255. Polygon(L_4, M_4, N_4) → poly3''_3, n_5, l_5, m_5
  256. Translate(I, w_1) → I'_9
  257. Translate(B, w_1) → B'_3
  258. Translate(G, w_1) → G'_{12}
  259. Polygon(I'_9, B'_3, G'_{12}) → poly4'_3, g'_{12}, i'_9, b'_3
  260. Intersect(k_1, r) → O_4
  261. Intersect(n_1, j_1) → P_4
  262. Intersect(k_1, s) → Q_4
  263. Intersect(n_1, t) → R_4
  264. Intersect(c, r_1) → S_4, T_4
  265. Line(P_4, B) → p_4
  266. Intersect(g_1, p_4) → U_4

AI Drawing Prompt Reference

Draw a square with a circle centered at the intersection of perpendicular bisectors of two adjacent sides, passing through a point on one side. Add auxiliary segments connecting vertices, side points, and circle intersections to form internal triangles. Animate key points (vertices, center, intersections) sliding along specific segments toward target points using a slider, while keeping connecting lines, to show a continuous transformation between the circle and the square. The final figure includes the square, the circle, multiple line segments, and moving points forming a dynamic geometric construction.
The prompt above was compiled from the original drawing commands and is for reference only.