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Road Intersection Thoughts

Road Intersection Thoughts
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2016-04-18 23:14
Solid GeometryTransformationGeometric Construction
This example demonstrates the geometric construction of road intersections. Key mathematical concepts include solid geometry, specifically cuboids and pyramids. Using GeoGebra commands such as sliders, loci, and geometric transformations, the example dynamically illustrates the geometric properties of road intersections. Suitable for high school and above students or teachers.

GeoGebra Commands & Parameters

  1. slide(C, B, J) → c, d
  2. Point(c) → L
  3. Point(d) → M
  4. OBounds(C, B, J, L, M) → N, O, P, Q
  5. slide(K, B, A) → e, f
  6. Point(e) → Z
  7. Point(f) → A_1
  8. OBounds(K, B, A, Z, A_1) → B_1, C_1, D_1, E_1
  9. slide(J, B, K) → g, h
  10. Point(h) → Q_1
  11. Point(h) → P_1
  12. Point(g) → O_1
  13. Point(g) → N_1
  14. Point(g) → R
  15. Point(h) → S
  16. OBounds(J, B, K, R, S) → T, U, V, W
  17. LimitINT(W, T, D_1, C_1, B) → poly4, l
  18. LimitINT(Q, N, V, U, B) → poly3, k
  19. CornerX(J, B, l, K, N_1, O_1, R, P_1, Q_1, S, k) → s_1, t_1, i_2, poly7, k_2, a_2, b_2, c_2, d_2, e_2, f_2, g_2, h_2
  20. Point(d_2) → B_2
  21. Point(e_2) → Z_1
  22. Circle(Z_1, B_2) → f_3
  23. Intersect(f_3, a_2) → K_2,
  24. OrthogonalLine(K_2, t_1) → k_3
  25. Intersect(f_3, e_2) → J_2,
  26. OrthogonalLine(J_2, d_2) → j_3
  27. Intersect(j_3, k_3) → N_2
  28. Ray(N_2, B) → b_{12}
  29. Distance(N_2, K_2) → distanceN2K2
  30. Circle(N_2, distanceN2K2 + 0.7) → d_{12}
  31. Intersect(d_{12}, b_{12}) → K_7,
  32. Circle(N_2, distanceN2K2 + 5.2) → g_{12}
  33. Intersect(g_{12}, b_{12}) → Z_6,
  34. Segment(Z_6, K_7) → j_{12}
  35. Point(j_{12}) → F_8
  36. Segment(B, J) → s_8
  37. slide(A, B, C) → a, b
  38. Point(a) → D
  39. Point(b) → E
  40. OBounds(A, B, C, D, E) → F, G, H, I
  41. LimitINT(I, F, P, O, B) → poly2, j
  42. Point(d) → M_1
  43. Point(d) → L_1
  44. Midpoint(J, L_1) → P_5
  45. Midpoint(L_1, P_5) → E_6
  46. Point(c) → K_1
  47. Point(c) → J_1
  48. Segment(B, K) → b_9
  49. Point(e) → R_1
  50. Midpoint(R_1, K) → D_6
  51. Midpoint(D_6, R_1) → F_6
  52. Point(e) → S_1
  53. Point(f) → T_1
  54. Point(f) → U_1
  55. LimitINT(E_1, B_1, H, G, B) → poly1, i
  56. Midpoint(N_1, J) → Q_5
  57. Midpoint(Q_5, N_1) → G_7
  58. Intersect(l, b_9) → V_5
  59. CornerX(K, B, i, A, R_1, S_1, Z, T_1, U_1, A_1, l) → j_2, l_2, b_3, poly8, c_3, m_2, n_2, p_2, q_2, r_2, s_2, t_2, a_3
  60. Point(r_2) → V_1
  61. Segment(R_1, V_1) → g_8
  62. Intersect(g_8, l) → K_5
  63. Midpoint(V_5, K_5) → Z_5
  64. Midpoint(Z_5, K_5) → G_6
  65. Line(G_6, F_6) → n_{10}
  66. Intersect(s_8, k) → G_5
  67. Center(k_2) → P_7
  68. Segment(Z_1, N_1) → b_8
  69. Intersect(k, b_8) → C_5
  70. Midpoint(C_5, G_5) → S_5
  71. Midpoint(S_5, C_5) → F_7
  72. Line(F_7, G_7) → s_{10}
  73. Line(P_7, K_2) → h_{11}
  74. Intersect(s_{10}, h_{11}) → Q_7
  75. Circle(P_7, Q_7) → k_{11}
  76. CornerX(C, B, k, J, J_1, K_1, L, L_1, M_1, M, j) → f_1, g_1, q_1, poly6, r_1, h_1, i_1, j_1, k_1, l_1, m_1, n_1, p_1
  77. Point(l_1) → A_5
  78. Midpoint(G_5, A_5) → R_5
  79. Midpoint(A_5, R_5) → B_5
  80. Line(B_5, E_6) → m_{10}
  81. AngularBisector(m_{10}, n_{10}) → t_{10}, a_{11}
  82. Intersect(k_{11}, t_{10}) → R_7, S_7
  83. Point(l_1) → A_2
  84. Point(k_1) → C_2
  85. Circle(A_2, C_2) → g_3
  86. Intersect(g_3, l_1) → L_2,
  87. OrthogonalLine(L_2, k_1) → l_3
  88. Intersect(g_3, h_1) → M_2,
  89. OrthogonalLine(M_2, g_1) → m_3
  90. Intersect(m_3, l_3) → Q_2
  91. Segment(Q_2, A_2) → f_4
  92. Intersect(r_1, f_4) → T_2,
  93. Ray(T_2, A_2) → f_6
  94. Circle(T_2, o) → s_4
  95. Intersect(s_4, f_6) → U_4,
  96. Point(b) → I_1
  97. Point(b) → H_1
  98. Point(a) → G_1
  99. Point(a) → F_1
  100. CornerX(A, B, j, C, F_1, G_1, D, H_1, I_1, E, i) → m, n, d_1, poly5, e_1, p, q, r, s, t, a_1, b_1, c_1
  101. Point(p) → E_2
  102. Point(n) → D_2
  103. Circle(E_2, D_2) → e_3
  104. Intersect(e_3, t) → G_2,
  105. OrthogonalLine(G_2, s) → n_3
  106. Intersect(e_3, p) → F_2,
  107. OrthogonalLine(F_2, n) → p_3
  108. Intersect(n_3, p_3) → P_2
  109. Segment(P_2, E_2) → g_4
  110. Intersect(e_1, g_4) → S_2,
  111. Ray(S_2, E_2) → e_6
  112. Circle(S_2, o) → r_4
  113. Intersect(r_4, e_6) → T_4,
  114. Point(q_2) → W_1
  115. Circle(V_1, W_1) → d_3
  116. Intersect(d_3, m_2) → I_2,
  117. OrthogonalLine(I_2, l_2) → i_3
  118. Intersect(d_3, r_2) → H_2,
  119. OrthogonalLine(H_2, q_2) → h_3
  120. Intersect(i_3, h_3) → O_2
  121. Segment(O_2, V_1) → h_4
  122. Intersect(c_3, h_4) → R_2,
  123. Ray(R_2, V_1) → b_6
  124. Circle(R_2, o) → q_4
  125. Intersect(q_4, b_6) → S_4,
  126. Segment(N_2, Z_1) → e_4
  127. Intersect(k_2, e_4) → U_2,
  128. Ray(U_2, Z_1) → a_6
  129. Circle(U_2, o) → p_4
  130. Intersect(p_4, a_6) → R_4,
  131. Line(E_2, G_2) → k_4
  132. Line(A_2, M_2) → l_4
  133. OrthogonalLine(Z_2, l_4) → l_6
  134. Intersect(l_6, l_4) → G_4
  135. OrthogonalLine(A_3, k_4) → m_6
  136. Intersect(m_6, k_4) → I_4
  137. Angle(I_4, A_3, Z_2) → δ
  138. Line(A_2, L_2) → g_6
  139. Line(Z_1, K_2) → h_6
  140. OrthogonalLine(A_3, h_6) → n_6
  141. Intersect(n_6, h_6) → H_4
  142. OrthogonalLine(V_2, g_6) → s_6
  143. Intersect(s_6, g_6) → B_3
  144. Angle(B_3, V_2, A_3) → γ
  145. Line(Z_1, J_2) → i_6
  146. Line(V_1, I_2) → j_6
  147. OrthogonalLine(V_2, j_6) → r_6
  148. Intersect(r_6, j_6) → S_3
  149. OrthogonalLine(W_2, i_6) → q_6
  150. Intersect(i_6, q_6) → E_4
  151. Angle(E_4, W_2, V_2) → β
  152. Line(E_2, F_2) → j_4
  153. Line(V_1, H_2) → i_4
  154. OrthogonalLine(Z_2, i_4) → k_6
  155. Intersect(k_6, i_4) → F_4
  156. OrthogonalLine(W_2, j_4) → p_6
  157. Intersect(j_4, p_6) → T_3
  158. Angle(F_4, Z_2, W_2) → α
  159. Segment(J_2, N_2) → q_3
  160. Segment(N_2, K_2) → r_3
  161. Segment(M_2, Q_2) → s_3
  162. Segment(Q_2, L_2) → t_3
  163. Segment(P_2, F_2) → a_4
  164. Segment(P_2, G_2) → b_4
  165. Segment(O_2, H_2) → c_4
  166. Segment(O_2, I_2) → d_4
  167. If(β < π / 6, Line[W_2, V_2], If[Distance[B, W_2] > Distance[B, V_2], Line[W_2, E_4], Line[V_2, S_3]]) → m_4
  168. Mirror(B, m_4) → B'
  169. If(α < π / 6, Line[W_2, Z_2], If[Distance[B, Z_2] > Distance[B, W_2], Line[F_4, Z_2], Line[W_2, T_3]]) → n_4
  170. Mirror(B, n_4) → B'_1
  171. If(δ < π / 6, Line[Z_2, A_3], If[Distance[B, A_3] > Distance[B, Z_2], Line[I_4, A_3], Line[Z_2, G_4]]) → t_4
  172. Mirror(B, t_4) → B'_2
  173. If(γ < π / 6, Line[V_2, A_3], If[Distance[B, V_2] > Distance[B, A_3], Line[V_2, B_3], Line[A_3, H_4]]) → a_5
  174. Mirror(B, a_5) → B'_3
  175. Midpoint(B, B'_1) → C_3
  176. Midpoint(B, B'_2) → D_3
  177. Midpoint(B, B'_3) → E_3
  178. Midpoint(B, B') → F_3
  179. Ray(F_3, B') → b_5
  180. Ray(C_3, B'_1) → c_5
  181. Ray(D_3, B'_2) → d_5
  182. Ray(E_3, B'_3) → e_5
  183. Circle(F_3, u) → f_5
  184. Circle(C_3, u) → g_5
  185. Circle(D_3, u) → h_5
  186. Circle(E_3, u) → k_5
  187. Intersect(f_5, b_5) → G_3,
  188. Intersect(g_5, c_5) → H_3,
  189. Intersect(h_5, d_5) → I_3,
  190. Intersect(k_5, e_5) → J_3,
  191. Line(G_3, m_4) → i_5
  192. Line(H_3, n_4) → j_5
  193. Line(I_3, t_4) → l_5
  194. Line(J_3, a_5) → m_5
  195. Intersect(e_2, i_5) → K_3
  196. Intersect(a_2, m_5) → L_3
  197. Intersect(l_1, m_5) → M_3
  198. Intersect(h_1, l_5) → N_3
  199. Intersect(t, l_5) → O_3
  200. Intersect(p, j_5) → P_3
  201. Intersect(r_2, j_5) → Q_3
  202. Intersect(m_2, i_5) → R_3
  203. Segment(R_3, K_3) → n_5
  204. Segment(L_3, M_3) → p_5
  205. Segment(N_3, O_3) → q_5
  206. Segment(P_3, Q_3) → r_5
  207. Vector(F_3, G_3) → v
  208. Vector(C_3, H_3) → w
  209. Vector(D_3, I_3) → u_1
  210. Vector(E_3, J_3) → v_1
  211. CircleSector(E_2, G_2, F_2) → s_5
  212. CircleSector(V_1, H_2, I_2) → t_5
  213. CircleSector(Z_1, J_2, K_2) → c_6
  214. CircleSector(A_2, L_2, M_2) → d_6
  215. Intersect(d_2, q_3) → U_3
  216. Intersect(t_1, r_3) → V_3
  217. Intersect(k_1, t_3) → W_3
  218. Intersect(s_3, g_1) → Z_3
  219. Intersect(b_4, s) → A_4
  220. Intersect(a_4, n) → B_4
  221. Intersect(q_2, c_4) → C_4
  222. Intersect(l_2, d_4) → D_4
  223. Intersect(m_4, i_6) → J_4
  224. Intersect(m_4, j_6) → K_4
  225. Intersect(n_4, i_4) → L_4
  226. Intersect(a_5, h_6) → M_4
  227. Intersect(n_4, j_4) → N_4
  228. Intersect(t_4, k_4) → O_4
  229. Intersect(t_4, l_4) → P_4
  230. Intersect(g_6, a_5) → Q_4
  231. CircleSector(N_2, K_2, J_2) → t_6
  232. CircleSector(O_2, I_2, H_2) → c_7
  233. CircleSector(P_2, F_2, G_2) → d_7
  234. CircleSector(Q_2, M_2, L_2) → e_7
  235. Polygon(O_4, P_4, N_3, O_3) → poly10, o_4, p_7, n_7, o_3
  236. Polygon(Q_3, L_4, N_4, P_3) → poly11, q_8, l_7, n_8, p_8
  237. Polygon(R_3, K_3, J_4, K_4) → poly12, r_7, k_8, j_7, k_9
  238. Polygon(M_4, L_3, M_3, Q_4) → poly9, m_7, l_8, m_8, q_7
  239. Point(h_1) → V_4
  240. Point(g_1) → W_4
  241. Point(k_1) → Z_4
  242. Polygon(V_4, W_4, C_2, Z_4, A_5, A_2) → poly13, v_4, w_4, c_8, z_4, a_7, a_8
  243. Vector(S_2, T_4) → w_1
  244. Vector(T_2, U_4) → u_2
  245. Vector(R_4, Z_1) → v_2
  246. Vector(R_2, S_4) → w_2
  247. Segment(I_1, D_2) → b_7
  248. Segment(H_1, E_2) → f_7
  249. Segment(G_1, D_2) → g_7
  250. Segment(F_1, E_2) → h_7
  251. Segment(J_1, A_2) → i_7
  252. Segment(C_2, K_1) → k_7
  253. Segment(C_2, M_1) → s_7
  254. Segment(L_1, A_2) → t_7
  255. Segment(B_2, O_1) → d_8
  256. Segment(Q_1, B_2) → e_8
  257. Segment(Z_1, P_1) → f_8
  258. Segment(W_1, S_1) → h_8
  259. Segment(V_1, T_1) → i_8
  260. Segment(U_1, W_1) → j_8
  261. Segment(B, C) → r_8
  262. Segment(B, K) → t_8
  263. Segment(B, A) → a_9
  264. Intersect(j, f_7) → D_5
  265. Intersect(i, h_7) → E_5
  266. Intersect(i, i_8) → F_5
  267. Intersect(j, r_8) → H_5
  268. Intersect(a_9, i) → I_5
  269. Intersect(l, f_8) → J_5
  270. Midpoint(E_5, I_5) → L_5
  271. Midpoint(I_5, F_5) → M_5
  272. Midpoint(A, F_1) → N_5
  273. Midpoint(T_1, A) → O_5
  274. Midpoint(H_5, V_4) → T_5
  275. Midpoint(H_5, D_5) → U_5
  276. Segment(V_5, K) → c_9
  277. Segment(I_5, A) → d_9
  278. Segment(H_5, C) → e_9
  279. Segment(G_5, J) → f_9
  280. Midpoint(V_5, J_5) → W_5
  281. Midpoint(J_1, C) → A_6
  282. Midpoint(C, H_1) → B_6
  283. Midpoint(P_1, K) → C_6
  284. Segment(U_5, B_6) → g_9
  285. Segment(T_5, A_6) → h_9
  286. Segment(P_5, R_5) → i_9
  287. Segment(Q_5, S_5) → j_9
  288. Segment(L_5, N_5) → l_9
  289. Segment(M_5, O_5) → m_9
  290. Segment(C_6, W_5) → n_9
  291. Segment(D_6, Z_5) → p_9
  292. Line(E_6, B_5) → q_9
  293. Line(F_6, G_6) → r_9
  294. Intersect(q_9, r_9) → H_6
  295. Circle(H_6, If[Distance[H_6, B_5] < Distance[H_6, G_6], Distance[H_6, B_5], Distance[H_6, G_6]]) → s_9
  296. Intersect(s_9, q_9) → I_6, J_6
  297. Intersect(s_9, r_9) → K_6, L_6
  298. OrthogonalLine(I_6, q_9) → t_9
  299. OrthogonalLine(K_6, r_9) → a_{10}
  300. Intersect(a_{10}, t_9) → M_6
  301. CircleArc(M_6, I_6, K_6) → c_{10}
  302. Segment(E_6, I_6) → b_{10}
  303. Segment(F_6, K_6) → d_{10}
  304. Midpoint(G_6, V_5) → N_6
  305. Midpoint(K, F_6) → O_6
  306. Line(O_6, N_6) → e_{10}
  307. Circle(M_6, K_6) → f_{10}
  308. Midpoint(B_5, G_5) → R_6
  309. Midpoint(E_6, J) → S_6
  310. Line(R_6, S_6) → g_{10}
  311. Segment(B_7, C_7) → l_{10}
  312. Intersect(n_{10}, m_{10}) → E_7
  313. Midpoint(P_1, C_6) → H_7
  314. Midpoint(J_5, W_5) → I_7
  315. Line(H_7, I_7) → r_{10}
  316. OrthogonalLine(APEX, t_{10}) → b_{11}
  317. Intersect(b_{11}, m_{10}) → L_7
  318. AngularBisector(m_{10}, b_{11}) → c_{11}, d_{11}
  319. AngularBisector(n_{10}, b_{11}) → e_{11}, f_{11}
  320. Intersect(d_{11}, e_{11}) → O_7
  321. Line(H_7, I_7) → j_{10}
  322. Line(G_7, F_7) → k_{10}
  323. OrthogonalLine(APEX, e_4) → p_{10}
  324. AngularBisector(k_{10}, p_{10}) → q_{10}, i_{11}
  325. AngularBisector(j_{10}, p_{10}) → j_{11}, l_{11}
  326. Mirror(APEX, i_{11}) → P_6
  327. Mirror(APEX, j_{11}) → Q_6
  328. Intersect(j_{11}, i_{11}) → T_6
  329. CircleArc(T_6, P_6, Q_6) → p_{11}
  330. Segment(P_6, G_7) → m_{11}
  331. Segment(Q_6, H_7) → n_{11}
  332. Point(k) → U_6
  333. Point(l) → V_6
  334. Line(U_6, i_9) → h_{10}
  335. Line(V_6, p_9) → i_{10}
  336. Intersect(i_{10}, h_{10}) → W_6
  337. OrthogonalLine(J, h_{10}) → q_{11}
  338. OrthogonalLine(D_7, i_{10}) → r_{11}
  339. Intersect(i_{10}, r_{11}) → M_7
  340. Intersect(h_{10}, q_{11}) → N_7
  341. AppoLLP(W_6, U_6, V_6, APEX) → c_{12}, T_7, U_7
  342. Segment(M_7, T_7) → g_{11}
  343. Segment(N_7, U_7) → s_{11}
  344. Segment(K_2, N_2) → t_{11}
  345. Segment(N_2, J_2) → a_{12}
  346. Circle(N_2, distanceN2K2 + 2.2) → e_{12}
  347. Circle(N_2, distanceN2K2 + 3.7) → f_{12}
  348. Ray(N_2, K_2) → h_{12}
  349. Ray(N_2, J_2) → i_{12}
  350. Intersect(d_{12}, h_{12}) → V_7,
  351. Intersect(e_{12}, h_{12}) → W_7,
  352. Intersect(f_{12}, h_{12}) → Z_7,
  353. Intersect(g_{12}, h_{12}) → A_8,
  354. Intersect(g_{12}, i_{12}) → B_8,
  355. Intersect(f_{12}, i_{12}) → C_8,
  356. Intersect(e_{12}, i_{12}) → D_8,
  357. Intersect(d_{12}, i_{12}) → E_8,
  358. Intersect(f_{12}, b_{12}) → A_7,
  359. Intersect(e_{12}, b_{12}) → J_7,
  360. CircumcircleArc(D_8, J_7, W_7) → k_{12}
  361. CircumcircleArc(C_8, A_7, Z_7) → p_{12}
  362. CircumcircleArc(B_8, Z_6, A_8) → q_{12}
  363. CircumcircleArc(E_8, K_7, V_7) → r_{12}