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动态圆锥与概率分布可视化

动态圆锥与概率分布可视化
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2018-02-04 00:17
三角函数立体几何概率统计几何作图
①本范例动态演示圆锥、单位圆与圆柱的几何关系及概率分布现象；②涉及核心数学概念包括立体几何、三角函数、概率统计与随机过程；③作图要点运用了Circle（圆）、Ray（射线）、Point（点）、Semicircle（半圆）、Segment（线段）、Polygon（多边形）、Midpoint（中点）、Angle（角）、OrthogonalLine（垂线）、Intersect（交点）、Rotate（旋转）、Line（直线）、Distance（距离）、Dilate（缩放）、Min（最小值）、CircleSector（扇形）、Vector（向量）、Translate（平移）等GeoGebra命令与对象；④适合高中及以上学段，用于几何建模与概率统计教学。

GeoGebra 命令及参数

  1. Circle(A, B) → c
  2. Ray(A, B) → f
  3. Point(f) → C
  4. Semicircle(A, C) → d
  5. Point(d) → D
  6. Segment(A, D) → g
  7. Segment(D, C) → h
  8. Segment(C, A) → i
  9. Polygon(A, C, D) → t1, d_1, a, c_1
  10. Midpoint(A, C) → E
  11. Angle(A, D, C) → α
  12. OrthogonalLine(D, i) → j
  13. Intersect(j, i) → F
  14. Angle(D, F, A) → β
  15. Segment(D, F) → k
  16. Angle(D, C, F) → γ
  17. Angle(F, A, D) → δ
  18. Rotate(F, δ / 2, A) → F'
  19. Angle(F, A, F') → ε
  20. Rotate(D, γ / 2, C) → D'
  21. Angle(D, C, D') → ζ
  22. Ray(A, F') → l
  23. Ray(C, D') → m
  24. Rotate(D, π / 4, F) → D'_1
  25. Angle(D, F, D'_1) → η
  26. Ray(F, D'_1) → n
  27. Rotate(C, π / 4, F) → C'
  28. Angle(C, F, C') → θ
  29. Ray(F, C') → p
  30. Intersect(l, n) → G
  31. Intersect(m, p) → H
  32. OrthogonalLine(G, g) → q
  33. Intersect(g, q) → I
  34. OrthogonalLine(H, i) → r
  35. Intersect(i, xAxis) → J
  36. Intersect(r, i) → K
  37. Line(D, C) → s
  38. Ray(G, H) → t
  39. Intersect(s, t) → L
  40. Distance(G, L) → distanceGL
  41. Distance(F, D) → distanceFD
  42. Distance(D, L) → distanceDL
  43. Distance(D, F) → distanceDF
  44. Dilate(D, If[0 < b ≤ 1, 1 - b, 0], F) → D'_2
  45. Segment(D, D'_2) → e
  46. Dilate(F, If[1 < b ≤ 2, 2 - b, 0], H) → F'_1
  47. Dilate(C, If[1 < b ≤ 2, 2 - b, 0], H) → C'_1
  48. Segment(F, F'_1) → f_1
  49. Segment(C, C'_1) → g_1
  50. Segment(C, L) → h_1
  51. Segment(F, K) → i_1
  52. Intersect(t, e) → M
  53. Segment(F, M) → j_1
  54. Segment(F, A) → k_1
  55. Segment(A, I) → l_1
  56. Min({h_1, i_1, j_1, k_1, l_1}) → o
  57. Circle(C, (0.7 * o)) → e_1
  58. Circle(F, (0.7 * o)) → p_1
  59. Circle(A, (0.7 * o)) → q_1
  60. Intersect(i, xAxis) → N
  61. Intersect(i, xAxis) → O
  62. Intersect(i, xAxis) → P
  63. Intersect(q_1, i) → Q,
  64. Intersect(p_1, i) → R, S
  65. Intersect(p_1, e_1) → T, , ,
  66. Intersect(e_1, s) → U, V
  67. Rotate(U, If[2 < b ≤ 3, ((b - 2) * ζ), ζ], C) → U'
  68. Rotate(R, If[2 < b ≤ 3, ((b - 2) * π / 4), π / 4], F) → R'
  69. CircleSector(C, U, U') → r_1
  70. CircleSector(F, R, R') → s_1
  71. Rotate(r_1, If[4 < b ≤ 5, ((b - 4) * ζ), ζ], C) → t_1
  72. Rotate(s_1, If[4 < b ≤ 5, ((b - 4) * π / 4), π / 4], F) → c_2
  73. Dilate(A, If[5 < b ≤ 6, 6 - b, 0], G) → A'
  74. Dilate(F, If[5 < b ≤ 6, 6 - b, 0], G) → F'_2
  75. Segment(A, A') → m_1
  76. Segment(F, F'_2) → n_1
  77. Rotate(Q, If[6 < b ≤ 7, ((b - 6) * ε), ε], A) → Q'
  78. CircleSector(A, Q, Q') → d_2
  79. Rotate(d_2, If[8 < b ≤ 9, ((b - 8) * ε), ε], A) → e_2
  80. Rotate(c_2, If[8 < b ≤ 9, ((b - 8) * π / 4), π / 4], F) → f_2
  81. Dilate(G, If[9 < b ≤ 10, 10 - b, 0], H) → G'
  82. Segment(G, G') → a_1
  83. Dilate(H, If[10 < b ≤ 11, 11 - b, 0], L) → H'
  84. Segment(H, H') → b_1
  85. Dilate(D, If[11 < b ≤ 12, 12 - b, 0], L) → D'_3
  86. Angle(F, D, C) → ρ
  87. Rotate(D, If[13 < b ≤ 14, ((b - 13) * ρ), ρ], D) → D'_4
  88. Rotate(D'_2, If[13 < b ≤ 14, ((b - 13) * ρ), ρ], D) → W
  89. Line(G, H) → h_2
  90. Line(A, D) → i_2
  91. Line(D, C) → j_2
  92. Intersect(i_2, h_2) → Z
  93. Intersect(j_2, h_2) → A_1
  94. Dilate(G, If[10 < b ≤ 11, 11 - b, 0], Z) → G'_1
  95. Segment(G, G'_1) → k_2
  96. Dilate(F, If[11 < b ≤ 12, 12 - b, 0], Z) → F'_3
  97. Dilate(F, If[11 < b ≤ 12, 12 - b, 0], L) → F'_4
  98. Segment(F, F'_3) → g_2
  99. Segment(F, F'_4) → l_2
  100. Ray(A, D) → m_2
  101. Ray(C, D) → n_2
  102. Circle(D, 1) → p_2
  103. Intersect(p_2, n_2, 2) → B_1
  104. Angle(B_1, D, A) → ι
  105. Circle(D, (0.7 * o)) → q_2
  106. Intersect(q_2, m_2, 2) → C_1
  107. Rotate(C_1, ((-π)) / 4, D) → D_1
  108. Ray(D, D_1) → r_2
  109. Circle(D, (1.4 * o)) → s_2
  110. Intersect(s_2, r_2, 1) → E_1
  111. Rotate(C_1, If[13 < b ≤ 14, ((-(b - 13)) * π / 4), ((-π)) / 4], D) → F_1
  112. CircleSector(D, F_1, C_1) → t_2
  113. Dilate(D, If[13 < b ≤ 14, 14 - b, 0], E_1) → D'_5
  114. Segment(D, D'_5) → a_2
  115. Rotate(t_2, If[15.5 < b ≤ 16.5, ((-(b - 15.5)) * π / 4), ((-π)) / 4], D) → c_3
  116. Midpoint(D, F) → G_1
  117. Line(D, l_2) → b_2
  118. Point(m_2) → H_1
  119. Dilate(D, If[17.5 < b ≤ 18.5, 18.5 - b, 0], G_1) → D'_6
  120. Vector(D, D'_6) → u
  121. Translate(c_3, u) → d_3
  122. Translate(D, u) → D'_7
  123. Translate(D'_5, u) → I_1
  124. Segment(D'_7, I_1) → f_3
  125. Rotate(D'_7, If[18.5 < b ≤ 19.5, ((b - 18.5) * π / 2), π / 2], G_1) → J_1
  126. Rotate(I_1, If[18.5 < b ≤ 19.5, ((b - 18.5) * π / 2), π / 2], G_1) → K_1
  127. Point(b_2) → L_1
  128. Angle(C, D, L_1) → κ
  129. Rotate(d_3, If[18.5 < b ≤ 19.5, ((b - 18.5) * κ), κ], G_1) → e_3
  130. Rotate(D'_7, If[18.5 < b ≤ 19.5, ((b - 18.5) * κ), κ], G_1) → M_1
  131. Rotate(I_1, If[18.5 < b ≤ 19.5, ((b - 18.5) * κ), κ], G_1) → N_1
  132. Segment(M_1, N_1) → g_3
  133. Dilate(G_1, If[19.5 < b ≤ 20.5, 20.5 - b, 0], F) → O_1
  134. Vector(G_1, O_1) → v
  135. Translate(e_3, v) → h_3
  136. Translate(M_1, v) → P_1
  137. Translate(N_1, v) → Q_1
  138. Segment(P_1, Q_1) → i_3
  139. Dilate(D, If[20.5 < b ≤ 21.5, 21.5 - b, 0], G_1) → D'_8
  140. Vector(D, D'_8) → w
  141. Translate(D, w) → D'_9
  142. Translate(D'_5, w) → R_1
  143. Segment(D'_9, R_1) → j_3
  144. Translate(c_3, w) → k_3
  145. Rotate(Q_1, π / 4, F) → S_1
  146. Segment(F, S_1) → l_3
  147. Line(G_1, l_3) → m_3
  148. Point(m_3) → T_1
  149. Angle(R_1, G_1, T_1) → λ
  150. Rotate(k_3, If[21.5 < b ≤ 22.5, ((b - 21.5) * λ), λ], G_1) → p_3
  151. Rotate(D'_9, If[21.5 < b ≤ 22.5, ((b - 21.5) * λ), λ], G_1) → U_1
  152. Rotate(R_1, If[21.5 < b ≤ 22.5, ((b - 21.5) * λ), λ], G_1) → V_1
  153. Segment(U_1, V_1) → n_3
  154. Dilate(G_1, If[22.5 < b ≤ 23.5, 23.5 - b, 0], F) → W_1
  155. Vector(G_1, W_1) → u_1
  156. Translate(p_3, u_1) → q_3
  157. Translate(U_1, u_1) → Z_1
  158. Translate(V_1, u_1) → A_2
  159. Segment(Z_1, A_2) → r_3
  160. Midpoint(D, Z) → B_2
  161. Dilate(D, If[23.5 < b ≤ 24.5, 24.5 - b, 0], B_2) → D'_{10}
  162. Vector(D, D'_{10}) → v_1
  163. Translate(c_3, v_1) → s_3
  164. Translate(D, v_1) → D'_{11}
  165. Translate(D'_5, v_1) → C_2
  166. Segment(D'_{11}, C_2) → t_3
  167. Line(B_2, g_2) → a_3
  168. Point(a_3) → D_2
  169. Angle(C_2, B_2, D_2) → μ
  170. Rotate(s_3, If[24.5 < b ≤ 25.5, ((b - 24.5) * μ), μ], B_2) → c_4
  171. Rotate(D'_{11}, If[24.5 < b ≤ 25.5, ((b - 24.5) * μ), μ], B_2) → E_2
  172. Rotate(C_2, If[24.5 < b ≤ 25.5, ((b - 24.5) * μ), μ], B_2) → F_2
  173. Segment(E_2, F_2) → b_3
  174. Dilate(B_2, If[25.5 < b ≤ 26.5, 26.5 - b, 0], F) → G_2
  175. Vector(B_2, G_2) → w_1
  176. Translate(c_4, w_1) → d_4
  177. Translate(E_2, w_1) → H_2
  178. Translate(F_2, w_1) → I_2
  179. Segment(H_2, I_2) → f_4

AI 绘图提示语参考

画一个圆锥和圆柱的动态关系图，包含一个单位圆和相关的概率分布。首先绘制一个单位圆和一条射线，然后基于射线创建一个半圆和三角形。接着，找到半圆的中点和角度，绘制垂直线和交点。通过旋转和角度计算，创建多个新的点和射线，并找到它们的交点。最后，绘制连接这些点的线段，并展示它们之间的距离关系，以展示几何和概率统计的关联。
以上提示语为根据原始绘图命令整理而来，仅供参考。