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Rolling Bridge

Rolling Bridge
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2021-12-31 19:16
Solid GeometryTransformationGeometric Construction
This example demonstrates the rolling process of a bridge. It involves concepts of geometric transformations and loci. Using GeoGebra commands such as Rotate, Intersect, and Polygon, the dynamic path of the bridge's rolling can be shown. Suitable for high school geometry and coordinate geometry courses.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Line(B, yAxis) → k
  4. Line(A, k) → l
  5. Rotate(B, (-α), l) → B'
  6. Line(B', l) → m
  7. Rotate(A, α, m) → A'
  8. Line(A', m) → n
  9. Rotate(B', α, n) → B''
  10. Line(B'', n) → p
  11. Rotate(A', α, p) → A''
  12. Line(A'', p) → q
  13. Rotate(B'', α, q) → B'''
  14. Line(B''', q) → r
  15. Rotate(A'', α, r) → A'''
  16. Line(A''', r) → s
  17. Rotate(B''', α, s) → E
  18. Intersect(k, g) → C
  19. Intersect(l, g) → D
  20. Intersect(m, g) → F
  21. Intersect(n, g) → G
  22. Intersect(p, q) → H
  23. Intersect(r, g) → I
  24. Intersect(p, g) → J
  25. Intersect(q, g) → K
  26. Intersect(s, g) → L
  27. Line(E, s) → h
  28. Intersect(h, g) → M
  29. Polygon(C, B, A, D) → q1, c, b, a, d
  30. Polygon(D, F, B', A) → q2, d_1, f_1, b', a_1
  31. Polygon(F, G, A', B') → q3, f_2, g_1, a', i
  32. Polygon(G, J, B'', A') → q4, g_2, j, b'', t
  33. Polygon(J, K, A'', B'') → q5, j_1, k_1, a'', e
  34. Polygon(K, I, B''', A'') → q6, k_2, i_1, b''', h_1
  35. Polygon(I, L, A''', B''') → q7, i_2, l_1, a''', m_1
  36. Polygon(L, M, E, A''') → q8, l_2, m_2, e_1, n_1
  37. Rotate(A, 67.5°, k) → A'_{1}
  38. Rotate(B, -67.5°, l) → B'_{1}
  39. Ray(A, B'_{1}) → q_1
  40. Rotate(B', 65°, l) → B''_{1}
  41. Ray(A, B''_{1}) → s_1
  42. Rotate(A, 67.5°, m) → A'_{2}
  43. Ray(B', A'_{2}) → t_1
  44. Rotate(A', -67.5°, m) → A''_{1}
  45. Rotate(A', -67.5°, m) → A''_{2}
  46. Rotate(B', 67.5°, n) → B''_{2}
  47. Rotate(B'', -67.5°, n) → B'''_{1}
  48. Ray(B', A''_{2}) → b_1
  49. Ray(A', B''_{2}) → c_1
  50. Ray(A', B'''_{1}) → h_2
  51. Rotate(A', 67.5°, p) → A''_{3}
  52. Ray(B'', A''_{3}) → j_2
  53. Rotate(A'', -67.5°, p) → A'''_{1}
  54. Rotate(B''', -67.5°, q) → R
  55. Rotate(A''', -67.5°, r) → S
  56. Rotate(E, -67.5°, s) → E'
  57. Ray(B'', A'''_{1}) → n_2
  58. Ray(A'', R) → q_2
  59. Ray(B''', S) → s_2
  60. Ray(A''', E') → a_2
  61. Rotate(B'', 67.5°, q) → B'''_{2}
  62. Ray(A'', B'''_{2}) → r_1
  63. Rotate(A'', 67.5°, r) → A'''_{2}
  64. Ray(B''', A'''_{2}) → p_2
  65. Rotate(B''', 67.5°, s) → O
  66. Ray(A''', O) → r_2
  67. Rotate(A''', 67.5°, h) → P
  68. Ray(E, P) → t_2
  69. Ray(B, A'_{1}) → p_1
  70. Sphere(B, sradius) → o
  71. Intersect(p_1, o) → N,
  72. Sphere(A, sradius) → u
  73. Intersect(q_1, u) → Q,
  74. Intersect(s_1, u) → T,
  75. Sphere(B', sradius) → v
  76. Sphere(A', sradius) → w
  77. Sphere(B'', sradius) → o_1
  78. Sphere(A'', sradius) → u_1
  79. Sphere(B''', sradius) → v_1
  80. Sphere(A''', sradius) → w_1
  81. Sphere(E, sradius) → b_2
  82. Intersect(t_1, v) → U,
  83. Intersect(b_1, v) → V,
  84. Intersect(c_1, w) → W,
  85. Intersect(h_2, w) → Z,
  86. Intersect(j_2, o_1) → A_1,
  87. Intersect(n_2, o_1) → B_1,
  88. Intersect(r_1, u_1) → C_1,
  89. Intersect(q_2, u_1) → D_1,
  90. Intersect(p_2, v_1) → E_1,
  91. Intersect(s_2, v_1) → F_1,
  92. Intersect(r_2, w_1) → G_1,
  93. Intersect(a_2, w_1) → H_1,
  94. Intersect(t_2, b_2) → I_1,
  95. Midpoint(H_1, G_1) → J_1
  96. Midpoint(F_1, E_1) → K_1
  97. Midpoint(D_1, C_1) → L_1
  98. Midpoint(B_1, A_1) → M_1
  99. Midpoint(Z, W) → N_1
  100. Midpoint(V, U) → O_1
  101. Midpoint(T, Q) → P_1
  102. Segment(I_1, E) → j_3
  103. Ray(A, P_1) → c_2
  104. Ray(A', N_1) → e_2
  105. Ray(B'', M_1) → f_3
  106. Ray(A'', L_1) → g_3
  107. Ray(B''', K_1) → h_3
  108. Ray(A''', J_1) → i_3
  109. Ray(B', O_1) → d_2
  110. Sphere(P_1, adjustment) → o_2
  111. Intersect(c_2, o_2) → Q_1, R_1
  112. Sphere(O_1, adjustment) → u_2
  113. Sphere(N_1, adjustment) → v_2
  114. Sphere(M_1, adjustment) → w_2
  115. Sphere(L_1, adjustment) → a_3
  116. Sphere(K_1, adjustment) → b_3
  117. Sphere(J_1, adjustment) → c_3
  118. Intersect(d_2, u_2) → S_1, T_1
  119. Intersect(e_2, v_2) → U_1, V_1
  120. Intersect(f_3, w_2) → W_1, Z_1
  121. Intersect(g_3, a_3) → A_2, B_2
  122. Intersect(h_3, b_3) → C_2, D_2
  123. Intersect(i_3, c_3) → E_2, F_2
  124. Segment(R_1, Q) → k_3
  125. Segment(R_1, T) → l_3
  126. Segment(T_1, U) → m_3
  127. Segment(T_1, V) → n_3
  128. Segment(V_1, W) → p_3
  129. Segment(V_1, Z) → q_3
  130. Segment(Z_1, A_1) → r_3
  131. Segment(Z_1, B_1) → s_3
  132. Segment(B_2, C_1) → t_3
  133. Segment(B_2, D_1) → d_3
  134. Segment(D_2, E_1) → e_3
  135. Segment(D_2, F_1) → f_4
  136. Segment(F_2, G_1) → g_4
  137. Segment(F_2, H_1) → h_4
  138. Segment(N, Q) → i_4
  139. Segment(T, U) → j_4
  140. Segment(V, W) → k_4
  141. Segment(Z, A_1) → l_4
  142. Segment(B_1, C_1) → m_4
  143. Segment(D_1, E_1) → n_4
  144. Segment(F_1, G_1) → p_4
  145. Segment(H_1, I_1) → q_4
  146. Segment(H_1, A''') → r_4
  147. Segment(G_1, A''') → s_4
  148. Segment(F_1, B''') → t_4
  149. Segment(B''', E_1) → a_4
  150. Segment(D_1, A'') → b_4
  151. Segment(A'', C_1) → c_4
  152. Segment(B_1, B'') → d_4
  153. Segment(B'', A_1) → e_4
  154. Segment(Z, A') → f_5
  155. Segment(A', W) → g_5
  156. Segment(V, B') → h_5
  157. Segment(B', U) → i_5
  158. Segment(T, A) → j_5
  159. Segment(A, Q) → k_5
  160. Segment(N, B) → l_5
  161. Segment(F_2, A''') → m_5
  162. Segment(D_2, B''') → n_5
  163. Segment(B_2, A'') → p_5
  164. Segment(Z_1, B'') → q_5
  165. Segment(V_1, A') → r_5
  166. Segment(T_1, B') → s_5
  167. Segment(R_1, A) → t_5
  168. Ray(B, N) → b_5
  169. Line(C, B) → c_5
  170. Line(B, N) → a_5
  171. Sphere(N, adjustment) → o_3
  172. Intersect(a_5, o_3) → G_2, H_2
  173. Segment(B, H_2) → d_5
  174. Ray(E, I_1) → e_5
  175. Sphere(I_1, adjustment) → u_3
  176. Intersect(e_5, u_3) → I_2, J_2
  177. Segment(J_2, E) → f_6
  178. Segment(R_1, A) → g_6
  179. Mirror(J_2, h_6) → K_2
  180. Mirror(J_2, h_6) → L_2
  181. Mirror(E, h_6) → E'_{1}
  182. Segment(L_2, E'_{1}) → i_6
  183. Mirror(H_1, h_6) → M_2
  184. Mirror(H_1, h_6) → N_2
  185. Mirror(I_1, h_6) → O_2
  186. Segment(N_2, O_2) → j_6
  187. Mirror(H_1, h_6) → P_2
  188. Mirror(A''', h_6) → Q_2
  189. Segment(P_2, Q_2) → k_6
  190. Mirror(F_2, h_6) → R_2
  191. Mirror(A''', h_6) → S_2
  192. Segment(R_2, S_2) → l_6
  193. Mirror(F_2, h_6) → T_2
  194. Mirror(G_1, h_6) → U_2
  195. Segment(T_2, U_2) → m_6
  196. Mirror(G_1, h_6) → V_2
  197. Mirror(A''', h_6) → W_2
  198. Segment(V_2, W_2) → n_6
  199. Mirror(F_1, h_6) → Z_2
  200. Mirror(G_1, h_6) → A_3
  201. Segment(Z_2, A_3) → p_6
  202. Mirror(D_2, h_6) → B_3
  203. Mirror(F_1, h_6) → C_3
  204. Segment(B_3, C_3) → q_6
  205. Mirror(D_2, h_6) → D_3
  206. Mirror(E_1, h_6) → E_3
  207. Segment(D_3, E_3) → r_6
  208. Mirror(D_2, h_6) → F_3
  209. Mirror(B''', h_6) → G_3
  210. Segment(F_3, G_3) → s_6
  211. Mirror(B''', h_6) → H_3
  212. Mirror(E_1, h_6) → I_3
  213. Segment(H_3, I_3) → t_6
  214. Mirror(D_1, h_6) → J_3
  215. Mirror(E_1, h_6) → K_3
  216. Segment(J_3, K_3) → a_6
  217. Mirror(B_2, h_6) → L_3
  218. Mirror(C_1, h_6) → M_3
  219. Segment(L_3, M_3) → b_6
  220. Mirror(B_2, h_6) → N_3
  221. Mirror(D_1, h_6) → O_3
  222. Segment(N_3, O_3) → c_6
  223. Mirror(B_2, h_6) → P_3
  224. Mirror(A'', h_6) → A'''_{3}
  225. Segment(P_3, A'''_{3}) → d_6
  226. Mirror(A'', h_6) → A'''_{4}
  227. Mirror(C_1, h_6) → Q_3
  228. Segment(A'''_{4}, Q_3) → e_6
  229. Mirror(B_1, h_6) → R_3
  230. Mirror(C_1, h_6) → S_3
  231. Segment(R_3, S_3) → f_7
  232. Mirror(Z_1, h_6) → T_3
  233. Mirror(B_1, h_6) → U_3
  234. Segment(T_3, U_3) → g_7
  235. Mirror(Z_1, h_6) → V_3
  236. Mirror(A_1, h_6) → W_3
  237. Segment(V_3, W_3) → h_7
  238. Mirror(Z_1, h_6) → Z_3
  239. Mirror(B'', h_6) → B'''_{3}
  240. Segment(Z_3, B'''_{3}) → i_7
  241. Mirror(B'', h_6) → B'''_{4}
  242. Mirror(A_1, h_6) → A_4
  243. Segment(B'''_{4}, A_4) → j_7
  244. Mirror(B_1, h_6) → B_4
  245. Mirror(B'', h_6) → B'''_{5}
  246. Segment(B_4, B'''_{5}) → k_7
  247. Mirror(Z, h_6) → Z'
  248. Mirror(A_1, h_6) → C_4
  249. Segment(Z', C_4) → l_7
  250. Mirror(V_1, h_6) → D_4
  251. Mirror(W, h_6) → W'
  252. Segment(D_4, W') → m_7
  253. Mirror(Z, h_6) → Z'_{1}
  254. Mirror(V_1, h_6) → E_4
  255. Mirror(Z, h_6) → Z'_{2}
  256. Segment(E_4, Z'_{2}) → n_7
  257. Mirror(V_1, h_6) → F_4
  258. Mirror(A', h_6) → A''_{4}
  259. Segment(F_4, A''_{4}) → p_7
  260. Mirror(A', h_6) → A''_{5}
  261. Mirror(W, h_6) → W'_{1}
  262. Segment(A''_{5}, W'_{1}) → q_7
  263. Mirror(Z, h_6) → Z'_{3}
  264. Mirror(A', h_6) → A''_{6}
  265. Segment(Z'_{3}, A''_{6}) → r_7
  266. Mirror(A''_{6}, h_6) → G_4
  267. Mirror(V, h_6) → V'
  268. Mirror(B', h_6) → B''_{3}
  269. Segment(V', B''_{3}) → s_7
  270. Mirror(B', h_6) → B''_{4}
  271. Mirror(U, h_6) → U'
  272. Segment(B''_{4}, U') → t_7
  273. Mirror(T_1, h_6) → H_4
  274. Mirror(U, h_6) → U'_{1}
  275. Segment(H_4, U'_{1}) → a_7
  276. Mirror(T_1, h_6) → I_4
  277. Mirror(B', h_6) → B''_{5}
  278. Segment(I_4, B''_{5}) → b_7
  279. Mirror(T_1, h_6) → J_4
  280. Mirror(V, h_6) → V'_{1}
  281. Segment(J_4, V'_{1}) → c_7
  282. Mirror(T, h_6) → T'
  283. Mirror(U, h_6) → U'_{2}
  284. Segment(T', U'_{2}) → d_7
  285. Mirror(R_1, h_6) → K_4
  286. Mirror(Q, h_6) → Q'
  287. Segment(K_4, Q') → e_7
  288. Mirror(T, h_6) → T'_{1}
  289. Mirror(R_1, h_6) → L_4
  290. Mirror(A, h_6) → A'_{3}
  291. Segment(L_4, A'_{3}) → f_8
  292. Mirror(R_1, h_6) → M_4
  293. Mirror(T, h_6) → T'_{2}
  294. Segment(M_4, T'_{2}) → g_8
  295. Mirror(A, h_6) → A'_{4}
  296. Mirror(Q, h_6) → Q'_{1}
  297. Segment(A'_{4}, Q'_{1}) → h_8
  298. Mirror(T, h_6) → T'_{3}
  299. Mirror(A, h_6) → A'_{5}
  300. Segment(T'_{3}, A'_{5}) → i_8
  301. Mirror(B, h_6) → B'_{2}
  302. Mirror(H_2, h_6) → N_4
  303. Segment(B'_{2}, N_4) → j_8
  304. Mirror(N, h_6) → N'
  305. Mirror(Q, h_6) → Q'_{2}
  306. Segment(N', Q'_{2}) → k_8
  307. Mirror(D_1, h_6) → O_4
  308. Mirror(A'', h_6) → A'''_{5}
  309. Segment(O_4, A'''_{5}) → l_8
  310. Mirror(V, h_6) → V'_{2}
  311. Mirror(W, h_6) → W'_{2}
  312. Segment(V'_{2}, W'_{2}) → m_8
  313. Mirror(F_2, h_6) → P_4
  314. Mirror(H_1, h_6) → Q_4
  315. Segment(P_4, Q_4) → n_8
  316. Mirror(F_1, h_6) → R_4
  317. Mirror(B''', h_6) → S_4
  318. Segment(R_4, S_4) → p_8
  319. Polygon(T_4, U_4, V_4, Z_4) → q9, q_8, u_4, v_4, z_4
  320. Prism(q9, -7) → w_3, D_5, edge9, edge10, edge12, edge11, edge13, edge15, edge16, edge14, E_5, F_5, G_5, face6, face8, face9, face7, face10
  321. Mirror(w_3, eq1) → v_3

AI Drawing Prompt Reference

On a coordinate plane, draw two perpendicular axes and place two initial points on the x-axis to set the starting geometry. Construct baseline lines by connecting each point to the appropriate axis line so that you get several reference lines. Then perform a step-by-step rolling motion: rotate the points around the corresponding axes/intersections by a fixed angle (including both positive and negative directions) to generate successive vertex positions and their associated lines. Take intersections of the rotated lines to obtain the key corner points, and connect these corner points in sequence to form multiple quadrilateral panels that together depict the rolling bridge outline. Add a few extra line segments from selected key endpoints to show the extended structure.
The prompt above was compiled from the original drawing commands and is for reference only.