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非灯几何体演示

非灯几何体演示
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2019-09-20 23:52
解析几何立体几何几何作图
①本范例演示一个非传统几何体（十二面体）的构造与特性；②涉及核心数学概念包括立体几何、中点、线面关系、正态分布（用于点生成）、坐标几何；③构造中运用了点、十二面体、中点、线、交点、平移、射线、旋转、垂线、路径交点、距离、点在路径上、圆、镜像、多边形、序列等GeoGebra命令；④适合高中及以上学生用于立体几何学习和空间想象能力培养。

GeoGebra 命令及参数

  1. Point(Circle[((A * (1 - sqrt(5))) + (B * (3 + sqrt(5)))) / 4, (Distance[A, B] * sqrt(10 + (2 * sqrt(5))) / 4), Segment[A, B]]) → C
  2. Dodecahedron(A, B, C) → a, D, M, N, O, P, Q, R, S, T, 면ABCDE, 면AEIOJ, E, 면ABFKJ, 면BCGLF, 면CDHMG, 면DEINH, 면INSTO, 면JKPTO, 면FKPQL, 면GLQRM, 면HMRSN, 면PQRST, F, 모서리AE, 모서리AB, 모서리BC, 모서리CD, 모서리DE, 모서리AJ, 모서리EI, 모서리IO, 모서리JO, 모서리JK, G, 모서리FK, 모서리BF, 모서리FL, 모서리GL, 모서리CG, 모서리GM, 모서리HM, 모서리DH, 모서리HN, 모서리IN, H, 모서리NS, 모서리ST, 모서리OT, 모서리PT, 모서리KP, 모서리PQ, 모서리LQ, 모서리QR, 모서리MR, 모서리RS, I, J, K, L
  3. Midpoint(A, R) → U
  4. Line(P, L) → h
  5. Line(Q, G) → g
  6. Line(Q, K) → f
  7. Intersect(h, f) → V
  8. Translate(V, Vector[(0.9 * u)]) → W
  9. Translate(W, Vector[(-5.2 * u) - (3.3 * v)]) → W'_1
  10. Translate(W, Vector[(2 * v)]) → W'
  11. Translate(W', Vector[(0.6 * u)]) → W''
  12. Plane(g, f) → p
  13. Ray(U, l3(A12)) → B12
  14. Ray(U, Centroid[l2(A6)]) → C6
  15. Rotate(Rotate[W'', (-(72°)), C6], 120°, B12) → P_1
  16. Rotate(Rotate[W', (-(72°)), C6], 120°, B12) → O_1
  17. Rotate(Rotate[p, (-(72°)), C6], 120°, B12) → q
  18. OrthogonalLine(O_1, q) → n
  19. Plane(P_1, n) → r
  20. IntersectPath(p, r) → s
  21. Point(s) → U_1
  22. OrthogonalLine(U_1, s, p) → c
  23. Distance(U, J) → b
  24. Translate(W, Vector[(0.2 * u)]) → Z
  25. Translate(W, Vector[(0.6 * u)]) → A_1
  26. Translate(W, Vector[(1.7 * u)]) → C_1
  27. Translate(V, Vector[(3.4 * v)]) → B_1
  28. Translate(C_1, Vector[(3.4 * v)]) → D_1
  29. Translate(C_1, Vector[(-1.1 * v)]) → E_1
  30. Translate(V, Vector[(-1.1 * v)]) → F_1
  31. Translate(V, Vector[(-0.5 * v)]) → V'
  32. Translate(V, Vector[(-0.9 * v)]) → V'_1
  33. Translate(C_1, Vector[(-0.9 * v)]) → G_1
  34. Translate(C_1, Vector[(-0.5 * v)]) → H_1
  35. Translate(Q, Vector[(-((Distance[Q, W] * v)))]) → I_1
  36. Translate(I_1, Vector[(0.3 * u)]) → J_1
  37. Translate(I_1, Vector[(0.3 * u) + (0.2 * v)]) → K_1
  38. Translate(I_1, Vector[(0.2 * v)]) → L_1
  39. Translate(J_1, Vector[(-((0.3 * v)))]) → M_1
  40. Translate(J_1, Vector[(1.1 * v)]) → N_1
  41. Ray(U, l3(A1)) → B1
  42. Ray(U, l3(A2)) → B2
  43. Ray(U, l3(A3)) → B3
  44. Ray(U, l3(A4)) → B4
  45. Ray(U, l3(A5)) → B5
  46. Ray(U, l3(A6)) → B6
  47. Ray(U, l3(A7)) → B7
  48. Ray(U, l3(A8)) → B8
  49. Ray(U, l3(A9)) → B9
  50. Ray(U, l3(A10)) → B10
  51. Ray(U, l3(A11)) → B11
  52. Ray(U, l3(A13)) → B13
  53. Ray(U, l3(A14)) → B14
  54. Ray(U, l3(A15)) → B15
  55. Ray(U, l3(A16)) → B16
  56. Ray(U, l3(A17)) → B17
  57. Ray(U, l3(A18)) → B18
  58. Ray(U, l3(A19)) → B19
  59. Ray(U, l3(A20)) → B20
  60. Ray(U, Centroid[l2(A1)]) → C1
  61. Ray(U, Centroid[l2(A2)]) → C2
  62. Ray(U, Centroid[l2(A3)]) → C3
  63. Ray(U, Centroid[l2(A4)]) → C4
  64. Ray(U, Centroid[l2(A5)]) → C5
  65. Ray(U, Centroid[l2(A7)]) → C7
  66. Ray(U, Centroid[l2(A8)]) → C8
  67. Ray(U, Centroid[l2(A9)]) → C9
  68. Ray(U, Centroid[l2(A10)]) → C10
  69. Ray(U, Centroid[l2(A11)]) → C11
  70. Ray(U, Centroid[l2(A12)]) → C12
  71. Translate(W, Vector[(-3.5 * u) - (1.4 * v)]) → W'_2
  72. Translate(W'_1, Vector[(0.4 * UnitVector[c])]) → R_1
  73. Line(R_1, W'_1) → t
  74. Translate(W'_1, Vector[(0.3 * UnitVector[t])]) → S_1
  75. Translate(R_1, Vector[(-0.3 * UnitVector[t])]) → T_1
  76. Translate(U_1, Vector[(0.3 / tan(36°) * UnitVector[c])]) → V_1
  77. Point(s) → W_1
  78. Translate(V_1, Vector[U_1, W_1]) → Z_1
  79. Point(Circle[W'_2, 0.4, p]) → Q_1
  80. OrthogonalLine(S_1, t, p) → d
  81. Translate(S_1, Vector[(1.3 * UnitVector[d])]) → A_2
  82. Translate(T_1, Vector[(1.3 * UnitVector[d])]) → B_2
  83. PointIn(p) → C_2
  84. PointIn(p) → D_2
  85. PointIn(p) → E_2
  86. Point(d) → F_2
  87. PointIn(p) → G_2
  88. PointIn(p) → H_2
  89. PointIn(p) → I_2
  90. PointIn(p) → J_2
  91. PointIn(p) → K_2
  92. PointIn(p) → L_2
  93. Translate(N_1, Vector[(1.25 * u) - (0.5 * v)]) → M_2
  94. Translate(N_1, Vector[(1.25 * u) - (0.9 * v)]) → N_2
  95. PointIn(p) → O_2
  96. PointIn(p) → P_2
  97. PointIn(p) → Q_2
  98. PointIn(p) → R_2
  99. PointIn(p) → S_2
  100. PointIn(p) → V_2
  101. PointIn(p) → W_2
  102. PointIn(p) → Z_2
  103. PointIn(p) → A_3
  104. PointIn(p) → B_3
  105. Rotate(Rotate[Rotate[W', 72°, C6], (-(120°)), B16], (-(144°)), C2) → C_3
  106. Rotate(Rotate[Rotate[W'', 72°, C6], (-(120°)), B16], (-(144°)), C2) → D_3
  107. Line(C_3, D_3) → e
  108. Line(O_1, P_1) → i_1
  109. Translate(P_1, Vector[(0.2 * UnitVector[i_1])]) → H_3
  110. Translate(O_1, Vector[(0.2 * UnitVector[i_1])]) → I_3
  111. OrthogonalLine(P_1, i_1, r) → j_1
  112. Translate(P_1, Vector[(-0.3 * UnitVector[j_1])]) → J_3
  113. Translate(H_3, Vector[(-0.3 * UnitVector[j_1])]) → K_3
  114. Translate(I_3, Vector[(-0.3 * UnitVector[j_1])]) → L_3
  115. Translate(O_1, Vector[(-0.3 * UnitVector[j_1])]) → M_3
  116. Circle(V, 4.2, r) → c_1
  117. Translate(O_1, Vector[(-0.3 * UnitVector[i_1])]) → N_3
  118. Translate(H_3, Vector[(0.3 * UnitVector[i_1])]) → O_3
  119. Translate(I_3, Vector[(1.25 * UnitVector[j_1])]) → P_3
  120. Translate(P_1, Vector[(1.25 * UnitVector[j_1])]) → Q_3
  121. Translate(Q_3, Vector[(-0.4 * UnitVector[j_1]) + (0.4 * UnitVector[i_1])]) → H_4
  122. Translate(P_3, Vector[(-0.4 * UnitVector[j_1]) - (0.4 * UnitVector[i_1])]) → I_4
  123. Midpoint(D_3, P_1) → E_3
  124. Translate(V_1, Vector[(-0.3 * UnitOrthogonalVector[p])]) → F_3
  125. Line(Q_3, j_1) → k_1
  126. Mirror(W_1, E_3) → G_3
  127. Line(G_3, F_3) → l_1
  128. Line(W_1, l_1) → a_1
  129. Line(P_3, j_1) → f_3
  130. Line(P_3, Q_3) → g_3
  131. Intersect(s, g_3) → R_3
  132. Intersect(k_1, l_1) → S_3
  133. Intersect(a_1, f_3) → T_3
  134. Mirror(G_3, E_3) → U_3
  135. Mirror(F_3, E_3) → V_3
  136. Mirror(S_3, E_3) → W_3
  137. Mirror(Q_3, E_3) → Z_3
  138. Mirror(H_4, E_3) → A_4
  139. Mirror(O_3, E_3) → B_4
  140. Mirror(P_1, E_3) → C_4
  141. Mirror(J_3, E_3) → D_4
  142. Mirror(K_3, E_3) → E_4
  143. Mirror(H_3, E_3) → F_4
  144. Mirror(O_1, E_3) → G_4
  145. Mirror(M_3, E_3) → J_4
  146. Mirror(L_3, E_3) → K_4
  147. Mirror(I_3, E_3) → L_4
  148. Mirror(N_3, E_3) → M_4
  149. Mirror(I_4, E_3) → N_4
  150. Mirror(P_3, E_3) → O_4
  151. Mirror(T_3, E_3) → P_4
  152. Mirror(W_1, E_3) → Q_4
  153. Translate(M_2, Vector[(-0.4 * u) + (0.4 * v)]) → T_2
  154. Translate(N_2, Vector[(-0.4 * u) - (0.4 * v)]) → U_2
  155. Translate(J_1, Vector[(0.6 * v)]) → R_4
  156. Translate(I_1, Vector[(0.6 * v)]) → S_4
  157. Translate(L_1, Vector[(0.6 * v)]) → T_4
  158. Translate(K_1, Vector[(0.6 * v)]) → U_4
  159. Polygon(V', B_1, U_2, M_1, J_1, I_1, L_1, K_1, R_4, S_4, T_4, U_4, D_1, N_1, T_2, M_2, S_2, R_2, Q_2, P_2, O_2, G_1, E_1, H_1, F_1, V'_1, L_2, Q_1, R_1, T_1, B_2, K_2, J_2, I_2, B_3, H_2, G_2, F_2, E_2, W_1, Z_1, V_1, U_1, D_2, A_2, A_3, S_1, W'_1, W'_2, C_2, Z_2, W_2, V_2, N_2) → 다각형1, i, n_1, p_1, q_1, r_1, s_1, t_1, b_1, d_1, e_1, f_2, j, g_2, h_2, i_2, j_2, k_2, l_2, m_2, n_2, p_2, q_2, k, r_2, s_2, t_2, a_2, b_2, c_2, d_2, e_2, h_3, a_3, l, b_3, r_4, s_4, t_4, l_5, m_5, n_5, p_5, q_5, r_5, m, s_5, t_5, a_5, b_5, c_5, f_1, g_1, h_1, m_1
  160. Polygon(O_3, H_4, L_4, K_4, J_4, C_3, F_4, E_4, D_4, C_4, B_4, A_4, Q_3, Z_3, W_3, V_3, W_1, T_3, P_3, I_4, N_3, I_3, L_3, S_3, M_3, O_1, H_3, K_3, J_3, P_1, F_3, G_3, P_4, O_4, N_4, M_4) → 다각형2, i_3, s_3, t_3, c_3, d_3, e_3, f_4, g_4, h_4, i_4, j_4, j_3, k_4, l_4, m_4, n_4, p_4, q_4, a_4, b_4, c_4, d_4, k_3, e_4, f_5, g_5, h_5, i_5, j_5, k_5, l_3, m_3, n_3, p_3, q_3, r_3
  161. Sequence(Rotate[Prism[다각형1, -0.3], ((ω * 2) * π / 5), C6], ω, 0, 4) → l4
  162. Rotate(Prism[다각형2, 0.3], 120°, B11) → o
  163. Rotate(l4, 120°, B17) → l4'
  164. Rotate(l4', 120°, B17) → l4''
  165. Rotate(l4', 120°, B13) → l4''_1
  166. Rotate(l4''_1, 120°, B13) → l5
  167. Rotate(l5, 120°, B14) → l5'
  168. Rotate(l5', 120°, B14) → l5''
  169. Rotate(l5'', 120°, B15) → l5'''
  170. Rotate(l5''', 120°, B15) → l6
  171. Rotate(l4, 120°, B6) → l4'_1
  172. Rotate(l4'_1, 120°, B6) → l7
  173. Rotate(l7, 120°, B3) → l7'
  174. Rotate(o, 120°, B16) → o'
  175. Rotate(o', 120°, B16) → o''
  176. Rotate(o', 120°, B17) → o''_1
  177. Rotate(o''_1, 120°, B17) → w
  178. Rotate(w, 120°, B18) → w'
  179. Rotate(w', 120°, B18) → w''
  180. Rotate(w'', 120°, B19) → w'''
  181. Rotate(w''', 120°, B19) → o_1
  182. Rotate(w''', 120°, B14) → u_1
  183. Rotate(u_1, 120°, B14) → v_1
  184. Rotate(v_1, 120°, B9) → w_1
  185. Rotate(w_1, 120°, B9) → o_2
  186. Rotate(o_2, 120°, B15) → u_2
  187. Rotate(u_2, 120°, B15) → v_2
  188. Rotate(u_2, 120°, B10) → w_2
  189. Rotate(w_2, 120°, B10) → o_3
  190. Rotate(o_3, 120°, B11) → u_3
  191. Rotate(u_3, 120°, B6) → v_3
  192. Rotate(v_3, 120°, B6) → w_3
  193. Rotate(w_3, 120°, B12) → o_4
  194. Rotate(o_4, 120°, B7) → u_4
  195. Rotate(u_4, 120°, B7) → v_4
  196. Rotate(v_4, 120°, B13) → w_4
  197. Rotate(w_4, 120°, B8) → d_5
  198. Rotate(w_1, 120°, B5) → e_5
  199. Rotate(e_5, 120°, B5) → o_5
  200. Rotate(o_5, 120°, B1) → u_5
  201. Rotate(u_5, 120°, B2) → v_5
  202. Rotate(v_5, 120°, B3) → w_5