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动画演示几何变换与优化

动画演示几何变换与优化
几何画板及绘图内容加载中，请稍等……
2017-04-13 09:53
测试改名0多边形向量最值优化几何作图
①本范例通过动画演示几何图形的变换与优化过程；②涉及核心数学概念包括角度、几何图形（正方形、三角形、多边形）、线性规划、立体几何等；③作图要点运用了 Ray, Segment, Circle, Intersect, Point, Rotate, Dilate, Polygon, Midpoint, CircleArc, Vector, Translate, Line, Angle, Mirror, OrthogonalLine 等GeoGebra命令；④适合高中及以上学生及数学爱好者学习几何变换与优化方法。

GeoGebra 命令及参数

  1. Ray(A, B_2) → m_2
  2. Segment(A, B_2) → g
  3. Circle(B_2, g) → q_2
  4. Intersect(q_2, m_2, 2) → J_1
  5. Segment(B_2, J_1) → n_2
  6. Point(n_2) → K_1
  7. Segment(B_2, K_1) → r_2
  8. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  9. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  10. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  11. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  12. Midpoint(A, C) → V
  13. CircleArc(V, C, D) → g_2
  14. Point(g_2) → B
  15. Segment(C, B) → j_1
  16. Segment(A, B) → n
  17. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  18. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  19. Segment(R, A''') → f''
  20. Intersect(n, f'') → Z
  21. Segment(B, Z) → l_1
  22. Segment(A, A') → f
  23. Segment(A'_1, A'') → f'
  24. Segment(B_2, C) → h
  25. Segment(C, D) → i
  26. Circle(B_2, (0.2 * r_2)) → p
  27. Segment(D, A) → j
  28. Circle(C, Radius[p]) → k
  29. Circle(A, (0.1 * j)) → q
  30. Circle(D, (0.1 * j)) → r
  31. Segment(A, C) → l
  32. Segment(D, B_2) → m
  33. Intersect(r, j, 1) → F
  34. Intersect(r, m, 1) → G
  35. Intersect(r, i, 1) → H
  36. Intersect(k, i, 1) → I
  37. Intersect(k, l, 1) → J
  38. Intersect(k, h, 1) → K
  39. Intersect(p, h, 1) → L
  40. Intersect(p, m, 1) → M
  41. Intersect(p, g, 1) → N
  42. Intersect(q, g, 1) → O
  43. Intersect(q, l, 1) → P
  44. Intersect(q, j, 1) → Q
  45. CircleSector(B_2, L, N) → s
  46. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  47. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  48. Segment(R', S) → f'''
  49. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  50. Vector(B_2, B') → u
  51. Translate(s, u) → s'
  52. Rotate(s', If[5 < e ≤ 6, ((e - 5) * π / 2), 1.5707963267948966], C) → s''
  53. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  54. Vector(C, C') → v
  55. Translate(s'', v) → s'''
  56. Rotate(s''', If[7 < e ≤ 8, ((e - 7) * π / 2), 1.5707963267948966], D) → t
  57. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  58. Vector(D, D') → w
  59. Translate(t, w) → t'
  60. Rotate(t', If[9 < e ≤ 10, ((e - 9) * π / 2), 1.5707963267948966], A) → t''
  61. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  62. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  63. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  64. Vector(B_2, B'_1) → v_1
  65. Vector(B_2, B'_3) → w_1
  66. Vector(B_2, B'_2) → i_1
  67. Circle(A, n - j_1) → h_2
  68. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  69. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  70. Vector(B_2, B'_4) → u_1
  71. Translate(s, u_1) → s'_1
  72. Line(B, f') → a_1
  73. Point(a_1) → U
  74. Angle(U, B, C) → α
  75. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  76. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  77. Line(A, B_2) → h_1
  78. Angle(B, A, D) → β
  79. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  80. Mirror(C, D) → C'_1
  81. Rotate(C, ((-π)) / 3, B_2) → C'_2
  82. Segment(B_2, C'_2) → f_1
  83. Segment(C, C'_2) → g_1
  84. Intersect(k, g_1, 1) → E
  85. Intersect(k, h) → T,
  86. CircleSector(C, T, E) → d_1
  87. Rotate(D, 15°, C) → D'_2
  88. Angle(D, C, D'_2) → γ
  89. Ray(C, D'_2) → k_1
  90. Rotate(C, (-(15°)), D) → C'_3
  91. Angle(C'_3, D, C) → δ
  92. Ray(D, C'_3) → m_1
  93. Intersect(k_1, m_1) → W
  94. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  95. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  96. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  97. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  98. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  99. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  100. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  101. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  102. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  103. Rotate(T, 15°, C) → T'
  104. Angle(T, C, T') → ε
  105. Ray(C, T') → q_1
  106. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  107. Vector(C, C'_8) → u_2
  108. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  109. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  110. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  111. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  112. Segment(A, A'_2) → j_2
  113. Segment(B_2, G_1) → l_2
  114. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  115. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  116. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  117. Polygon(H_1, I_1, C'_{10}) → poly6', c'_{10}, h_3, i_3
  118. OrthogonalLine(K_1, r_2) → s_2
  119. Circle(K_1, r_2) → t_2
  120. Intersect(t_2, s_2, 2) → L_1
  121. OrthogonalLine(L_1, h) → f_3
  122. Intersect(f_3, h) → M_1
  123. Dilate(B_2, If[10 < e ≤ 11, 11 - e, 0], K_1) → C_1
  124. Segment(B_2, C_1) → n_1
  125. Rotate(B_2, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → D_1
  126. Rotate(C_1, If[11 < e ≤ 12, ((-(e - 11)) * π / 2), ((-π)) / 2], K_1) → E_1
  127. Segment(D_1, E_1) → p_1
  128. Rotate(D_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → F_1
  129. Rotate(E_1, If[12 < e ≤ 13, ((-(e - 12)) * π / 2), ((-π)) / 2], L_1) → N_1
  130. Segment(F_1, N_1) → r_1
  131. Rotate(F_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → O_1
  132. Rotate(N_1, If[13 < e ≤ 14, ((-(e - 13)) * π / 2), ((-π)) / 2], M_1) → P_1
  133. Segment(O_1, P_1) → s_1
  134. Polygon(B_2, K_1, L_1, M_1) → poly6, b_3, k_2, l_3, m_3
  135. Dilate(B_2, If[14 < e ≤ 15, 15 - e, 0], K_1) → Q_1
  136. Vector(B_2, Q_1) → c_1
  137. Translate(s, c_1) → s'_2
  138. Segment(C, K_1) → e_1
  139. Line(A, M_1) → t_1
  140. Intersect(t_1, e_1) → R_1
  141. Segment(A, R_1) → f_2
  142. Dilate(C, If[15 < e ≤ 16, 16 - e, 0], K_1) → C'_6
  143. Segment(C, C'_6) → i_2
  144. Dilate(A, If[16 < e ≤ 17, 17 - e, 0], M_1) → A'_3
  145. Segment(A, A'_3) → p_2
  146. Dilate(M_1, If[17 < e ≤ 18, 18 - e, 0], R_1) → S_1
  147. Segment(M_1, S_1) → e_2
  148. Dilate(B_2, If[18.5 < e ≤ 19.5, 19.5 - e, 0], R_1) → T_1
  149. Vector(B_2, T_1) → v_2
  150. Translate(s, v_2) → s'_3
  151. Line(R_1, p_1) → g_3
  152. Point(g_3) → U_1
  153. Angle(U_1, R_1, K_1) → ζ
  154. Rotate(s'_3, If[18.5 < e ≤ 19.5, ((e - 18.5) * (π / 2 + ζ)), π / 2 + ζ], T_1) → e_3
  155. Polygon(A, B_2, C, D) → poly7, a_4, b_4, c_4, d_4