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Dynamic Cone and Probability Distribution Visualization

Dynamic Cone and Probability Distribution Visualization
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2018-02-04 00:17
TrigonometrySolid GeometryProbability & StatisticsGeometric Construction
This example dynamically demonstrates the geometric relationships between a cone, unit circle, and cylinder, as well as stochastic processes and probability distributions. Key math concepts include solid geometry, trigonometry, probability & statistics, and random processes. It is constructed using GeoGebra commands like Circle, Ray, Point, Semicircle, Segment, Polygon, Midpoint, Angle, OrthogonalLine, Intersect, Rotate, Line, Distance, Dilate, Min, CircleSector, Vector, and Translate. Suitable for high school and above, ideal for geometric modeling and probability/statistics education.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Visualize the dynamic relationship between a cone and a cylinder, including a unit circle and related probability distributions. Start by drawing a unit circle and a ray, then create a semicircle and a triangle based on the ray. Find the midpoint and angles, draw perpendicular lines and intersection points. Through rotation and angle calculations, create multiple new points and rays, and locate their intersections. Finally, draw line segments connecting these points and display their distance relationships to illustrate the connection between geometry and probability statistics.
The prompt above was compiled from the original drawing commands and is for reference only.