积极 AI Logo 积极 AI

无限正方形级数

无限正方形级数
几何画板及绘图内容加载中，请稍等……
2019-07-24 19:08
三角函数数列解析几何微积分几何作图
该范例通过GeoGebra动态构建，展示了无限正方形排列的几何图形及其面积、边长的无限级数行为。核心数学概念包括级数极限、几何级数求和、三角函数应用与解析几何坐标构图。作图要点利用Polygon、Sequence、Sum、IntegralBetween、OrthogonalLine等命令，结合Point、Line、Conic等对象实现正方形的递进排列与性质动态变化。适合高中及大学微积分、级数收敛性探究的教学演示与自学。

GeoGebra 命令及参数

  1. Polygon(A, B, 4) → polígono1, f, g, h, i, C, D
  2. CircleArc(D, A, C) → c
  3. Segment(D, B) → j
  4. Intersect(c, j, 1) → E
  5. OrthogonalLine(E, g) → k
  6. OrthogonalLine(E, f) → l
  7. Intersect(k, g) → F
  8. Intersect(l, f) → G
  9. Line(G, j) → m
  10. Intersect(c, m, 1) → H
  11. Line(H, k) → n
  12. Line(H, l) → p
  13. Intersect(l, n) → I
  14. Intersect(p, f) → J
  15. Line(J, m) → q
  16. Intersect(c, q, 1) → K
  17. Line(K, n) → r
  18. Line(K, p) → s
  19. Intersect(p, r) → L
  20. Intersect(s, f) → M
  21. Line(M, q) → t
  22. Intersect(c, t, 1) → N
  23. Line(N, r) → a
  24. Line(N, s) → b
  25. Intersect(a, s) → O
  26. Intersect(f, b) → P
  27. Line(P, t) → d
  28. Intersect(c, d, 1) → Q
  29. Line(Q, a) → e
  30. Line(Q, b) → f_1
  31. Intersect(b, e) → R
  32. Intersect(f_1, f) → S
  33. Line(S, d) → g_1
  34. Intersect(c, g_1, 1) → T
  35. Line(T, e) → h_1
  36. Line(T, f_1) → i_1
  37. Intersect(f_1, h_1) → U
  38. Intersect(i_1, f) → V
  39. Line(V, g_1) → j_1
  40. Intersect(c, j_1, 1) → W
  41. Line(W, h_1) → k_1
  42. Line(W, i_1) → l_1
  43. Intersect(k_1, i_1) → Z
  44. Intersect(f, l_1) → A_1
  45. Line(A_1, j_1) → m_1
  46. Intersect(c, m_1, 1) → B_1
  47. Line(B_1, k_1) → n_1
  48. Line(B_1, l_1) → p_1
  49. Intersect(l_1, n_1) → C_1
  50. Intersect(p_1, f) → D_1
  51. Line(D_1, m_1) → q_1
  52. Intersect(c, q_1) → E_1,
  53. Line(E_1, n_1) → r_1
  54. Line(E_1, p_1) → s_1
  55. Intersect(p_1, r_1) → F_1
  56. Intersect(s_1, f) → G_1
  57. Line(G_1, q_1) → t_1
  58. Intersect(c, t_1, 1) → H_1
  59. Line(H_1, s_1) → a_1
  60. Line(H_1, r_1) → b_1
  61. Intersect(b_1, s_1) → I_1
  62. Intersect(a_1, f) → J_1
  63. Line(J_1, t_1) → c_1
  64. Intersect(c, c_1, 1) → K_1
  65. Line(K_1, b_1) → d_1
  66. Line(K_1, a_1) → e_1
  67. Intersect(a_1, d_1) → L_1
  68. Intersect(e_1, f) → M_1
  69. Line(M_1, c_1) → f_2
  70. Intersect(c, f_2, 1) → N_1
  71. Line(N_1, d_1) → g_2
  72. Line(N_1, e_1) → h_2
  73. Intersect(e_1, g_2) → O_1
  74. Intersect(h_2, f) → P_1
  75. Line(P_1, f_2) → i_2
  76. Intersect(c, i_2) → Q_1,
  77. Line(Q_1, g_2) → j_2
  78. Line(Q_1, h_2) → k_2
  79. Intersect(h_2, j_2) → R_1
  80. Intersect(k_2, f) → S_1
  81. Polygon(S_1, P_1, 4) → p2, l_2, m_2, n_2, p_2, T_1, U_1
  82. Polygon(P_1, M_1, 4) → p3, q_2, r_2, s_2, t_2, V_1, W_1
  83. Polygon(M_1, J_1, 4) → p4, a_2, b_2, c_2, d_2, Z_1, A_2
  84. Polygon(J_1, G_1, 4) → p5, e_2, f_3, g_3, h_3, B_2, C_2
  85. Polygon(G_1, D_1, 4) → p6, i_3, j_3, k_3, l_3, D_2, E_2
  86. Polygon(D_1, A_1, 4) → p7, m_3, n_3, p_3, q_3, F_2, G_2
  87. Polygon(A_1, V, 4) → p8, r_3, s_3, t_3, a_3, H_2, I_2
  88. Polygon(V, S, 4) → p9, b_3, c_3, d_3, e_3, J_2, K_2
  89. Polygon(S, P, 4) → p10, f_4, g_4, h_4, i_4, L_2, M_2
  90. Polygon(P, M, 4) → p11, j_4, k_4, l_4, m_4, N_2, O_2
  91. Polygon(M, J, 4) → p12, n_4, p_4, q_4, r_4, P_2, Q_2
  92. Polygon(J, G, 4) → p13, s_4, t_4, a_4, b_4, R_2, S_2
  93. Polygon(G, B, 4) → p14, c_4, d_4, e_4, f_5, T_2, U_2
  94. Sequence(2, 14) → l2
  95. Sum(lista1) → o
  96. Intersect(h, i) → V_2
  97. Intersect(g, h) → W_2
  98. Angle(E, V_2, W_2) → α
  99. Angle(H, V_2, E) → β
  100. Angle(K, V_2, H) → γ
  101. Angle(N, V_2, K) → δ
  102. Sequence((Element[l2, i], Element[lista1, i]), i, 1, 13) → l3
  103. Intersect(h, i) → Z_2
  104. Intersect(c, f, 1) → A_3
  105. Point(f) → B_3
  106. Line(A_3, B_3) → g_5
  107. Parabola((-4, 2), y = -10) → c_5
  108. Conic(F, I, L, O, R) → d_5
  109. Tangent((-4, -4), d_5) → h_5,
  110. Midpoint(E, B_3) → C_3
  111. Midpoint(H, G) → D_3
  112. Midpoint(K, J) → E_3
  113. Midpoint(N, M) → F_3
  114. Midpoint(Q, P) → G_3
  115. Midpoint(T, S) → H_3
  116. Conic(H_3, G_3, F_3, E_3, D_3) → e_5
  117. Intersect(e_5, q_5) → I_3, J_3
  118. Tangent(I_3, e_5) → i_5,
  119. Midpoint(J_3, I_3) → K_3
  120. Line(K_3, i_5) → j_5
  121. Intersect(e_5, j_5) → L_3, M_3
  122. Intersect(Y, d_5) → N_3, O_3
  123. Intersect(d_5, p_5) → P_3, Q_3
  124. Segment(N_3, O_3) → k_5
  125. Segment(Q_3, P_3) → l_5
  126. Segment(Z_2, B) → r_5
  127. Slope(l_5) → m_6
  128. Slope(k_5) → m_7
  129. Segment(W_2, A) → m_5
  130. Intersect(r_5, m_5) → R_3
  131. Parabola((-4, 0), y = -8) → s_5
  132. IntegralBetween(s_5, (0 * x) - 4, -4, 4) → u
  133. Midpoint(Q, R) → S_3
  134. Midpoint(N, O) → T_3
  135. Midpoint(K, L) → U_3
  136. Midpoint(H, I) → V_3
  137. Midpoint(E, F) → W_3
  138. Conic(W_3, V_3, U_3, T_3, S_3) → t_5
  139. Parabola((-4, 4), y = -12) → c_6
  140. IntegralBetween(c_6, (0 * x) - 4, -4, 4) → v
  141. Intersect(e_5, g) → B_4, C_4
  142. Intersect(e_5, i) → Z_3, A_4
  143. Intersect(f, g) → D_4
  144. Segment(Z_3, D_4) → n_5
  145. Segment(D_4, C_4) → a_5
  146. CurveCartesian((-8.908 * cos(t)) + (0.803 * sin(t)), (-2.103 * cos(t)) - (3.403 * sin(t)), t, 1.03, 1.521) → w
  147. Intersect(b_4, f) → E_4
  148. Segment(E_4, Z_3) → b_5
  149. Intersect(e_5, b_4) → F_4,
  150. Segment(F_4, E_4) → f_6
  151. Vector(G_4, H_4) → v_1

AI 绘图提示语参考

绘制一个无限序列的正方形图形，首先绘制一个正方形，然后通过正方形的顶点构造一系列正方形，每个新正方形的边与前一个正方形的边垂直，且逐渐缩小。从一个正方形开始，通过旋转和交点构造出一系列正方形，直到形成一个无限序列的正方形图案。
以上提示语为根据原始绘图命令整理而来，仅供参考。