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GoGeometry 191: 多重几何变换综合范例

GoGeometry 191: 多重几何变换综合范例
几何画板及绘图内容加载中，请稍等……
2021-05-03 10:05
测试改名0多边形向量立体几何几何作图
本范例演示如何运用GeoGebra构建一个包含平行四边形、正方形、等腰三角形、菱形和三维向量的综合几何图形。核心数学概念涵盖平面几何图形性质、向量运算、几何变换（旋转、缩放、平移、镜像）以及三维空间向量表示。作图要点：使用Point创建关键点，通过Polygon构建多边形，用Dilate实现缩放变换，用Rotate进行旋转操作，用Vector和Translate处理向量平移，用Mirror实现镜像对称，用CircleArc和CircleSector绘制圆弧和扇形，用Midpoint找中点，用OrthogonalLine作垂线。本范例适合中学几何教学、向量概念教学及GeoGebra技能训练，可用于演示几何图形之间的关系和变换规律。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Point(Ray[A, A + (1, 0)]) → B_2
  3. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  4. Segment(B_2, C) → h
  5. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  6. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  7. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  8. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  9. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  10. Segment(R, A''') → f''
  11. Segment(A, B_2) → g
  12. Circle(B_2, g) → q_2
  13. Ray(A, B_2) → m_2
  14. Intersect(q_2, m_2, 2) → J_1
  15. Segment(B_2, J_1) → n_2
  16. Point(Ray[B_2, B_2 + (1, 0)]) → K_1
  17. Segment(B_2, K_1) → r_2
  18. Circle(K_1, r_2) → t_2
  19. OrthogonalLine(K_1, r_2) → s_2
  20. Intersect(t_2, s_2, 2) → L_1
  21. OrthogonalLine(L_1, h) → f_3
  22. Intersect(f_3, h) → M_1
  23. Dilate(A, If[17 < e ≤ 18, 18 - e, 0], M_1) → A'_4
  24. Line(K_1, M_1) → i_2
  25. Midpoint(A, C) → V
  26. CircleArc(V, C, D) → g_2
  27. Point(g_2) → B
  28. Segment(C, B) → j_1
  29. Segment(A, B) → n
  30. Intersect(n, f'') → Z
  31. Segment(B, Z) → l_1
  32. Segment(A, A') → f
  33. Segment(A'_1, A'') → f'
  34. Segment(C, D) → i
  35. Circle(B_2, (0.25 * r_2)) → p
  36. Segment(D, A) → j
  37. Circle(C, Radius[p]) → k
  38. Circle(A, (0.1 * j)) → q
  39. Circle(D, (0.1 * j)) → r
  40. Segment(A, C) → l
  41. Segment(D, B_2) → m
  42. Intersect(r, j, 1) → F
  43. Intersect(r, m, 1) → G
  44. Intersect(r, i, 1) → H
  45. Intersect(k, i, 1) → I
  46. Intersect(k, l, 1) → J
  47. Intersect(k, h, 1) → K
  48. Intersect(p, h, 1) → L
  49. Intersect(p, m, 1) → M
  50. Intersect(p, g, 1) → N
  51. Intersect(q, g, 1) → O
  52. Intersect(q, l, 1) → P
  53. Intersect(q, j, 1) → Q
  54. CircleSector(B_2, L, N) → s
  55. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  56. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  57. Segment(R', S) → f'''
  58. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  59. Vector(B_2, B') → u
  60. Translate(s, u) → s'
  61. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  62. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  63. Vector(C, C') → v
  64. Translate(s'', v) → s'''
  65. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  66. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  67. Vector(D, D') → w
  68. Translate(t, w) → t'
  69. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  70. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  71. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  72. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  73. Vector(B_2, B'_1) → v_1
  74. Vector(B_2, B'_3) → w_1
  75. Vector(B_2, B'_2) → i_1
  76. Circle(A, n - j_1) → h_2
  77. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  78. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  79. Vector(B_2, B'_4) → u_1
  80. Translate(s, u_1) → s'_1
  81. Line(B, f') → a_1
  82. Point(a_1) → U
  83. Angle(U, B, C) → α
  84. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  85. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  86. Line(A, B_2) → h_1
  87. Angle(B, A, D) → β
  88. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  89. Mirror(C, D) → C'_1
  90. Rotate(C, ((-π)) / 3, B_2) → C'_2
  91. Segment(B_2, C'_2) → f_1
  92. Segment(C, C'_2) → g_1
  93. Intersect(k, g_1, 1) → E
  94. Intersect(k, h) → T,
  95. CircleSector(C, T, E) → d_1
  96. Rotate(D, 15°, C) → D'_2
  97. Angle(D, C, D'_2) → γ
  98. Ray(C, D'_2) → k_1
  99. Rotate(C, (-(15°)), D) → C'_3
  100. Angle(C'_3, D, C) → δ
  101. Ray(D, C'_3) → m_1
  102. Intersect(k_1, m_1) → W
  103. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  104. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  105. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  106. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  107. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  108. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  109. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  110. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  111. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  112. Rotate(T, 15°, C) → T'
  113. Angle(T, C, T') → ε
  114. Ray(C, T') → q_1
  115. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  116. Vector(C, C'_8) → u_2
  117. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  118. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  119. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  120. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  121. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  122. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  123. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  124. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  125. Segment(B_2, C_1) → n_1
  126. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  127. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  128. Segment(D_1, E_1) → p_1
  129. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  130. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  131. Segment(F_1, N_1) → r_1
  132. Rotate(F_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → O_1
  133. Rotate(N_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → P_1
  134. Segment(O_1, P_1) → s_1
  135. Polygon(B_2, K_1, L_1, M_1) → poly6, b_3, k_2, l_3, m_3
  136. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  137. Vector(B_2, Q_1) → c_1
  138. Translate(s, c_1) → s'_2
  139. Segment(C, K_1) → e_1
  140. Line(A, M_1) → t_1
  141. Intersect(t_1, e_1) → R_1
  142. Segment(A, R_1) → f_2
  143. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  144. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], M_1) → A'_3
  145. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], R_1) → S_1
  146. Line(R_1, p_1) → g_3
  147. Point(g_3) → U_1
  148. Angle(U_1, R_1, K_1) → ζ
  149. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4
  150. Dilate(K_1, If[15 < e ≤ 16, 16 - e, 0], M_1) → W_1
  151. Line(A, K_1) → k_3
  152. Rotate(D_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → H_2
  153. Rotate(E_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → I_2
  154. Segment(H_2, I_2) → f_4
  155. Rotate(B_2, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → J_2
  156. Rotate(K_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → K_2
  157. Rotate(L_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → L_2
  158. Rotate(M_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → M_2
  159. Polygon(J_2, K_2, L_2, M_2) → poly6'_{1}, g_4, h_4, i_4, j_4
  160. Rotate(F_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → N_2
  161. Rotate(N_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → O_2
  162. Segment(N_2, O_2) → k_4
  163. Rotate(O_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → P_2
  164. Rotate(P_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → Q_2
  165. Segment(P_2, Q_2) → l_4
  166. Rotate(B_2, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → R_2
  167. Rotate(C_1, If[15 < e ≤ 16, ((-(e - 15)) * π / 2), ((-π)) / 2], K_1) → S_2
  168. Segment(R_2, S_2) → m_4
  169. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], D_1) → A'_{7}
  170. Segment(A, A'_{7}) → p_2
  171. Rotate(A, If[17 < e ≤ 18, (((-π)) / 2 * (e - 17)), ((-π)) / 2], D_1) → A'_{6}
  172. Segment(D_1, A'_{6}) → e_2
  173. Dilate(D_1, If[18 < e < -19, 19 - e, 0], A'_{6}) → T_1
  174. Dilate(D_1, If[18 < e < 19, 19 - e, 0], A'_{6}) → V_1
  175. Vector(D_1, V_1) → v_2
  176. Translate(A, v_2) → A'_{5}
  177. Translate(A'_{7}, v_2) → Z_1
  178. Segment(A'_{5}, Z_1) → j_3
  179. Dilate(D_1, If[19 < e ≤ 20, 20 - e, 0], A) → A_2
  180. Vector(D_1, A_2) → u_3
  181. Translate(D_1, u_3) → C_2
  182. Translate(A'_{6}, u_3) → D_2
  183. Segment(C_2, D_2) → n_3
  184. Dilate(R, If[20 < e < 21, 21 - e, 0], T_1) → R'_{1}
  185. Vector(R, R'_{1}) → v_3
  186. Translate(s'', v_3) → s'''_{1}
  187. Line(T_1, f'') → p_3
  188. Point(p_3) → E_2
  189. Angle(E_2, T_1, A'_{5}) → η
  190. Rotate(s'''_{1}, If[20 < e ≤ 21, ((-(e - 20)) * ((2 * π) - η)), (-((2 * π) - η))], R'_{1}) → e_3
  191. Dilate(A'_{5}, If[21 < e ≤ 22, 22 - e, 0], D) → F_2
  192. Vector(A'_{5}, F_2) → q_3
  193. Segment(A'_{5}, F_2) → r_3
  194. Dilate(D, If[22 < e ≤ 23, 23 - e, 0], H_2) → D'_{4}
  195. Segment(D, D'_{4}) → s_3
  196. Dilate(H_2, If[23 < e ≤ 24, 24 - e, 0], D_1) → G_2
  197. Vector(H_2, G_2) → t_3
  198. Translate(D, t_3) → D'_{5}
  199. Translate(D'_{4}, t_3) → T_2
  200. Segment(D'_{5}, T_2) → n_4
  201. Dilate(A'_{5}, If[24 < e ≤ 25, 25 - e, 0], D_1) → U_2
  202. Vector(A'_{5}, U_2) → u_4
  203. Translate(A'_{5}, u_4) → V_2
  204. Translate(F_2, u_4) → W_2
  205. Segment(V_2, W_2) → p_4
  206. Polygon(A'_{5}, D, H_2, D_1) → q1, a'_{5}, d_5, j_2, l_2

AI 绘图提示语参考

绘制一个综合几何图形：先画一个正方形，以其一边为底边构造一个动态旋转的三角形，该三角形随参数变化经历缩放、旋转和反射。在正方形外侧构造一个圆和一条垂线，形成交点，并连接相关线段。再画一个以正方形对角线中点为圆心的圆弧，在圆弧上取一点，连接该点与正方形顶点形成三角形。最后在正方形各边附近添加多个小圆和交点，形成完整的几何变换演示图。
以上提示语为根据原始绘图命令整理而来，仅供参考。