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Parametric Surfaces with Curve Layering

Parametric Surfaces with Curve Layering
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2013-11-04 03:07
Coordinate GeometrySolid GeometryParametric Equation
In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1
This example demonstrates the drawing of parametric surfaces and their curve layering effects. Key mathematical concepts include parametric equations, surfaces, and curves. Using commands like Surface and CurveCartesian in GeoGebra, complex surface graphics can be constructed. Suitable for high school and above students or teachers.

GeoGebra Commands & Parameters

  1. Surface(fx(u, v), fy(u, v), fz(u, v), u, u_{min}, u_{max}, v, v_{min}, v_{max}) → f3
  2. Textfield(NN) → inputBox5
  3. Textfield(fx) → inputBox1
  4. Textfield(fy) → inputBox2
  5. Textfield(fz) → inputBox3
  6. Textfield(u_{min}) → inputBox4
  7. Textfield(u_{max}) → inputBox6
  8. Textfield(v_{min}) → inputBox7
  9. Textfield(v_{max}) → inputBox8
  10. CurveCartesian(fx(t, 0), fy(t, 0), fz(t, 0), t, u_{min}, u_{max}) → M4_{1}
  11. CurveCartesian(fx(0, t), fy(0, t), fz(0, t), t, v_{min}, v_{max}) → M5_{1}
  12. CurveCartesian(fx(t, 0.15), fy(t, 0.15), fz(t, 0.15), t, u_{min}, u_{max}) → M4_{2}
  13. CurveCartesian(fx(0.06283185307179587, t), fy(0.06283185307179587, t), fz(0.06283185307179587, t), t, v_{min}, v_{max}) → M5_{2}
  14. CurveCartesian(fx(t, 0.3), fy(t, 0.3), fz(t, 0.3), t, u_{min}, u_{max}) → M4_{3}
  15. CurveCartesian(fx(0.12566370614359174, t), fy(0.12566370614359174, t), fz(0.12566370614359174, t), t, v_{min}, v_{max}) → M5_{3}
  16. CurveCartesian(fx(t, 0.44999999999999996), fy(t, 0.44999999999999996), fz(t, 0.44999999999999996), t, u_{min}, u_{max}) → M4_{4}
  17. CurveCartesian(fx(0.1884955592153876, t), fy(0.1884955592153876, t), fz(0.1884955592153876, t), t, v_{min}, v_{max}) → M5_{4}
  18. CurveCartesian(fx(t, 0.6), fy(t, 0.6), fz(t, 0.6), t, u_{min}, u_{max}) → M4_{5}
  19. CurveCartesian(fx(0.25132741228718347, t), fy(0.25132741228718347, t), fz(0.25132741228718347, t), t, v_{min}, v_{max}) → M5_{5}
  20. CurveCartesian(fx(t, 0.75), fy(t, 0.75), fz(t, 0.75), t, u_{min}, u_{max}) → M4_{6}
  21. CurveCartesian(fx(0.3141592653589793, t), fy(0.3141592653589793, t), fz(0.3141592653589793, t), t, v_{min}, v_{max}) → M5_{6}
  22. CurveCartesian(fx(t, 0.8999999999999999), fy(t, 0.8999999999999999), fz(t, 0.8999999999999999), t, u_{min}, u_{max}) → M4_{7}
  23. CurveCartesian(fx(0.3769911184307752, t), fy(0.3769911184307752, t), fz(0.3769911184307752, t), t, v_{min}, v_{max}) → M5_{7}
  24. CurveCartesian(fx(t, 1.05), fy(t, 1.05), fz(t, 1.05), t, u_{min}, u_{max}) → M4_{8}
  25. CurveCartesian(fx(0.4398229715025711, t), fy(0.4398229715025711, t), fz(0.4398229715025711, t), t, v_{min}, v_{max}) → M5_{8}
  26. CurveCartesian(fx(t, 1.2), fy(t, 1.2), fz(t, 1.2), t, u_{min}, u_{max}) → M4_{9}
  27. CurveCartesian(fx(0.5026548245743669, t), fy(0.5026548245743669, t), fz(0.5026548245743669, t), t, v_{min}, v_{max}) → M5_{9}
  28. CurveCartesian(fx(t, 1.3499999999999999), fy(t, 1.3499999999999999), fz(t, 1.3499999999999999), t, u_{min}, u_{max}) → M4_{10}
  29. CurveCartesian(fx(0.5654866776461628, t), fy(0.5654866776461628, t), fz(0.5654866776461628, t), t, v_{min}, v_{max}) → M5_{10}
  30. CurveCartesian(fx(t, 1.5), fy(t, 1.5), fz(t, 1.5), t, u_{min}, u_{max}) → M4_{11}
  31. CurveCartesian(fx(0.6283185307179586, t), fy(0.6283185307179586, t), fz(0.6283185307179586, t), t, v_{min}, v_{max}) → M5_{11}
  32. CurveCartesian(fx(t, 1.65), fy(t, 1.65), fz(t, 1.65), t, u_{min}, u_{max}) → M4_{12}
  33. CurveCartesian(fx(0.6911503837897546, t), fy(0.6911503837897546, t), fz(0.6911503837897546, t), t, v_{min}, v_{max}) → M5_{12}
  34. CurveCartesian(fx(t, 1.7999999999999998), fy(t, 1.7999999999999998), fz(t, 1.7999999999999998), t, u_{min}, u_{max}) → M4_{13}
  35. CurveCartesian(fx(0.7539822368615504, t), fy(0.7539822368615504, t), fz(0.7539822368615504, t), t, v_{min}, v_{max}) → M5_{13}
  36. CurveCartesian(fx(t, 1.95), fy(t, 1.95), fz(t, 1.95), t, u_{min}, u_{max}) → M4_{14}
  37. CurveCartesian(fx(0.8168140899333463, t), fy(0.8168140899333463, t), fz(0.8168140899333463, t), t, v_{min}, v_{max}) → M5_{14}
  38. CurveCartesian(fx(t, 2.1), fy(t, 2.1), fz(t, 2.1), t, u_{min}, u_{max}) → M4_{15}
  39. CurveCartesian(fx(0.8796459430051422, t), fy(0.8796459430051422, t), fz(0.8796459430051422, t), t, v_{min}, v_{max}) → M5_{15}
  40. CurveCartesian(fx(t, 2.25), fy(t, 2.25), fz(t, 2.25), t, u_{min}, u_{max}) → M4_{16}
  41. CurveCartesian(fx(0.942477796076938, t), fy(0.942477796076938, t), fz(0.942477796076938, t), t, v_{min}, v_{max}) → M5_{16}
  42. CurveCartesian(fx(t, 2.4), fy(t, 2.4), fz(t, 2.4), t, u_{min}, u_{max}) → M4_{17}
  43. CurveCartesian(fx(1.0053096491487339, t), fy(1.0053096491487339, t), fz(1.0053096491487339, t), t, v_{min}, v_{max}) → M5_{17}
  44. CurveCartesian(fx(t, 2.55), fy(t, 2.55), fz(t, 2.55), t, u_{min}, u_{max}) → M4_{18}
  45. CurveCartesian(fx(1.0681415022205298, t), fy(1.0681415022205298, t), fz(1.0681415022205298, t), t, v_{min}, v_{max}) → M5_{18}
  46. CurveCartesian(fx(t, 2.6999999999999997), fy(t, 2.6999999999999997), fz(t, 2.6999999999999997), t, u_{min}, u_{max}) → M4_{19}
  47. CurveCartesian(fx(1.1309733552923256, t), fy(1.1309733552923256, t), fz(1.1309733552923256, t), t, v_{min}, v_{max}) → M5_{19}
  48. CurveCartesian(fx(t, 2.85), fy(t, 2.85), fz(t, 2.85), t, u_{min}, u_{max}) → M4_{20}
  49. CurveCartesian(fx(1.1938052083641215, t), fy(1.1938052083641215, t), fz(1.1938052083641215, t), t, v_{min}, v_{max}) → M5_{20}
  50. CurveCartesian(fx(t, 3), fy(t, 3), fz(t, 3), t, u_{min}, u_{max}) → M4_{21}
  51. CurveCartesian(fx(1.2566370614359172, t), fy(1.2566370614359172, t), fz(1.2566370614359172, t), t, v_{min}, v_{max}) → M5_{21}
  52. CurveCartesian(fx(t, 3.15), fy(t, 3.15), fz(t, 3.15), t, u_{min}, u_{max}) → M4_{22}
  53. CurveCartesian(fx(1.3194689145077132, t), fy(1.3194689145077132, t), fz(1.3194689145077132, t), t, v_{min}, v_{max}) → M5_{22}
  54. CurveCartesian(fx(t, 3.3), fy(t, 3.3), fz(t, 3.3), t, u_{min}, u_{max}) → M4_{23}
  55. CurveCartesian(fx(1.3823007675795091, t), fy(1.3823007675795091, t), fz(1.3823007675795091, t), t, v_{min}, v_{max}) → M5_{23}
  56. CurveCartesian(fx(t, 3.4499999999999997), fy(t, 3.4499999999999997), fz(t, 3.4499999999999997), t, u_{min}, u_{max}) → M4_{24}
  57. CurveCartesian(fx(1.4451326206513049, t), fy(1.4451326206513049, t), fz(1.4451326206513049, t), t, v_{min}, v_{max}) → M5_{24}
  58. CurveCartesian(fx(t, 3.5999999999999996), fy(t, 3.5999999999999996), fz(t, 3.5999999999999996), t, u_{min}, u_{max}) → M4_{25}
  59. CurveCartesian(fx(1.5079644737231008, t), fy(1.5079644737231008, t), fz(1.5079644737231008, t), t, v_{min}, v_{max}) → M5_{25}
  60. CurveCartesian(fx(t, 3.75), fy(t, 3.75), fz(t, 3.75), t, u_{min}, u_{max}) → M4_{26}
  61. CurveCartesian(fx(1.5707963267948968, t), fy(1.5707963267948968, t), fz(1.5707963267948968, t), t, v_{min}, v_{max}) → M5_{26}
  62. CurveCartesian(fx(t, 3.9), fy(t, 3.9), fz(t, 3.9), t, u_{min}, u_{max}) → M4_{27}
  63. CurveCartesian(fx(1.6336281798666925, t), fy(1.6336281798666925, t), fz(1.6336281798666925, t), t, v_{min}, v_{max}) → M5_{27}
  64. CurveCartesian(fx(t, 4.05), fy(t, 4.05), fz(t, 4.05), t, u_{min}, u_{max}) → M4_{28}
  65. CurveCartesian(fx(1.6964600329384885, t), fy(1.6964600329384885, t), fz(1.6964600329384885, t), t, v_{min}, v_{max}) → M5_{28}
  66. CurveCartesian(fx(t, 4.2), fy(t, 4.2), fz(t, 4.2), t, u_{min}, u_{max}) → M4_{29}
  67. CurveCartesian(fx(1.7592918860102844, t), fy(1.7592918860102844, t), fz(1.7592918860102844, t), t, v_{min}, v_{max}) → M5_{29}
  68. CurveCartesian(fx(t, 4.35), fy(t, 4.35), fz(t, 4.35), t, u_{min}, u_{max}) → M4_{30}
  69. CurveCartesian(fx(1.8221237390820801, t), fy(1.8221237390820801, t), fz(1.8221237390820801, t), t, v_{min}, v_{max}) → M5_{30}
  70. CurveCartesian(fx(t, 4.5), fy(t, 4.5), fz(t, 4.5), t, u_{min}, u_{max}) → M4_{31}
  71. CurveCartesian(fx(1.884955592153876, t), fy(1.884955592153876, t), fz(1.884955592153876, t), t, v_{min}, v_{max}) → M5_{31}
  72. CurveCartesian(fx(t, 4.6499999999999995), fy(t, 4.6499999999999995), fz(t, 4.6499999999999995), t, u_{min}, u_{max}) → M4_{32}
  73. CurveCartesian(fx(1.9477874452256718, t), fy(1.9477874452256718, t), fz(1.9477874452256718, t), t, v_{min}, v_{max}) → M5_{32}
  74. CurveCartesian(fx(t, 4.8), fy(t, 4.8), fz(t, 4.8), t, u_{min}, u_{max}) → M4_{33}
  75. CurveCartesian(fx(2.0106192982974678, t), fy(2.0106192982974678, t), fz(2.0106192982974678, t), t, v_{min}, v_{max}) → M5_{33}
  76. CurveCartesian(fx(t, 4.95), fy(t, 4.95), fz(t, 4.95), t, u_{min}, u_{max}) → M4_{34}
  77. CurveCartesian(fx(2.0734511513692637, t), fy(2.0734511513692637, t), fz(2.0734511513692637, t), t, v_{min}, v_{max}) → M5_{34}
  78. CurveCartesian(fx(t, 5.1), fy(t, 5.1), fz(t, 5.1), t, u_{min}, u_{max}) → M4_{35}
  79. CurveCartesian(fx(2.1362830044410597, t), fy(2.1362830044410597, t), fz(2.1362830044410597, t), t, v_{min}, v_{max}) → M5_{35}
  80. CurveCartesian(fx(t, 5.25), fy(t, 5.25), fz(t, 5.25), t, u_{min}, u_{max}) → M4_{36}
  81. CurveCartesian(fx(2.199114857512855, t), fy(2.199114857512855, t), fz(2.199114857512855, t), t, v_{min}, v_{max}) → M5_{36}
  82. CurveCartesian(fx(t, 5.3999999999999995), fy(t, 5.3999999999999995), fz(t, 5.3999999999999995), t, u_{min}, u_{max}) → M4_{37}
  83. CurveCartesian(fx(2.261946710584651, t), fy(2.261946710584651, t), fz(2.261946710584651, t), t, v_{min}, v_{max}) → M5_{37}
  84. CurveCartesian(fx(t, 5.55), fy(t, 5.55), fz(t, 5.55), t, u_{min}, u_{max}) → M4_{38}
  85. CurveCartesian(fx(2.324778563656447, t), fy(2.324778563656447, t), fz(2.324778563656447, t), t, v_{min}, v_{max}) → M5_{38}
  86. CurveCartesian(fx(t, 5.7), fy(t, 5.7), fz(t, 5.7), t, u_{min}, u_{max}) → M4_{39}
  87. CurveCartesian(fx(2.387610416728243, t), fy(2.387610416728243, t), fz(2.387610416728243, t), t, v_{min}, v_{max}) → M5_{39}
  88. CurveCartesian(fx(t, 5.85), fy(t, 5.85), fz(t, 5.85), t, u_{min}, u_{max}) → M4_{40}
  89. CurveCartesian(fx(2.450442269800039, t), fy(2.450442269800039, t), fz(2.450442269800039, t), t, v_{min}, v_{max}) → M5_{40}
  90. CurveCartesian(fx(t, 6), fy(t, 6), fz(t, 6), t, u_{min}, u_{max}) → M4_{41}
  91. CurveCartesian(fx(2.5132741228718345, t), fy(2.5132741228718345, t), fz(2.5132741228718345, t), t, v_{min}, v_{max}) → M5_{41}
  92. CurveCartesian(fx(t, 6.1499999999999995), fy(t, 6.1499999999999995), fz(t, 6.1499999999999995), t, u_{min}, u_{max}) → M4_{42}
  93. CurveCartesian(fx(2.5761059759436304, t), fy(2.5761059759436304, t), fz(2.5761059759436304, t), t, v_{min}, v_{max}) → M5_{42}
  94. CurveCartesian(fx(t, 6.3), fy(t, 6.3), fz(t, 6.3), t, u_{min}, u_{max}) → M4_{43}
  95. CurveCartesian(fx(2.6389378290154264, t), fy(2.6389378290154264, t), fz(2.6389378290154264, t), t, v_{min}, v_{max}) → M5_{43}
  96. CurveCartesian(fx(t, 6.45), fy(t, 6.45), fz(t, 6.45), t, u_{min}, u_{max}) → M4_{44}
  97. CurveCartesian(fx(2.7017696820872223, t), fy(2.7017696820872223, t), fz(2.7017696820872223, t), t, v_{min}, v_{max}) → M5_{44}
  98. CurveCartesian(fx(t, 6.6), fy(t, 6.6), fz(t, 6.6), t, u_{min}, u_{max}) → M4_{45}
  99. CurveCartesian(fx(2.7646015351590183, t), fy(2.7646015351590183, t), fz(2.7646015351590183, t), t, v_{min}, v_{max}) → M5_{45}
  100. CurveCartesian(fx(t, 6.75), fy(t, 6.75), fz(t, 6.75), t, u_{min}, u_{max}) → M4_{46}
  101. CurveCartesian(fx(2.8274333882308142, t), fy(2.8274333882308142, t), fz(2.8274333882308142, t), t, v_{min}, v_{max}) → M5_{46}
  102. CurveCartesian(fx(t, 6.8999999999999995), fy(t, 6.8999999999999995), fz(t, 6.8999999999999995), t, u_{min}, u_{max}) → M4_{47}
  103. CurveCartesian(fx(2.8902652413026098, t), fy(2.8902652413026098, t), fz(2.8902652413026098, t), t, v_{min}, v_{max}) → M5_{47}
  104. CurveCartesian(fx(t, 7.05), fy(t, 7.05), fz(t, 7.05), t, u_{min}, u_{max}) → M4_{48}
  105. CurveCartesian(fx(2.9530970943744057, t), fy(2.9530970943744057, t), fz(2.9530970943744057, t), t, v_{min}, v_{max}) → M5_{48}
  106. CurveCartesian(fx(t, 7.199999999999999), fy(t, 7.199999999999999), fz(t, 7.199999999999999), t, u_{min}, u_{max}) → M4_{49}
  107. CurveCartesian(fx(3.0159289474462017, t), fy(3.0159289474462017, t), fz(3.0159289474462017, t), t, v_{min}, v_{max}) → M5_{49}
  108. CurveCartesian(fx(t, 7.35), fy(t, 7.35), fz(t, 7.35), t, u_{min}, u_{max}) → M4_{50}
  109. CurveCartesian(fx(3.0787608005179976, t), fy(3.0787608005179976, t), fz(3.0787608005179976, t), t, v_{min}, v_{max}) → M5_{50}
  110. CurveCartesian(fx(t, 7.5), fy(t, 7.5), fz(t, 7.5), t, u_{min}, u_{max}) → M4_{51}
  111. CurveCartesian(fx(3.1415926535897936, t), fy(3.1415926535897936, t), fz(3.1415926535897936, t), t, v_{min}, v_{max}) → M5_{51}
  112. CurveCartesian(fx(t, 7.6499999999999995), fy(t, 7.6499999999999995), fz(t, 7.6499999999999995), t, u_{min}, u_{max}) → M4_{52}
  113. CurveCartesian(fx(3.204424506661589, t), fy(3.204424506661589, t), fz(3.204424506661589, t), t, v_{min}, v_{max}) → M5_{52}
  114. CurveCartesian(fx(t, 7.8), fy(t, 7.8), fz(t, 7.8), t, u_{min}, u_{max}) → M4_{53}
  115. CurveCartesian(fx(3.267256359733385, t), fy(3.267256359733385, t), fz(3.267256359733385, t), t, v_{min}, v_{max}) → M5_{53}
  116. CurveCartesian(fx(t, 7.949999999999999), fy(t, 7.949999999999999), fz(t, 7.949999999999999), t, u_{min}, u_{max}) → M4_{54}
  117. CurveCartesian(fx(3.330088212805181, t), fy(3.330088212805181, t), fz(3.330088212805181, t), t, v_{min}, v_{max}) → M5_{54}
  118. CurveCartesian(fx(t, 8.1), fy(t, 8.1), fz(t, 8.1), t, u_{min}, u_{max}) → M4_{55}
  119. CurveCartesian(fx(3.392920065876977, t), fy(3.392920065876977, t), fz(3.392920065876977, t), t, v_{min}, v_{max}) → M5_{55}
  120. CurveCartesian(fx(t, 8.25), fy(t, 8.25), fz(t, 8.25), t, u_{min}, u_{max}) → M4_{56}
  121. CurveCartesian(fx(3.455751918948773, t), fy(3.455751918948773, t), fz(3.455751918948773, t), t, v_{min}, v_{max}) → M5_{56}
  122. CurveCartesian(fx(t, 8.4), fy(t, 8.4), fz(t, 8.4), t, u_{min}, u_{max}) → M4_{57}
  123. CurveCartesian(fx(3.518583772020569, t), fy(3.518583772020569, t), fz(3.518583772020569, t), t, v_{min}, v_{max}) → M5_{57}
  124. CurveCartesian(fx(t, 8.549999999999999), fy(t, 8.549999999999999), fz(t, 8.549999999999999), t, u_{min}, u_{max}) → M4_{58}
  125. CurveCartesian(fx(3.5814156250923643, t), fy(3.5814156250923643, t), fz(3.5814156250923643, t), t, v_{min}, v_{max}) → M5_{58}
  126. CurveCartesian(fx(t, 8.7), fy(t, 8.7), fz(t, 8.7), t, u_{min}, u_{max}) → M4_{59}
  127. CurveCartesian(fx(3.6442474781641603, t), fy(3.6442474781641603, t), fz(3.6442474781641603, t), t, v_{min}, v_{max}) → M5_{59}
  128. CurveCartesian(fx(t, 8.85), fy(t, 8.85), fz(t, 8.85), t, u_{min}, u_{max}) → M4_{60}
  129. CurveCartesian(fx(3.7070793312359562, t), fy(3.7070793312359562, t), fz(3.7070793312359562, t), t, v_{min}, v_{max}) → M5_{60}
  130. CurveCartesian(fx(t, 9), fy(t, 9), fz(t, 9), t, u_{min}, u_{max}) → M4_{61}
  131. CurveCartesian(fx(3.769911184307752, t), fy(3.769911184307752, t), fz(3.769911184307752, t), t, v_{min}, v_{max}) → M5_{61}
  132. CurveCartesian(fx(t, 9.15), fy(t, 9.15), fz(t, 9.15), t, u_{min}, u_{max}) → M4_{62}
  133. CurveCartesian(fx(3.832743037379548, t), fy(3.832743037379548, t), fz(3.832743037379548, t), t, v_{min}, v_{max}) → M5_{62}
  134. CurveCartesian(fx(t, 9.299999999999999), fy(t, 9.299999999999999), fz(t, 9.299999999999999), t, u_{min}, u_{max}) → M4_{63}
  135. CurveCartesian(fx(3.8955748904513436, t), fy(3.8955748904513436, t), fz(3.8955748904513436, t), t, v_{min}, v_{max}) → M5_{63}
  136. CurveCartesian(fx(t, 9.45), fy(t, 9.45), fz(t, 9.45), t, u_{min}, u_{max}) → M4_{64}
  137. CurveCartesian(fx(3.9584067435231396, t), fy(3.9584067435231396, t), fz(3.9584067435231396, t), t, v_{min}, v_{max}) → M5_{64}
  138. CurveCartesian(fx(t, 9.6), fy(t, 9.6), fz(t, 9.6), t, u_{min}, u_{max}) → M4_{65}
  139. CurveCartesian(fx(4.0212385965949355, t), fy(4.0212385965949355, t), fz(4.0212385965949355, t), t, v_{min}, v_{max}) → M5_{65}
  140. CurveCartesian(fx(t, 9.75), fy(t, 9.75), fz(t, 9.75), t, u_{min}, u_{max}) → M4_{66}
  141. CurveCartesian(fx(4.084070449666731, t), fy(4.084070449666731, t), fz(4.084070449666731, t), t, v_{min}, v_{max}) → M5_{66}
  142. CurveCartesian(fx(t, 9.9), fy(t, 9.9), fz(t, 9.9), t, u_{min}, u_{max}) → M4_{67}
  143. CurveCartesian(fx(4.1469023027385274, t), fy(4.1469023027385274, t), fz(4.1469023027385274, t), t, v_{min}, v_{max}) → M5_{67}
  144. CurveCartesian(fx(t, 10.049999999999999), fy(t, 10.049999999999999), fz(t, 10.049999999999999), t, u_{min}, u_{max}) → M4_{68}
  145. CurveCartesian(fx(4.209734155810323, t), fy(4.209734155810323, t), fz(4.209734155810323, t), t, v_{min}, v_{max}) → M5_{68}
  146. CurveCartesian(fx(t, 10.2), fy(t, 10.2), fz(t, 10.2), t, u_{min}, u_{max}) → M4_{69}
  147. CurveCartesian(fx(4.272566008882119, t), fy(4.272566008882119, t), fz(4.272566008882119, t), t, v_{min}, v_{max}) → M5_{69}
  148. CurveCartesian(fx(t, 10.35), fy(t, 10.35), fz(t, 10.35), t, u_{min}, u_{max}) → M4_{70}
  149. CurveCartesian(fx(4.335397861953915, t), fy(4.335397861953915, t), fz(4.335397861953915, t), t, v_{min}, v_{max}) → M5_{70}
  150. CurveCartesian(fx(t, 10.5), fy(t, 10.5), fz(t, 10.5), t, u_{min}, u_{max}) → M4_{71}
  151. CurveCartesian(fx(4.39822971502571, t), fy(4.39822971502571, t), fz(4.39822971502571, t), t, v_{min}, v_{max}) → M5_{71}
  152. CurveCartesian(fx(t, 10.65), fy(t, 10.65), fz(t, 10.65), t, u_{min}, u_{max}) → M4_{72}
  153. CurveCartesian(fx(4.461061568097507, t), fy(4.461061568097507, t), fz(4.461061568097507, t), t, v_{min}, v_{max}) → M5_{72}
  154. CurveCartesian(fx(t, 10.799999999999999), fy(t, 10.799999999999999), fz(t, 10.799999999999999), t, u_{min}, u_{max}) → M4_{73}
  155. CurveCartesian(fx(4.523893421169302, t), fy(4.523893421169302, t), fz(4.523893421169302, t), t, v_{min}, v_{max}) → M5_{73}
  156. CurveCartesian(fx(t, 10.95), fy(t, 10.95), fz(t, 10.95), t, u_{min}, u_{max}) → M4_{74}
  157. CurveCartesian(fx(4.586725274241099, t), fy(4.586725274241099, t), fz(4.586725274241099, t), t, v_{min}, v_{max}) → M5_{74}
  158. CurveCartesian(fx(t, 11.1), fy(t, 11.1), fz(t, 11.1), t, u_{min}, u_{max}) → M4_{75}
  159. CurveCartesian(fx(4.649557127312894, t), fy(4.649557127312894, t), fz(4.649557127312894, t), t, v_{min}, v_{max}) → M5_{75}
  160. CurveCartesian(fx(t, 11.25), fy(t, 11.25), fz(t, 11.25), t, u_{min}, u_{max}) → M4_{76}
  161. CurveCartesian(fx(4.71238898038469, t), fy(4.71238898038469, t), fz(4.71238898038469, t), t, v_{min}, v_{max}) → M5_{76}
  162. CurveCartesian(fx(t, 11.4), fy(t, 11.4), fz(t, 11.4), t, u_{min}, u_{max}) → M4_{77}
  163. CurveCartesian(fx(4.775220833456486, t), fy(4.775220833456486, t), fz(4.775220833456486, t), t, v_{min}, v_{max}) → M5_{77}
  164. CurveCartesian(fx(t, 11.549999999999999), fy(t, 11.549999999999999), fz(t, 11.549999999999999), t, u_{min}, u_{max}) → M4_{78}
  165. CurveCartesian(fx(4.838052686528282, t), fy(4.838052686528282, t), fz(4.838052686528282, t), t, v_{min}, v_{max}) → M5_{78}
  166. CurveCartesian(fx(t, 11.7), fy(t, 11.7), fz(t, 11.7), t, u_{min}, u_{max}) → M4_{79}
  167. CurveCartesian(fx(4.900884539600078, t), fy(4.900884539600078, t), fz(4.900884539600078, t), t, v_{min}, v_{max}) → M5_{79}
  168. CurveCartesian(fx(t, 11.85), fy(t, 11.85), fz(t, 11.85), t, u_{min}, u_{max}) → M4_{80}
  169. CurveCartesian(fx(4.9637163926718735, t), fy(4.9637163926718735, t), fz(4.9637163926718735, t), t, v_{min}, v_{max}) → M5_{80}
  170. CurveCartesian(fx(t, 12), fy(t, 12), fz(t, 12), t, u_{min}, u_{max}) → M4_{81}
  171. CurveCartesian(fx(5.026548245743669, t), fy(5.026548245743669, t), fz(5.026548245743669, t), t, v_{min}, v_{max}) → M5_{81}
  172. CurveCartesian(fx(t, 12.15), fy(t, 12.15), fz(t, 12.15), t, u_{min}, u_{max}) → M4_{82}
  173. CurveCartesian(fx(5.089380098815465, t), fy(5.089380098815465, t), fz(5.089380098815465, t), t, v_{min}, v_{max}) → M5_{82}
  174. CurveCartesian(fx(t, 12.299999999999999), fy(t, 12.299999999999999), fz(t, 12.299999999999999), t, u_{min}, u_{max}) → M4_{83}
  175. CurveCartesian(fx(5.152211951887261, t), fy(5.152211951887261, t), fz(5.152211951887261, t), t, v_{min}, v_{max}) → M5_{83}
  176. CurveCartesian(fx(t, 12.45), fy(t, 12.45), fz(t, 12.45), t, u_{min}, u_{max}) → M4_{84}
  177. CurveCartesian(fx(5.215043804959057, t), fy(5.215043804959057, t), fz(5.215043804959057, t), t, v_{min}, v_{max}) → M5_{84}
  178. CurveCartesian(fx(t, 12.6), fy(t, 12.6), fz(t, 12.6), t, u_{min}, u_{max}) → M4_{85}
  179. CurveCartesian(fx(5.277875658030853, t), fy(5.277875658030853, t), fz(5.277875658030853, t), t, v_{min}, v_{max}) → M5_{85}
  180. CurveCartesian(fx(t, 12.75), fy(t, 12.75), fz(t, 12.75), t, u_{min}, u_{max}) → M4_{86}
  181. CurveCartesian(fx(5.340707511102649, t), fy(5.340707511102649, t), fz(5.340707511102649, t), t, v_{min}, v_{max}) → M5_{86}
  182. CurveCartesian(fx(t, 12.9), fy(t, 12.9), fz(t, 12.9), t, u_{min}, u_{max}) → M4_{87}
  183. CurveCartesian(fx(5.403539364174445, t), fy(5.403539364174445, t), fz(5.403539364174445, t), t, v_{min}, v_{max}) → M5_{87}
  184. CurveCartesian(fx(t, 13.049999999999999), fy(t, 13.049999999999999), fz(t, 13.049999999999999), t, u_{min}, u_{max}) → M4_{88}
  185. CurveCartesian(fx(5.46637121724624, t), fy(5.46637121724624, t), fz(5.46637121724624, t), t, v_{min}, v_{max}) → M5_{88}
  186. CurveCartesian(fx(t, 13.2), fy(t, 13.2), fz(t, 13.2), t, u_{min}, u_{max}) → M4_{89}
  187. CurveCartesian(fx(5.529203070318037, t), fy(5.529203070318037, t), fz(5.529203070318037, t), t, v_{min}, v_{max}) → M5_{89}
  188. CurveCartesian(fx(t, 13.35), fy(t, 13.35), fz(t, 13.35), t, u_{min}, u_{max}) → M4_{90}
  189. CurveCartesian(fx(5.592034923389832, t), fy(5.592034923389832, t), fz(5.592034923389832, t), t, v_{min}, v_{max}) → M5_{90}
  190. CurveCartesian(fx(t, 13.5), fy(t, 13.5), fz(t, 13.5), t, u_{min}, u_{max}) → M4_{91}
  191. CurveCartesian(fx(5.6548667764616285, t), fy(5.6548667764616285, t), fz(5.6548667764616285, t), t, v_{min}, v_{max}) → M5_{91}
  192. CurveCartesian(fx(t, 13.65), fy(t, 13.65), fz(t, 13.65), t, u_{min}, u_{max}) → M4_{92}
  193. CurveCartesian(fx(5.717698629533424, t), fy(5.717698629533424, t), fz(5.717698629533424, t), t, v_{min}, v_{max}) → M5_{92}
  194. CurveCartesian(fx(t, 13.799999999999999), fy(t, 13.799999999999999), fz(t, 13.799999999999999), t, u_{min}, u_{max}) → M4_{93}
  195. CurveCartesian(fx(5.7805304826052195, t), fy(5.7805304826052195, t), fz(5.7805304826052195, t), t, v_{min}, v_{max}) → M5_{93}
  196. CurveCartesian(fx(t, 13.95), fy(t, 13.95), fz(t, 13.95), t, u_{min}, u_{max}) → M4_{94}
  197. CurveCartesian(fx(5.843362335677016, t), fy(5.843362335677016, t), fz(5.843362335677016, t), t, v_{min}, v_{max}) → M5_{94}
  198. CurveCartesian(fx(t, 14.1), fy(t, 14.1), fz(t, 14.1), t, u_{min}, u_{max}) → M4_{95}
  199. CurveCartesian(fx(5.906194188748811, t), fy(5.906194188748811, t), fz(5.906194188748811, t), t, v_{min}, v_{max}) → M5_{95}
  200. CurveCartesian(fx(t, 14.25), fy(t, 14.25), fz(t, 14.25), t, u_{min}, u_{max}) → M4_{96}
  201. CurveCartesian(fx(5.969026041820608, t), fy(5.969026041820608, t), fz(5.969026041820608, t), t, v_{min}, v_{max}) → M5_{96}
  202. CurveCartesian(fx(t, 14.399999999999999), fy(t, 14.399999999999999), fz(t, 14.399999999999999), t, u_{min}, u_{max}) → M4_{97}
  203. CurveCartesian(fx(6.031857894892403, t), fy(6.031857894892403, t), fz(6.031857894892403, t), t, v_{min}, v_{max}) → M5_{97}
  204. CurveCartesian(fx(t, 14.549999999999999), fy(t, 14.549999999999999), fz(t, 14.549999999999999), t, u_{min}, u_{max}) → M4_{98}
  205. CurveCartesian(fx(6.094689747964199, t), fy(6.094689747964199, t), fz(6.094689747964199, t), t, v_{min}, v_{max}) → M5_{98}
  206. CurveCartesian(fx(t, 14.7), fy(t, 14.7), fz(t, 14.7), t, u_{min}, u_{max}) → M4_{99}
  207. CurveCartesian(fx(6.157521601035995, t), fy(6.157521601035995, t), fz(6.157521601035995, t), t, v_{min}, v_{max}) → M5_{99}
  208. CurveCartesian(fx(t, 14.85), fy(t, 14.85), fz(t, 14.85), t, u_{min}, u_{max}) → M4_{100}
  209. CurveCartesian(fx(6.220353454107791, t), fy(6.220353454107791, t), fz(6.220353454107791, t), t, v_{min}, v_{max}) → M5_{100}
  210. CurveCartesian(fx(t, 15), fy(t, 15), fz(t, 15), t, u_{min}, u_{max}) → M4_{101}
  211. CurveCartesian(fx(6.283185307179587, t), fy(6.283185307179587, t), fz(6.283185307179587, t), t, v_{min}, v_{max}) → M5_{101}