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圆锥的展开图（Net of a Cone）

圆锥的展开图（Net of a Cone）
几何画板及绘图内容加载中，请稍等……
2020-08-16 23:34
圆多边形角立体几何几何作图
该范例展示圆锥表面的“展开图”如何由圆盘扇形拼接而成，用于理解从立体到平面的对应关系。核心数学概念包括：圆锥的母线（侧面展开半径）、圆周在展开后对应扇形的圆心角与弧长匹配、以及锥体曲面与平面扇形之间的几何变换与切线/正交构造思路。作图要点来自构造摘要中的关键命令与对象：使用 Point、Line、Angle、Arc、Circle/​CircleSector 生成圆与圆弧边界，结合 Ellipse、Vector、Translate、Rotate、Mirror、Tangent、OrthogonalLine 做辅助几何约束与位置变换；再通过 Intersect、Segment、Midpoint、Polygon、Conic/​tangency 相关对象（如 u、n、h、d、e、α 等）确定展开扇形的边界、中心与对应线段；最终用 Polygon 等将扇形与圆等组成“网”的轮廓。适合中学到大学初级层面的立体几何教学、圆锥建模与展开图理解，也适用于几何作图训练与课堂演示。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Circle(A, r) → c
  3. Intersect(c, xAxis) → B, C
  4. Ellipse(E, D, C) → e
  5. Intersect(xAxis, yAxis) → F
  6. Vector(F, G) → u
  7. Translate(B, Vector[(u * h)]) → B'_1
  8. Translate(C, Vector[(u * h)]) → C'_1
  9. Translate(A, Vector[(u * h)]) → A'_1
  10. Translate(e, Vector[(u * h)]) → e'_1
  11. Arc(e'_1, C'_1, B'_1) → p_5
  12. Arc(e'_1, B'_1, C'_1) → q
  13. Segment(A'_1, A) → h_6
  14. Segment(A'_1, C'_1) → p
  15. Angle(A, A'_1, C'_1) → α
  16. Line(A, A'_1) → j
  17. Intersect(e'_1, j) → H, I
  18. Circle(I, r) → t
  19. Intersect(t, j, 2) → K
  20. Circle(K, I) → d_1
  21. Intersect(d_1, j, 2) → N
  22. Vector(H, N) → v
  23. Translate(H, Vector[(v * n)]) → H'_1
  24. Translate(D, Vector[(u * h)]) → D'
  25. Translate(E, Vector[(u * h)]) → E'
  26. Line(B, B'_1) → l
  27. Line(C, C'_1) → m
  28. Intersect(d_1, l, 1) → P
  29. Intersect(d_1, m, 1) → Q
  30. Vector(B'_1, P) → z
  31. Vector(C'_1, Q) → u_1
  32. Translate(B'_1, Vector[(z * n)]) → B''
  33. Translate(C'_1, Vector[(u_1 * n)]) → C''
  34. Midpoint(B'_1, A'_1) → R
  35. Midpoint(B, A) → S
  36. Line(R, S) → a_1
  37. Intersect(q, a_1) → T,
  38. Intersect(d_1, a_1, 1) → U
  39. Vector(T, U) → v_1
  40. Translate(T, Vector[(v_1 * n)]) → T'
  41. If(n ≟ 0.01, B'_1, If[n < 1, B'', P]) → B_1
  42. If(n ≟ 0.01, T, If[n < 1, T', U]) → D_1
  43. If(n ≟ 0.01, H, If[n < 1, H'_1, N]) → E_1
  44. If(n ≟ 0.01, C'_1, If[n < 1, C'', Q]) → F_1
  45. Conic(B_1, D_1, E_1, F_1, I) → g_1
  46. Conic(I, B_1, D_1, E_1, F_1) → h_1
  47. OrthogonalLine(I, j) → l_1
  48. OrthogonalLine(H, j) → b_2
  49. Intersect(m, l_1) → H_2
  50. Midpoint(H_2, C'_1) → I_2
  51. OrthogonalLine(C'_1, m) → i_2
  52. Segment(K, F_1) → m_4
  53. Translate(e, Vector[(u * h)]) → e'
  54. Tangent(A, e'_1) → a, b
  55. Intersect(e', b) → J, G_2
  56. Intersect(e', a) → L, R_2
  57. Segment(A, J) → f
  58. Segment(A, L) → g
  59. Circle(I, sqrt(h^(2) + r^(2))) → k
  60. Intersect(k, j, 1) → M
  61. Circle(M, sqrt(h^(2) + r^(2))) → s
  62. Segment(M, I) → i
  63. Rotate(I, γ, M) → I'_1
  64. Rotate(I, (-γ), M) → I'_2
  65. OrthogonalLine(I'_2, i_2) → b_1
  66. Intersect(b_2, b_1) → O
  67. Midpoint(H, O) → V
  68. Circle(V, O) → c_1
  69. Vector(O, I'_2) → w
  70. Center(g_1) → W
  71. Segment(W, F_1) → e_1
  72. Vector(H, A) → w_1
  73. Translate(H, Vector[(w_1 * (n - 1))]) → H'
  74. If(n < 1, H, If[n < 2, H', A]) → Z
  75. Segment(H, Z) → f_1
  76. Rotate(I, (-δ), M) → I'
  77. Vector(L, I') → z_1
  78. Translate(c_1, Vector[(w * (n - 2))]) → k_1
  79. Center(k_1) → A_1
  80. Intersect(k_1, j) → C_1, Q_1
  81. Rotate(C_1, (180° * (n - 2)), A_1) → G_1
  82. If(n < 2, H, If[n ≤ 3, G_1, I'_2]) → H_1
  83. Mirror(H_1, j) → I_1
  84. Translate(L, Vector[(z_1 * (n - 2))]) → L'
  85. If(n < 2, L, If[n ≤ 3, L', I']) → J_1
  86. Mirror(J_1, j) → K_1
  87. Conic(I_1, H_1, J_1, I, K_1) → p_1
  88. Vector(A, M) → i_1
  89. Translate(A, Vector[(i_1 * (n - 2))]) → A'
  90. If(n < 2, A, If[n ≤ 3, A', M]) → L_1
  91. Tangent(L_1, p_1) → j_1, m_1
  92. Intersect(p_1, m_1, 1) → M_1
  93. Intersect(p_1, j_1, 1) → N_1
  94. Polygon(L_1, M_1, N_1) → poly1, n_1, l_2, m_2
  95. Arc(p_1, N_1, M_1) → q_1
  96. Line(L_1, b_2) → r_1
  97. Intersect(p_1, r_1, 1) → O_1
  98. Intersect(p_1, r_1, 2) → P_1
  99. Arc(p_1, P_1, O_1) → s_1
  100. Polygon(L_1, O_1, I_1) → poly2, i_3, l_3, o_1
  101. Polygon(L_1, H_1, P_1) → poly3, p_2, l_4, h_2
  102. Arc(p_1, O_1, I_1) → t_1
  103. Arc(p_1, H_1, P_1) → c_2
  104. Segment(L_1, I_1) → a_2
  105. Segment(L_1, H_1) → d_2
  106. Arc(p_1, H_1, I_1) → e_2
  107. Polygon(I_1, L_1, H_1) → poly4, h_3, i_4, l_5
  108. CircleSector(M, I'_2, I'_1) → f_2
  109. CircleSector(M, I'_2, I'_1) → g_2
  110. Arc(p_1, H_1, I_1) → k_2
  111. Segment(M_1, L_1) → j_2
  112. Segment(N_1, L_1) → n_2
  113. Polygon(I_1, L_1, H_1) → poly5, h_4, i_5, l_6
  114. Arc(p_1, H_1, I_1) → q_2
  115. Polygon(J, A, L) → poly6, l_7, j_3, a_3
  116. Point(xAxis) → R_1
  117. OrthogonalLine(R_1, xAxis) → r_2
  118. Intersect(i_2, r_2) → S_1
  119. Circle(R_1, r) → s_2
  120. Intersect(s_2, xAxis) → T_1, U_1
  121. OrthogonalLine(T_1, i_2) → t_2
  122. OrthogonalLine(V_1, i_2) → b_3
  123. OrthogonalLine(W_1, i_2) → c_3
  124. Intersect(i_2, t_2) → Z_1
  125. Intersect(i_2, b_3) → A_2
  126. Intersect(i_2, c_3) → B_2
  127. Ellipse(A_2, B_2, Z_1) → d_3
  128. Tangent(R_1, d_3) → e_3, f_3
  129. Intersect(d_3, f_3, 1) → C_2
  130. Intersect(d_3, e_3, 1) → D_2
  131. Polygon(C_2, D_2, R_1) → poly7, r_3, c_4, d_4
  132. Ellipse(A_2, B_2, Z_1) → g_3
  133. OrthogonalLine(U_1, xAxis) → k_3
  134. Intersect(g_3, k_3) → E_2, J_2
  135. Segment(S_1, E_2) → m_3
  136. Segment(S_1, R_1) → n_3
  137. Intersect(d_3, m_3, 1) → F_2
  138. Midpoint(g) → K_2
  139. Midpoint(f_1) → L_2
  140. Point(j) → M_2
  141. Midpoint(I_1, A') → N_2
  142. Midpoint(A', H_1) → O_2
  143. Vector(M_2, N_2) → u_2
  144. Vector(M_2, O_2) → v_2
  145. Line(M_2, r_1) → p_3
  146. Point(p_3) → P_2
  147. Vector(P_2, K_2) → z_2
  148. Vector(P_2, L_2) → u_3
  149. Center(g_3) → Q_2
  150. Angle(F_2, Q_2, R_1) → ε
  151. Segment(I_1, H_1) → q_3
  152. Intersect(k_2, q_3, 2) → S_2
  153. Point(poly7) → T_2

AI 绘图提示语参考

绘制一个锥体的展开图，展示其侧面展开为平面扇形的过程。图形包含一个圆形扇形（代表展开后的锥体侧面），一个圆（代表锥体的底面），以及连接两者的弧线。扇形的半径对应锥体的斜高，扇形的弧长等于锥体底面周长。构造时，先确定扇形的半径和弧长，计算中心角，然后绘制扇形和圆，最后用弧线连接两者，形成完整的展开图。适用于几何教学和可视化演示。
以上提示语为根据原始绘图命令整理而来，仅供参考。