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Cylindrical Surface and Function Plotting: GeoGebra Geometric Construction Examp

Cylindrical Surface and Function Plotting: GeoGebra Geometric Construction Examp
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2019-01-21 20:57
FunctionSolid GeometryTransformationParametric EquationGeometric Construction
This example demonstrates the construction of a cylindrical surface in three-dimensional space and its relationship with function graphs. The key mathematical concepts include the equation of a cylinder, parametric equations, rotation transformations, and plotting function graphs. The construction involves using GeoGebra commands such as CurveCartesian, Rotate, and Surface to create cylindrical surface objects, and parameterizing different heights to generate cross-sections. It is suitable for high school students or college undergraduates to learn about the integration of solid geometry and function graphing.

GeoGebra Commands & Parameters

  1. CurveCartesian((3.2 * cos(t)), (3.2 * sin(t)), t, 0, (2 * π)) → curvaEXT
  2. CurveCartesian(cos(t) / 4, sin(t) / 4, t, 0, (2 * π)) → curvaINT
  3. CurveCartesian(cos(t) / 2, sin(t) / 2, t, 0, (2 * π)) → curvaINT2
  4. CurveCartesian(2 + (0.83 * cos(t)), 0.83 + (0.83 * sin(t)), t, -1.5, π - 0.9) → C1
  5. CurveCartesian(2 + (0.83 * cos(t)), 0.83 + (0.83 * sin(t)), t, -1.5, π - 0.9) → c1
  6. CurveCartesian(0.83 + (0.83 * cos(t)), 2 + (0.83 * sin(t)), t, -0.68, π - 0.2) → c2
  7. CurveCartesian(2 + (0.83 * cos(t)), -0.83 + (0.83 * sin(t)), t, -2.3, π - 1.65) → c3
  8. CurveCartesian(0.83 + (0.83 * cos(t)), -2 + (0.83 * sin(t)), t, -3, π - 2.4) → c4
  9. CurveCartesian(-0.83 + (0.83 * cos(t)), -2 + (0.83 * sin(t)), t, -3.96, π - 3.28) → c5
  10. CurveCartesian(-2 + (0.83 * cos(t)), -0.83 + (0.83 * sin(t)), t, -4.7, π - 4) → c6
  11. CurveCartesian(-2 + (0.83 * cos(t)), 0.83 + (0.83 * sin(t)), t, -5.5, π - 4.7) → c7
  12. CurveCartesian(-0.83 + (0.83 * cos(t)), 2 + (0.83 * sin(t)), t, -6.1, π - 5.5) → c8
  13. CurveCartesian(1.2 + (1.25 * cos(t)), 1.2 + (2.5 * sin(t)), t, 2.75, π + 0.28) → r1
  14. Rotate(r1, (0.25 * π)) → r2
  15. Rotate(r2, (0.25 * π)) → r3
  16. Rotate(r3, (0.25 * π)) → r4
  17. Rotate(r4, (0.25 * π)) → r5
  18. Rotate(r5, (0.25 * π)) → r6
  19. Rotate(r6, (0.25 * π)) → r7
  20. Rotate(r7, (0.25 * π)) → r8
  21. Surface(2 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), 0.83 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -1.5, π - 0.9) → a
  22. Surface(0.83 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), 2 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -0.68, π - 0.2) → b
  23. Surface(2 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), -0.83 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -2.3, π - 1.65) → c
  24. Surface(0.83 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), -2 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -3, π - 2.4) → d
  25. Surface(-0.83 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), -2 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -3.96, π - 3.28) → e
  26. Surface(-2 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), -0.83 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -4.7, π - 4) → f
  27. Surface(-2 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), 0.83 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -5.5, π - 4.7) → g
  28. Surface(-0.83 + (0.83 * cos(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * cos(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), 2 + (0.83 * sin(t)) - ((((0.83 * R) * sin(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))) * sin(t) / (((abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) * sqrt((0.6889 * abs(sin(t))^(2)) + (0.6889 * abs(cos(t))^(2)))) * sqrt(((0.6889 * abs(((0.6889 * cos(t)) * sin(t)^(2)) + (0.6889 * cos(t)^(3)))^(2)) + (0.6889 * abs((0.6889 * sin(t)^(3)) + ((0.6889 * sin(t)) * cos(t)^(2)))^(2))) / (((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((0.6889 * abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2))) / abs((0.6889 * sin(t)^(2)) + (0.6889 * cos(t)^(2)))), u, 0, (2 * π), t, -6.1, π - 5.5) → h
  29. Surface((3.2 * cos(t)) - ((((3.2 * MANU) * sin(u)) * ((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))) * cos(t) / (((abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2))) * sqrt((10.24 * abs(sin(t))^(2)) + (10.24 * abs(cos(t))^(2)))) * sqrt(((10.24 * abs(((10.24 * sin(t)^(2)) * cos(t)) + (10.24 * cos(t)^(3)))^(2)) + (10.24 * abs((10.24 * sin(t)^(3)) + ((10.24 * cos(t)^(2)) * sin(t)))^(2))) / (((10.24 * abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((10.24 * abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), (3.2 * sin(t)) - ((((3.2 * MANU) * sin(u)) * ((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))) * sin(t) / (((abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2))) * sqrt((10.24 * abs(sin(t))^(2)) + (10.24 * abs(cos(t))^(2)))) * sqrt(((10.24 * abs(((10.24 * sin(t)^(2)) * cos(t)) + (10.24 * cos(t)^(3)))^(2)) + (10.24 * abs((10.24 * sin(t)^(3)) + ((10.24 * cos(t)^(2)) * sin(t)))^(2))) / (((10.24 * abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((10.24 * abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((MANU * cos(u)) * ((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2))) / abs((10.24 * sin(t)^(2)) + (10.24 * cos(t)^(2)))), u, 0, MENCHU, t, 0, (2 * π)) → tuboEXT
  30. Surface((1 / 2 * cos(t)) - (((R * sin(u)) * ((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))) * cos(t) / (((abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2))) * sqrt(abs(sin(t))^(2) + abs(cos(t))^(2))) * sqrt((abs((cos(t) * sin(t)^(2)) + cos(t)^(3))^(2) + abs(sin(t)^(3) + (sin(t) * cos(t)^(2)))^(2)) / (((16 * abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((16 * abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), (1 / 2 * sin(t)) - (((R * sin(u)) * ((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))) * sin(t) / (((abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2))) * sqrt(abs(sin(t))^(2) + abs(cos(t))^(2))) * sqrt((abs((cos(t) * sin(t)^(2)) + cos(t)^(3))^(2) + abs(sin(t)^(3) + (sin(t) * cos(t)^(2)))^(2)) / (((16 * abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((16 * abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2))) / abs((1 / 4 * sin(t)^(2)) + (1 / 4 * cos(t)^(2)))), u, 0, (2 * π), t, 0, (2 * π)) → tuboINT2
  31. Surface(1.2 + (1.25 * cos(t)) - ((((2.5 * R) * sin(u)) * ((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))) * cos(t) / (((abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2))) * sqrt((1.5625 * abs(sin(t))^(2)) + (6.25 * abs(cos(t))^(2)))) * sqrt(((6.25 * abs(((3.125 * cos(t)) * sin(t)^(2)) + (3.125 * cos(t)^(3)))^(2)) + (1.5625 * abs((3.125 * sin(t)^(3)) + ((3.125 * sin(t)) * cos(t)^(2)))^(2))) / (((1.5625 * abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((6.25 * abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), 1.2 + (2.5 * sin(t)) - ((((1.25 * R) * sin(u)) * ((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))) * sin(t) / (((abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2))) * sqrt((1.5625 * abs(sin(t))^(2)) + (6.25 * abs(cos(t))^(2)))) * sqrt(((6.25 * abs(((3.125 * cos(t)) * sin(t)^(2)) + (3.125 * cos(t)^(3)))^(2)) + (1.5625 * abs((3.125 * sin(t)^(3)) + ((3.125 * sin(t)) * cos(t)^(2)))^(2))) / (((1.5625 * abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((6.25 * abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2))) / abs((3.125 * sin(t)^(2)) + (3.125 * cos(t)^(2)))), u, 0, (2 * π), t, 2.75, π + 0.28) → rec1
  32. Surface((1 / 4 * cos(t)) - (((R * sin(u)) * ((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))) * cos(t) / (((abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2))) * sqrt(abs(sin(t))^(2) + abs(cos(t))^(2))) * sqrt((abs((cos(t) * sin(t)^(2)) + cos(t)^(3))^(2) + abs(sin(t)^(3) + (sin(t) * cos(t)^(2)))^(2)) / (((256 * abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((256 * abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), (1 / 4 * sin(t)) - (((R * sin(u)) * ((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))) * sin(t) / (((abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2))) * sqrt(abs(sin(t))^(2) + abs(cos(t))^(2))) * sqrt((abs((cos(t) * sin(t)^(2)) + cos(t)^(3))^(2) + abs(sin(t)^(3) + (sin(t) * cos(t)^(2)))^(2)) / (((256 * abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))^(2)) * abs(sin(t))^(2)) + ((256 * abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))^(2)) * abs(cos(t))^(2))))))), ((R * cos(u)) * ((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2))) / abs((1 / 16 * sin(t)^(2)) + (1 / 16 * cos(t)^(2)))), u, 0, (2 * π), t, 0, (2 * π)) → i
  33. Rotate(rec1, (0.25 * π)) → rec2
  34. Rotate(rec2, (0.25 * π)) → rec3
  35. Rotate(rec3, (0.25 * π)) → rec4
  36. Rotate(rec4, (0.25 * π)) → rec5
  37. Rotate(rec5, (0.25 * π)) → rec6
  38. Rotate(rec1, (-0.25 * π)) → rec7
  39. Rotate(rec7, (-0.25 * π)) → rec8