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圆的面积作为平行四边形的变换

圆的面积作为平行四边形的变换
几何画板及绘图内容加载中，请稍等……
2020-12-19 16:04
圆向量几何作图
这个范例演示了如何通过变换将圆的面积转化为平行四边形的面积。核心数学概念包括圆、平行四边形和几何变换。使用了GeoGebra中的Min, Intersect, Point, Rotate, Vector, Translate等命令。适合初中及以上学生学习几何变换和面积计算。

GeoGebra 命令及参数

  1. Min(i, 1) → C1
  2. Intersect(xAxis, yAxis) → A
  3. Point(yAxis) → B
  4. Rotate(B, (360 * ° / n), A) → B'_L
  5. Rotate(B, (-360 * ° / n), A) → B'_R
  6. Rotate(B3, B5 / 2, B7) → C7
  7. Rotate(B7, (-B$4), C7) → B8
  8. Rotate(C7, B$5, B8) → C8
  9. Rotate(B8, (-B$4), C8) → B9
  10. Rotate(C8, B$5, B9) → C9
  11. Rotate(B9, (-B$4), C9) → B10
  12. Rotate(C9, B$5, B10) → C10
  13. Rotate(B10, (-B$4), C10) → B11
  14. Rotate(C10, B$5, B11) → C11
  15. Rotate(B11, (-B$4), C11) → B12
  16. Rotate(C11, B$5, B12) → C12
  17. Rotate(B12, (-B$4), C12) → B13
  18. Rotate(C12, B$5, B13) → C13
  19. Rotate(B13, (-B$4), C13) → B14
  20. Rotate(C13, B$5, B14) → C14
  21. Rotate(B14, (-B$4), C14) → B15
  22. Rotate(C14, B$5, B15) → C15
  23. Rotate(B15, (-B$4), C15) → B16
  24. Rotate(C15, B$5, B16) → C16
  25. Rotate(B16, (-B$4), C16) → B17
  26. Rotate(C16, B$5, B17) → C17
  27. Rotate(B17, (-B$4), C17) → B18
  28. Rotate(C17, B$5, B18) → C18
  29. Rotate(B18, (-B$4), C18) → B19
  30. Rotate(C18, B$5, B19) → C19
  31. Rotate(B19, (-B$4), C19) → B20
  32. Rotate(C19, B$5, B20) → C20
  33. Rotate(B20, (-B$4), C20) → B21
  34. Rotate(C20, B$5, B21) → C21
  35. Rotate(B21, (-B$4), C21) → B22
  36. Rotate(C21, B$5, B22) → C22
  37. Rotate(B22, (-B$4), C22) → B23
  38. Rotate(C22, B$5, B23) → C23
  39. Rotate(B23, (-B$4), C23) → B24
  40. Rotate(C23, B$5, B24) → C24
  41. Rotate(B24, (-B$4), C24) → B25
  42. Rotate(C24, B$5, B25) → C25
  43. Rotate(B25, (-B$4), C25) → B26
  44. Rotate(C25, B$5, B26) → C26
  45. Rotate(B26, (-B$4), C26) → B27
  46. Rotate(C26, B$5, B27) → C27
  47. Vector(B7, C7) → u
  48. Translate(CircleSector[C7, B8, B7], Vector[(C$2 * u)]) → D7
  49. Translate(CircleSector[C8, B9, B8], Vector[(C$2 * u)]) → D8
  50. Translate(CircleSector[F7, E7, E8], Vector[(C$2 * u)]) → G7
  51. Translate(CircleSector[F8, E8, E9], Vector[(C$2 * u)]) → G8
  52. Translate(CircleSector[I7, H7, H8], Vector[((-C$2) * u)]) → J7
  53. Translate(CircleSector[L7, K8, K7], Vector[((-C$2) * u)]) → M7
  54. Translate(CircleSector[L8, K9, K8], Vector[((-C$2) * u)]) → M8
  55. Translate(CircleSector[I8, H8, H9], Vector[((-C$2) * u)]) → J8
  56. Translate(CircleSector[C9, B10, B9], Vector[(C$2 * u)]) → D9
  57. Translate(CircleSector[F9, E9, E10], Vector[(C$2 * u)]) → G9
  58. Translate(CircleSector[L9, K10, K9], Vector[((-C$2) * u)]) → M9
  59. Translate(CircleSector[I9, H9, H10], Vector[((-C$2) * u)]) → J9
  60. Translate(CircleSector[C10, B11, B10], Vector[(C$2 * u)]) → D10
  61. Translate(CircleSector[F10, E10, E11], Vector[(C$2 * u)]) → G10
  62. Translate(CircleSector[L10, K11, K10], Vector[((-C$2) * u)]) → M10
  63. Translate(CircleSector[I10, H10, H11], Vector[((-C$2) * u)]) → J10
  64. Translate(CircleSector[C11, B12, B11], Vector[(C$2 * u)]) → D11
  65. Translate(CircleSector[F11, E11, E12], Vector[(C$2 * u)]) → G11
  66. Translate(CircleSector[L11, K12, K11], Vector[((-C$2) * u)]) → M11
  67. Translate(CircleSector[I11, H11, H12], Vector[((-C$2) * u)]) → J11
  68. Translate(CircleSector[C12, B13, B12], Vector[(C$2 * u)]) → D12
  69. Translate(CircleSector[F12, E12, E13], Vector[(C$2 * u)]) → G12
  70. Translate(CircleSector[L12, K13, K12], Vector[((-C$2) * u)]) → M12
  71. Translate(CircleSector[I12, H12, H13], Vector[((-C$2) * u)]) → J12
  72. Translate(CircleSector[C13, B14, B13], Vector[(C$2 * u)]) → D13
  73. Translate(CircleSector[F13, E13, E14], Vector[(C$2 * u)]) → G13
  74. Translate(CircleSector[L13, K14, K13], Vector[((-C$2) * u)]) → M13
  75. Translate(CircleSector[I13, H13, H14], Vector[((-C$2) * u)]) → J13
  76. Translate(CircleSector[C14, B15, B14], Vector[(C$2 * u)]) → D14
  77. Translate(CircleSector[F14, E14, E15], Vector[(C$2 * u)]) → G14
  78. Translate(CircleSector[L14, K15, K14], Vector[((-C$2) * u)]) → M14
  79. Translate(CircleSector[I14, H14, H15], Vector[((-C$2) * u)]) → J14
  80. Translate(CircleSector[C15, B16, B15], Vector[(C$2 * u)]) → D15
  81. Translate(CircleSector[F15, E15, E16], Vector[(C$2 * u)]) → G15
  82. Translate(CircleSector[L15, K16, K15], Vector[((-C$2) * u)]) → M15
  83. Translate(CircleSector[I15, H15, H16], Vector[((-C$2) * u)]) → J15
  84. Translate(CircleSector[C16, B17, B16], Vector[(C$2 * u)]) → D16
  85. Translate(CircleSector[F16, E16, E17], Vector[(C$2 * u)]) → G16
  86. Translate(CircleSector[L16, K17, K16], Vector[((-C$2) * u)]) → M16
  87. Translate(CircleSector[I16, H16, H17], Vector[((-C$2) * u)]) → J16
  88. Translate(CircleSector[C17, B18, B17], Vector[(C$2 * u)]) → D17
  89. Translate(CircleSector[F17, E17, E18], Vector[(C$2 * u)]) → G17
  90. Translate(CircleSector[L17, K18, K17], Vector[((-C$2) * u)]) → M17
  91. Translate(CircleSector[I17, H17, H18], Vector[((-C$2) * u)]) → J17
  92. Translate(CircleSector[C18, B19, B18], Vector[(C$2 * u)]) → D18
  93. Translate(CircleSector[F18, E18, E19], Vector[(C$2 * u)]) → G18
  94. Translate(CircleSector[L18, K19, K18], Vector[((-C$2) * u)]) → M18
  95. Translate(CircleSector[I18, H18, H19], Vector[((-C$2) * u)]) → J18
  96. Translate(CircleSector[C19, B20, B19], Vector[(C$2 * u)]) → D19
  97. Translate(CircleSector[F19, E19, E20], Vector[(C$2 * u)]) → G19
  98. Translate(CircleSector[L19, K20, K19], Vector[((-C$2) * u)]) → M19
  99. Translate(CircleSector[I19, H19, H20], Vector[((-C$2) * u)]) → J19
  100. Translate(CircleSector[C20, B21, B20], Vector[(C$2 * u)]) → D20
  101. Translate(CircleSector[F20, E20, E21], Vector[(C$2 * u)]) → G20
  102. Translate(CircleSector[L20, K21, K20], Vector[((-C$2) * u)]) → M20
  103. Translate(CircleSector[I20, H20, H21], Vector[((-C$2) * u)]) → J20
  104. Translate(CircleSector[C21, B22, B21], Vector[(C$2 * u)]) → D21
  105. Translate(CircleSector[F21, E21, E22], Vector[(C$2 * u)]) → G21
  106. Translate(CircleSector[L21, K22, K21], Vector[((-C$2) * u)]) → M21
  107. Translate(CircleSector[I21, H21, H22], Vector[((-C$2) * u)]) → J21
  108. Translate(CircleSector[C22, B23, B22], Vector[(C$2 * u)]) → D22
  109. Translate(CircleSector[F22, E22, E23], Vector[(C$2 * u)]) → G22
  110. Translate(CircleSector[L22, K23, K22], Vector[((-C$2) * u)]) → M22
  111. Translate(CircleSector[I22, H22, H23], Vector[((-C$2) * u)]) → J22
  112. Translate(CircleSector[C23, B24, B23], Vector[(C$2 * u)]) → D23
  113. Translate(CircleSector[F23, E23, E24], Vector[(C$2 * u)]) → G23
  114. Translate(CircleSector[L23, K24, K23], Vector[((-C$2) * u)]) → M23
  115. Translate(CircleSector[I23, H23, H24], Vector[((-C$2) * u)]) → J23
  116. Translate(CircleSector[C24, B25, B24], Vector[(C$2 * u)]) → D24
  117. Translate(CircleSector[F24, E24, E25], Vector[(C$2 * u)]) → G24
  118. Translate(CircleSector[L24, K25, K24], Vector[((-C$2) * u)]) → M24
  119. Translate(CircleSector[I24, H24, H25], Vector[((-C$2) * u)]) → J24
  120. Translate(CircleSector[C25, B26, B25], Vector[(C$2 * u)]) → D25
  121. Translate(CircleSector[F25, E25, E26], Vector[(C$2 * u)]) → G25
  122. Translate(CircleSector[L25, K26, K25], Vector[((-C$2) * u)]) → M25
  123. Translate(CircleSector[I25, H25, H26], Vector[((-C$2) * u)]) → J25
  124. Translate(CircleSector[C26, B27, B26], Vector[(C$2 * u)]) → D26
  125. Translate(CircleSector[F26, E26, E27], Vector[(C$2 * u)]) → G26
  126. Translate(CircleSector[L26, K27, K26], Vector[((-C$2) * u)]) → M26
  127. Translate(CircleSector[I26, H26, H27], Vector[((-C$2) * u)]) → J26

AI 绘图提示语参考

绘制一个圆，将其面积转化为平行四边形的演示图。先画出坐标轴，以原点为圆心，在纵轴上取一点作为半径端点。将该点绕圆心按固定角度（360度除以分割数）反复旋转，生成一系列扇形顶点，并依次连接相邻顶点形成多个三角形扇区。然后，将第一个扇区平移，使其与后续扇区拼接，最终构成一个近似平行四边形的形状。图中需包含圆、旋转生成的扇形序列、平移向量以及最终形成的平行四边形轮廓，突出面积转换的几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。