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参数化曲面集合（曲线铺层）

参数化曲面集合（曲线铺层）
几何画板及绘图内容加载中，请稍等……
2013-11-12 20:43
函数立体几何参数方程函数图像
所属专题 surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
①本范例演示了20种预定义函数的参数化曲面集合，展示了曲线铺层在三维空间中的形态与变化；②涉及的核心数学概念包括参数方程、曲面几何、函数图像、向量场等；③作图要点提炼自构造摘要，使用了Textfield、Surface、CurveCartesian等GeoGebra命令，以及Δu、Δv等参数控制曲面范围与形态；④适合高中及以上数学学习者，用于理解参数化曲面、函数可视化及三维几何作图。

GeoGebra 命令及参数

  1. Textfield(u0_{min}) → Textfeld3
  2. Textfield(u0_{max}) → Textfeld4
  3. Textfield(v0_{min}) → Textfeld5
  4. Textfield(v0_{max}) → Textfeld6
  5. Textfield(f0x) → inputBox1
  6. Textfield(f0y) → inputBox2
  7. Textfield(f0z) → inputBox3
  8. Element(lfy, Funktion) → fy
  9. Element(lfx, Funktion) → fx
  10. Element(lfz, Funktion) → fz
  11. Textfield(NN) → inputBox5
  12. Textfield(Funktion) → inputBox4
  13. Surface(fx(u, v), fy(u, v), fz(u, v), u, u_{min}, u_{max}, v, v_{min}, v_{max}) → f3
  14. CurveCartesian(fx(t, 0), fy(t, 0), fz(t, 0), t, u_{min}, u_{max}) → M4_{1}
  15. CurveCartesian(fx(0.2, t), fy(0.2, t), fz(0.2, t), t, v_{min}, v_{max}) → M5_{1}
  16. CurveCartesian(fx(t, 0.1176), fy(t, 0.1176), fz(t, 0.1176), t, u_{min}, u_{max}) → M4_{2}
  17. CurveCartesian(fx(0.322, t), fy(0.322, t), fz(0.322, t), t, v_{min}, v_{max}) → M5_{2}
  18. CurveCartesian(fx(t, 0.2352), fy(t, 0.2352), fz(t, 0.2352), t, u_{min}, u_{max}) → M4_{3}
  19. CurveCartesian(fx(0.444, t), fy(0.444, t), fz(0.444, t), t, v_{min}, v_{max}) → M5_{3}
  20. CurveCartesian(fx(t, 0.3528), fy(t, 0.3528), fz(t, 0.3528), t, u_{min}, u_{max}) → M4_{4}
  21. CurveCartesian(fx(0.5660000000000001, t), fy(0.5660000000000001, t), fz(0.5660000000000001, t), t, v_{min}, v_{max}) → M5_{4}
  22. CurveCartesian(fx(t, 0.4704), fy(t, 0.4704), fz(t, 0.4704), t, u_{min}, u_{max}) → M4_{5}
  23. CurveCartesian(fx(0.688, t), fy(0.688, t), fz(0.688, t), t, v_{min}, v_{max}) → M5_{5}
  24. CurveCartesian(fx(t, 0.588), fy(t, 0.588), fz(t, 0.588), t, u_{min}, u_{max}) → M4_{6}
  25. CurveCartesian(fx(0.81, t), fy(0.81, t), fz(0.81, t), t, v_{min}, v_{max}) → M5_{6}
  26. CurveCartesian(fx(t, 0.7056), fy(t, 0.7056), fz(t, 0.7056), t, u_{min}, u_{max}) → M4_{7}
  27. CurveCartesian(fx(0.9319999999999999, t), fy(0.9319999999999999, t), fz(0.9319999999999999, t), t, v_{min}, v_{max}) → M5_{7}
  28. CurveCartesian(fx(t, 0.8231999999999999), fy(t, 0.8231999999999999), fz(t, 0.8231999999999999), t, u_{min}, u_{max}) → M4_{8}
  29. CurveCartesian(fx(1.054, t), fy(1.054, t), fz(1.054, t), t, v_{min}, v_{max}) → M5_{8}
  30. CurveCartesian(fx(t, 0.9408), fy(t, 0.9408), fz(t, 0.9408), t, u_{min}, u_{max}) → M4_{9}
  31. CurveCartesian(fx(1.176, t), fy(1.176, t), fz(1.176, t), t, v_{min}, v_{max}) → M5_{9}
  32. CurveCartesian(fx(t, 1.0584), fy(t, 1.0584), fz(t, 1.0584), t, u_{min}, u_{max}) → M4_{10}
  33. CurveCartesian(fx(1.2979999999999998, t), fy(1.2979999999999998, t), fz(1.2979999999999998, t), t, v_{min}, v_{max}) → M5_{10}
  34. CurveCartesian(fx(t, 1.176), fy(t, 1.176), fz(t, 1.176), t, u_{min}, u_{max}) → M4_{11}
  35. CurveCartesian(fx(1.42, t), fy(1.42, t), fz(1.42, t), t, v_{min}, v_{max}) → M5_{11}
  36. CurveCartesian(fx(t, 1.2935999999999999), fy(t, 1.2935999999999999), fz(t, 1.2935999999999999), t, u_{min}, u_{max}) → M4_{12}
  37. CurveCartesian(fx(1.542, t), fy(1.542, t), fz(1.542, t), t, v_{min}, v_{max}) → M5_{12}
  38. CurveCartesian(fx(t, 1.4112), fy(t, 1.4112), fz(t, 1.4112), t, u_{min}, u_{max}) → M4_{13}
  39. CurveCartesian(fx(1.664, t), fy(1.664, t), fz(1.664, t), t, v_{min}, v_{max}) → M5_{13}
  40. CurveCartesian(fx(t, 1.5288), fy(t, 1.5288), fz(t, 1.5288), t, u_{min}, u_{max}) → M4_{14}
  41. CurveCartesian(fx(1.7859999999999998, t), fy(1.7859999999999998, t), fz(1.7859999999999998, t), t, v_{min}, v_{max}) → M5_{14}
  42. CurveCartesian(fx(t, 1.6463999999999999), fy(t, 1.6463999999999999), fz(t, 1.6463999999999999), t, u_{min}, u_{max}) → M4_{15}
  43. CurveCartesian(fx(1.908, t), fy(1.908, t), fz(1.908, t), t, v_{min}, v_{max}) → M5_{15}
  44. CurveCartesian(fx(t, 1.764), fy(t, 1.764), fz(t, 1.764), t, u_{min}, u_{max}) → M4_{16}
  45. CurveCartesian(fx(2.0300000000000002, t), fy(2.0300000000000002, t), fz(2.0300000000000002, t), t, v_{min}, v_{max}) → M5_{16}
  46. CurveCartesian(fx(t, 1.8816), fy(t, 1.8816), fz(t, 1.8816), t, u_{min}, u_{max}) → M4_{17}
  47. CurveCartesian(fx(2.152, t), fy(2.152, t), fz(2.152, t), t, v_{min}, v_{max}) → M5_{17}
  48. CurveCartesian(fx(t, 1.9991999999999999), fy(t, 1.9991999999999999), fz(t, 1.9991999999999999), t, u_{min}, u_{max}) → M4_{18}
  49. CurveCartesian(fx(2.274, t), fy(2.274, t), fz(2.274, t), t, v_{min}, v_{max}) → M5_{18}
  50. CurveCartesian(fx(t, 2.1168), fy(t, 2.1168), fz(t, 2.1168), t, u_{min}, u_{max}) → M4_{19}
  51. CurveCartesian(fx(2.396, t), fy(2.396, t), fz(2.396, t), t, v_{min}, v_{max}) → M5_{19}
  52. CurveCartesian(fx(t, 2.2344), fy(t, 2.2344), fz(t, 2.2344), t, u_{min}, u_{max}) → M4_{20}
  53. CurveCartesian(fx(2.5180000000000002, t), fy(2.5180000000000002, t), fz(2.5180000000000002, t), t, v_{min}, v_{max}) → M5_{20}
  54. CurveCartesian(fx(t, 2.352), fy(t, 2.352), fz(t, 2.352), t, u_{min}, u_{max}) → M4_{21}
  55. CurveCartesian(fx(2.64, t), fy(2.64, t), fz(2.64, t), t, v_{min}, v_{max}) → M5_{21}
  56. CurveCartesian(fx(t, 2.4696), fy(t, 2.4696), fz(t, 2.4696), t, u_{min}, u_{max}) → M4_{22}
  57. CurveCartesian(fx(2.762, t), fy(2.762, t), fz(2.762, t), t, v_{min}, v_{max}) → M5_{22}
  58. CurveCartesian(fx(t, 2.5871999999999997), fy(t, 2.5871999999999997), fz(t, 2.5871999999999997), t, u_{min}, u_{max}) → M4_{23}
  59. CurveCartesian(fx(2.8840000000000003, t), fy(2.8840000000000003, t), fz(2.8840000000000003, t), t, v_{min}, v_{max}) → M5_{23}
  60. CurveCartesian(fx(t, 2.7048), fy(t, 2.7048), fz(t, 2.7048), t, u_{min}, u_{max}) → M4_{24}
  61. CurveCartesian(fx(3.0060000000000002, t), fy(3.0060000000000002, t), fz(3.0060000000000002, t), t, v_{min}, v_{max}) → M5_{24}
  62. CurveCartesian(fx(t, 2.8224), fy(t, 2.8224), fz(t, 2.8224), t, u_{min}, u_{max}) → M4_{25}
  63. CurveCartesian(fx(3.128, t), fy(3.128, t), fz(3.128, t), t, v_{min}, v_{max}) → M5_{25}
  64. CurveCartesian(fx(t, 2.94), fy(t, 2.94), fz(t, 2.94), t, u_{min}, u_{max}) → M4_{26}
  65. CurveCartesian(fx(3.25, t), fy(3.25, t), fz(3.25, t), t, v_{min}, v_{max}) → M5_{26}
  66. CurveCartesian(fx(t, 3.0576), fy(t, 3.0576), fz(t, 3.0576), t, u_{min}, u_{max}) → M4_{27}
  67. CurveCartesian(fx(3.372, t), fy(3.372, t), fz(3.372, t), t, v_{min}, v_{max}) → M5_{27}
  68. CurveCartesian(fx(t, 3.1752), fy(t, 3.1752), fz(t, 3.1752), t, u_{min}, u_{max}) → M4_{28}
  69. CurveCartesian(fx(3.494, t), fy(3.494, t), fz(3.494, t), t, v_{min}, v_{max}) → M5_{28}
  70. CurveCartesian(fx(t, 3.2927999999999997), fy(t, 3.2927999999999997), fz(t, 3.2927999999999997), t, u_{min}, u_{max}) → M4_{29}
  71. CurveCartesian(fx(3.616, t), fy(3.616, t), fz(3.616, t), t, v_{min}, v_{max}) → M5_{29}
  72. CurveCartesian(fx(t, 3.4104), fy(t, 3.4104), fz(t, 3.4104), t, u_{min}, u_{max}) → M4_{30}
  73. CurveCartesian(fx(3.738, t), fy(3.738, t), fz(3.738, t), t, v_{min}, v_{max}) → M5_{30}
  74. CurveCartesian(fx(t, 3.528), fy(t, 3.528), fz(t, 3.528), t, u_{min}, u_{max}) → M4_{31}
  75. CurveCartesian(fx(3.8600000000000003, t), fy(3.8600000000000003, t), fz(3.8600000000000003, t), t, v_{min}, v_{max}) → M5_{31}
  76. CurveCartesian(fx(t, 3.6456), fy(t, 3.6456), fz(t, 3.6456), t, u_{min}, u_{max}) → M4_{32}
  77. CurveCartesian(fx(3.982, t), fy(3.982, t), fz(3.982, t), t, v_{min}, v_{max}) → M5_{32}
  78. CurveCartesian(fx(t, 3.7632), fy(t, 3.7632), fz(t, 3.7632), t, u_{min}, u_{max}) → M4_{33}
  79. CurveCartesian(fx(4.104, t), fy(4.104, t), fz(4.104, t), t, v_{min}, v_{max}) → M5_{33}
  80. CurveCartesian(fx(t, 3.8808), fy(t, 3.8808), fz(t, 3.8808), t, u_{min}, u_{max}) → M4_{34}
  81. CurveCartesian(fx(4.226, t), fy(4.226, t), fz(4.226, t), t, v_{min}, v_{max}) → M5_{34}
  82. CurveCartesian(fx(t, 3.9983999999999997), fy(t, 3.9983999999999997), fz(t, 3.9983999999999997), t, u_{min}, u_{max}) → M4_{35}
  83. CurveCartesian(fx(4.348, t), fy(4.348, t), fz(4.348, t), t, v_{min}, v_{max}) → M5_{35}
  84. CurveCartesian(fx(t, 4.116), fy(t, 4.116), fz(t, 4.116), t, u_{min}, u_{max}) → M4_{36}
  85. CurveCartesian(fx(4.47, t), fy(4.47, t), fz(4.47, t), t, v_{min}, v_{max}) → M5_{36}
  86. CurveCartesian(fx(t, 4.2336), fy(t, 4.2336), fz(t, 4.2336), t, u_{min}, u_{max}) → M4_{37}
  87. CurveCartesian(fx(4.592, t), fy(4.592, t), fz(4.592, t), t, v_{min}, v_{max}) → M5_{37}
  88. CurveCartesian(fx(t, 4.3511999999999995), fy(t, 4.3511999999999995), fz(t, 4.3511999999999995), t, u_{min}, u_{max}) → M4_{38}
  89. CurveCartesian(fx(4.714, t), fy(4.714, t), fz(4.714, t), t, v_{min}, v_{max}) → M5_{38}
  90. CurveCartesian(fx(t, 4.4688), fy(t, 4.4688), fz(t, 4.4688), t, u_{min}, u_{max}) → M4_{39}
  91. CurveCartesian(fx(4.836, t), fy(4.836, t), fz(4.836, t), t, v_{min}, v_{max}) → M5_{39}
  92. CurveCartesian(fx(t, 4.5864), fy(t, 4.5864), fz(t, 4.5864), t, u_{min}, u_{max}) → M4_{40}
  93. CurveCartesian(fx(4.958, t), fy(4.958, t), fz(4.958, t), t, v_{min}, v_{max}) → M5_{40}
  94. CurveCartesian(fx(t, 4.704), fy(t, 4.704), fz(t, 4.704), t, u_{min}, u_{max}) → M4_{41}
  95. CurveCartesian(fx(5.08, t), fy(5.08, t), fz(5.08, t), t, v_{min}, v_{max}) → M5_{41}
  96. CurveCartesian(fx(t, 4.8216), fy(t, 4.8216), fz(t, 4.8216), t, u_{min}, u_{max}) → M4_{42}
  97. CurveCartesian(fx(5.202, t), fy(5.202, t), fz(5.202, t), t, v_{min}, v_{max}) → M5_{42}
  98. CurveCartesian(fx(t, 4.9392), fy(t, 4.9392), fz(t, 4.9392), t, u_{min}, u_{max}) → M4_{43}
  99. CurveCartesian(fx(5.324, t), fy(5.324, t), fz(5.324, t), t, v_{min}, v_{max}) → M5_{43}
  100. CurveCartesian(fx(t, 5.0568), fy(t, 5.0568), fz(t, 5.0568), t, u_{min}, u_{max}) → M4_{44}
  101. CurveCartesian(fx(5.446, t), fy(5.446, t), fz(5.446, t), t, v_{min}, v_{max}) → M5_{44}
  102. CurveCartesian(fx(t, 5.1743999999999994), fy(t, 5.1743999999999994), fz(t, 5.1743999999999994), t, u_{min}, u_{max}) → M4_{45}
  103. CurveCartesian(fx(5.5680000000000005, t), fy(5.5680000000000005, t), fz(5.5680000000000005, t), t, v_{min}, v_{max}) → M5_{45}
  104. CurveCartesian(fx(t, 5.292), fy(t, 5.292), fz(t, 5.292), t, u_{min}, u_{max}) → M4_{46}
  105. CurveCartesian(fx(5.69, t), fy(5.69, t), fz(5.69, t), t, v_{min}, v_{max}) → M5_{46}
  106. CurveCartesian(fx(t, 5.4096), fy(t, 5.4096), fz(t, 5.4096), t, u_{min}, u_{max}) → M4_{47}
  107. CurveCartesian(fx(5.812, t), fy(5.812, t), fz(5.812, t), t, v_{min}, v_{max}) → M5_{47}
  108. CurveCartesian(fx(t, 5.5272), fy(t, 5.5272), fz(t, 5.5272), t, u_{min}, u_{max}) → M4_{48}
  109. CurveCartesian(fx(5.934, t), fy(5.934, t), fz(5.934, t), t, v_{min}, v_{max}) → M5_{48}
  110. CurveCartesian(fx(t, 5.6448), fy(t, 5.6448), fz(t, 5.6448), t, u_{min}, u_{max}) → M4_{49}
  111. CurveCartesian(fx(6.056, t), fy(6.056, t), fz(6.056, t), t, v_{min}, v_{max}) → M5_{49}
  112. CurveCartesian(fx(t, 5.7623999999999995), fy(t, 5.7623999999999995), fz(t, 5.7623999999999995), t, u_{min}, u_{max}) → M4_{50}
  113. CurveCartesian(fx(6.178, t), fy(6.178, t), fz(6.178, t), t, v_{min}, v_{max}) → M5_{50}
  114. CurveCartesian(fx(t, 5.88), fy(t, 5.88), fz(t, 5.88), t, u_{min}, u_{max}) → M4_{51}
  115. CurveCartesian(fx(6.3, t), fy(6.3, t), fz(6.3, t), t, v_{min}, v_{max}) → M5_{51}

AI 绘图提示语参考

绘制一个包含20个预定义参数化曲面的集合，展示三维空间中的曲线覆盖概念。每个曲面由三个函数fx、fy、fz定义，使用参数u和v控制范围和形状。同时绘制多条曲线，分别位于u和v的特定值处，以展示曲面的边界和覆盖效果。先画出所有曲面，再添加曲线，使图形层次分明。
以上提示语为根据原始绘图命令整理而来，仅供参考。