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Collection of Parametric Surfaces (Curve Covering)

Collection of Parametric Surfaces (Curve Covering)
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2013-11-12 20:43
FunctionSolid GeometryParametric EquationGraphing
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
This example demonstrates a collection of 20 predefined parametric surfaces, showcasing the concept of curve covering in three-dimensional space. It involves core mathematical concepts such as parametric equations, surface geometry, function visualization, and vector fields. The construction utilizes GeoGebra commands like Textfield, Surface, and CurveCartesian, with parameters like Δu and Δv to control the surface's range and shape. Suitable for high school and above students to understand parametric surfaces, function graphing, and 3D geometry plotting.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a collection of 20 predefined parametric surfaces to demonstrate curve covering in 3D space. Each surface is defined by functions fx, fy, fz and uses parameters u, v to control its range and shape. Also plot several curves at specific u and v values to show the boundaries and coverage of the surfaces. Start by drawing all surfaces, then add the curves to create a clear layered effect.
The prompt above was compiled from the original drawing commands and is for reference only.