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几何面积反射演示

几何面积反射演示
几何画板及绘图内容加载中，请稍等……
2020-10-22 05:14
向量三角函数解析几何几何作图
①本范例演示了通过反射变换探索几何图形面积变化的现象；②涉及的核心数学概念包括面积计算、反射变换、解析几何、三角函数；③作图要点提炼自构造摘要，使用了Line, Point, Vector, Translate, OrthogonalLine, Intersect, Segment, Polygon, Midpoint, Circle, Mirror, Semicircle, Angle, Rotate, CircumcircleArc, Sequence, Min, Max等GeoGebra命令与对象；④适合高中及以上学段，用于几何变换与面积计算的深入学习与可视化教学。

GeoGebra 命令及参数

  1. Line(A, xAxis) → b_3
  2. Point(b_3) → B
  3. Vector(A, O) → e
  4. Translate(B, e) → B'_4
  5. Line(A, B) → a_6
  6. OrthogonalLine(B, a_6) → c
  7. Translate(C, e) → C'_4
  8. OrthogonalLine(C, c) → d
  9. OrthogonalLine(A, a_6) → b
  10. Intersect(d, b) → D
  11. Translate(D, e) → D'_4
  12. Segment(E, E + (2, 0)) → a
  13. Vector(A, E_2) → u
  14. Vector(A, F_2) → v
  15. Vector(A, G) → w
  16. Vector(A, H) → z
  17. Polygon(A, B, C, D) → poly1, a_1, b_1, c_1, d_1
  18. Translate(A, u) → A'
  19. Translate(B, u) → B'
  20. Translate(C, u) → C'
  21. Translate(D, u) → D'
  22. Polygon(A', B', C', D') → poly1', a', b', c', d'
  23. Translate(A, v) → A'_1
  24. Translate(B, v) → B'_1
  25. Translate(C, v) → C'_1
  26. Translate(D, v) → D'_1
  27. Polygon(A'_1, B'_1, C'_1, D'_1) → poly1'_1, a'_1, b'_1, c'_1, d'_1
  28. Translate(A, w) → A'_2
  29. Translate(B, w) → B'_2
  30. Translate(C, w) → C'_2
  31. Translate(D, w) → D'_2
  32. Polygon(A'_2, B'_2, C'_2, D'_2) → poly1'_2, a'_2, b'_2, c'_2, d'_2
  33. Translate(A, z) → A'_3
  34. Translate(B, z) → B'_3
  35. Translate(C, z) → C'_3
  36. Translate(D, z) → D'_3
  37. Polygon(A'_3, B'_3, C'_3, D'_3) → poly1'_3, a'_3, b'_3, c'_3, d'_3
  38. Translate(A, e) → A'_4
  39. Polygon(O, B'_4, C'_4, D'_4) → poly1'_4, j_2, m_2, q_2, r_2
  40. Point(c') → J
  41. Line(J, a') → f
  42. Line(J, E_2) → g
  43. Line(B', g) → h
  44. Intersect(f, h) → K
  45. Midpoint(C', E_2) → L
  46. Midpoint(K, B') → N
  47. Midpoint(J, E_2) → M
  48. Segment(M, N) → i
  49. Point(i) → O_2
  50. Circle(O_2, Distance[N, K]) → q
  51. Line(O_2, h) → j
  52. Intersect(q, j) → P, Q
  53. Line(Q, d') → l_2
  54. Intersect(f, l_2) → R_2
  55. Polygon(R_2, P, Q) → poly3, q_1, r, p_1
  56. Midpoint(F_2, B'_1) → S
  57. Midpoint(c'_1) → T
  58. Midpoint(a'_1) → U
  59. Point(c'_2) → V
  60. Line(V, d'_2) → m
  61. Intersect(m, a'_2) → W
  62. Mirror(C, W) → C'_5
  63. Mirror(W, V) → W'
  64. Mirror(S, T) → S'
  65. Polygon(W', G, B'_2) → poly4, b'_6, w', g_1
  66. Polygon(S', D'_1, S, C'_1) → poly5, s', d'_5, s, c'_6
  67. Intersect(g_1, c'_2) → Z
  68. Intersect(c'_2, b'_6) → A_1
  69. Polygon(Z, C'_2, B'_2) → poly6, b'_7, z_1, c'_7
  70. Polygon(D'_2, A_1, G) → poly7, g_2, d'_6, a_2
  71. Line(D'_3, C'_3) → n_2
  72. Point(n_2) → B_1
  73. Point(n_2) → C_1
  74. Polygon(B_1, H, B'_3, C_1) → poly8, b_2, h_1, b'_8, c_2
  75. Polygon(A_1, Z, B'_2, G) → poly9, a_3, z_2, b'_9, g_4
  76. Polygon(J, K, B', E_2) → poly10, j_1, k_1, b'_{10}, e_2
  77. Segment(S', S) → p
  78. Point(p) → H_1
  79. Midpoint(c'_6) → I_1
  80. Midpoint(d'_5) → J_1
  81. Midpoint(s') → K_1
  82. Midpoint(s) → L_1
  83. Vector(I_1, J_1) → t
  84. Vector(K_1, L_1) → u_1
  85. Polygon(D'_1, T, S') → poly11, s'_1, d'_7, t_1
  86. Translate(D'_1, Vector[(Distance[S', H_1] / Distance[S', S] * u_1)]) → M_1
  87. Translate(T, Vector[(Distance[S', H_1] / Distance[S', S] * u_1)]) → T'
  88. Translate(S', Vector[(Distance[S', H_1] / Distance[S', S] * u_1)]) → S''
  89. Polygon(M_1, T', S'') → poly11', s'', m_1, t'
  90. Polygon(S', T, C'_1) → poly12, c'_8, s'_2, t_2
  91. Translate(S', Vector[(Distance[S', H_1] / Distance[S', S] * t)]) → S''_1
  92. Translate(T, Vector[(Distance[S', H_1] / Distance[S', S] * t)]) → T'_1
  93. Translate(C'_1, Vector[(Distance[S', H_1] / Distance[S', S] * t)]) → N_1
  94. Polygon(S''_1, T'_1, N_1) → poly12', n_1, s''_1, t'_1
  95. Polygon(D'_1, C'_1, S) → poly13, s_1, d'_8, c'_9
  96. Segment(V, W) → i_1
  97. Semicircle(W', B'_2) → r_1
  98. Semicircle(G, W') → f_2
  99. Point(r_1) → O_1
  100. Segment(O_1, Z) → l_1
  101. Semicircle(V, C'_2) → h_2
  102. Angle(B'_2, Z, C'_2) → α
  103. Rotate(O_1, α, Z) → D_1
  104. Segment(Z, D_1) → e_1
  105. Intersect(h_2, e_1) → E_1,
  106. Polygon(O_1, Z, E_1) → poly14, e_3, o_1, z_3
  107. Point(f_2) → F_1
  108. Semicircle(D'_2, V) → f_1
  109. Angle(D'_2, A_1, G) → β
  110. Rotate(F_1, (-β), A_1) → G_1
  111. Segment(G_1, A_1) → d_2
  112. Intersect(f_1, d_2) → P_1,
  113. Polygon(F_1, P_1, A_1) → poly15, a_4, f_3, p_2
  114. Polygon(A_1, V, W') → poly16, w'_1, a_5, v_1
  115. Midpoint(b'_8) → Q_1
  116. Mirror(H, B'_3) → H'
  117. Line(B_1, C_1) → i_2
  118. Mirror(H, Q_1) → H'_1
  119. CircumcircleArc(H, B_1, H'_1) → k_2
  120. Point(k_2) → R_1
  121. Angle(R_1, Q_1, H) → γ
  122. Rotate(B_1, (-γ), Q_1) → S_1
  123. Rotate(H, (-γ), Q_1) → H'_2
  124. Rotate(B'_3, (-γ), Q_1) → T_1
  125. Rotate(C_1, (-γ), Q_1) → U_1
  126. Polygon(S_1, H'_2, T_1, U_1) → poly8', s_2, h'_2, t_3, u_2
  127. Point(a) → F
  128. Sequence((x(O) + (R * sin((i * θ))), y(O) + R - (R * cos((i * θ)))), i, 0, (2 * n)) → list1
  129. Min((x(F) - x(E)) / (a / 2), 1) → k
  130. Sequence(If[k < 1, (x(O) + ((l_3 * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O) + ((l_3 * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O) + (l_3 * i), y(O))], i, 0, (2 * n)) → list2
  131. Max((x(F) - x(E)) / (a / 2) - 1, 0) → k'
  132. Sequence(Sector[Rotate[Translate[Circle[(x(O) + 0, y(O) + R), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]]], i, 1, n, 1) → liste1
  133. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O) + 0, y(O) + R), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l_3) - (R * sin(θ / 2) / 2), y(O) + (R * cos(θ / 2) / 2))], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l_3) - (R * sin(θ / 2) / 2), y(O) + (R * cos(θ / 2) / 2))], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O) + (n * l_3) - (R * sin(θ / 2) / 2), y(O) + (R * cos(θ / 2) / 2))]], i, n + 1, (2 * n)) → liste2
  134. Midpoint(m_2) → V_1
  135. Midpoint(r_2) → W_1
  136. Segment(V_1, W_1) → c_3
  137. Midpoint(C, A) → Z_1
  138. Midpoint(V_1, B'_4) → A_2
  139. Midpoint(j_2) → B_2
  140. Midpoint(J, B') → C_2
  141. Midpoint(a') → D_2
  142. Midpoint(p_1) → G_2
  143. Segment(H_1, S) → d_3
  144. Midpoint(b'_9) → H_2
  145. Midpoint(c_1) → I_2
  146. Midpoint(b_1) → J_2
  147. Midpoint(L_2, K_2) → M_2
  148. Vector(M_2, L_2) → l
  149. Vector(M_2, K_2) → w_1
  150. Midpoint(s_1) → N_2
  151. Vector(M_2, P_2) → v_2
  152. Midpoint(K_2, P_2) → Q_2
  153. Midpoint(S_2, T_2) → U_2
  154. Vector(U_2, T_2) → w_2
  155. Vector(U_2, S_2) → u_3
  156. OrthogonalLine(V, b'_9) → g_3
  157. Intersect(g_3, b'_9) → V_2
  158. Segment(V, V_2) → h_3
  159. Midpoint(h_3) → W_2
  160. OrthogonalLine(B_1, h_1) → i_3
  161. Intersect(i_3, h_1) → Z_2
  162. Segment(B_1, Z_2) → j_3
  163. Midpoint(j_3) → A_3
  164. Midpoint(B_1, C_1) → B_3
  165. Midpoint(h_1) → C_3
  166. Segment(W_1, O) → k_3
  167. Sequence(Segment[Element[list2, i], Element[list2, i + 1]], i, 1, Length[list2]) → liste3
  168. Segment(D_3, E_3) → m_3
  169. Segment(F_3, G_3) → n_3
  170. Segment(H_3, I_3) → p_3
  171. Midpoint(d_1) → J_3
  172. Midpoint(J_3, W_1) → K_3
  173. Midpoint(W_1, K_3) → L_3
  174. Midpoint(K_3, J_3) → M_3
  175. Midpoint(b_1) → N_3
  176. Midpoint(e_2) → O_3
  177. Midpoint(N_3, O_3) → P_3
  178. Midpoint(N_3, P_3) → Q_3
  179. Midpoint(P_3, O_3) → R_3
  180. Midpoint(N_3, D'_1) → S_3
  181. Midpoint(S_3, N_3) → T_3
  182. Midpoint(S_3, D'_1) → U_3
  183. Vector(Q_3, R_3) → v_3
  184. Vector(T_3, U_3) → w_3
  185. Vector(M_3, L_3) → q_3
  186. Midpoint(A, C_1) → V_3
  187. Midpoint(A, V_3) → W_3
  188. Midpoint(V_3, C_1) → Z_3
  189. Vector(W_3, Z_3) → r_3
  190. Midpoint(a_1) → A_4
  191. Midpoint(z_2) → B_4
  192. Midpoint(A_4, B_4) → C_4
  193. Midpoint(A_4, C_4) → D_4
  194. Midpoint(C_4, B_4) → E_4
  195. Vector(D_4, E_4) → s_3

AI 绘图提示语参考

绘制一个四边形，其顶点由一条水平线与一条斜线相交确定，并基于该四边形进行多种平移变换。首先，在坐标轴上取一点，构造一个平行四边形，再通过垂线构造出另一个点，形成完整的四边形。然后，以多个不同方向的向量平移该四边形的所有顶点，生成多个全等的平移副本，并分别连接对应顶点形成多边形。最后，在其中一个平移后的图形上，选取边上的点，构造辅助线，并标出中点，连接中点形成线段，以展示面积在反射和平移下的变化关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。