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Steinbach Helix and Catenoid

Steinbach Helix and Catenoid
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2022-02-17 18:15
FunctionSolid GeometryParametric EquationGeometric Construction
This example demonstrates the Steinbach helix, catenoid, and their relationship with the helicoid. It involves core mathematical concepts like parametric equations, surface generation, and geometric transformations. Constructed using commands such as Line, Point, Ray, Surface, and CurveCartesian, it is suitable for high school students and math enthusiasts studying surface geometry and parametric equations.

GeoGebra Commands & Parameters

  1. Line((0, 0, 0), (0, 0, -1)) → segSep
  2. Point(segSep) → POrigen
  3. Ray(POrigen, POrigen + (0, 0, -1)) → segSep2
  4. Textfield(vmin) → CasillaDeEntrada4
  5. Textfield(umax) → CasillaDeEntrada5
  6. Textfield(gCurva) → CasillaDeEntrada7
  7. Textfield(gSuperficie) → CasillaDeEntrada8
  8. Textfield(colSup) → CasillaDeEntrada9
  9. Textfield(colCurva) → CasillaDeEntrada10
  10. Textfield(n) → CasillaDeEntrada1
  11. Textfield(desarrolloVer) → CasillaDeEntrada11
  12. Textfield(colFondo) → CasillaDeEntrada12
  13. Surface((u * cos(v)), (u * sin(v)), zInc + v + ((v * k) * (cos(u) - 1)), v, 0, vmax, u, -4, 4) → sup
  14. CurveCartesian((u * cos(vmax)), (u * sin(vmax)), zInc + vmax + ((vmax * k) * (cos(u) - 1)), u, -4, 4) → curva1
  15. CurveCartesian(((-umax) * cos(v)), ((-umax) * sin(v)), zInc + v + ((v * k) * (cos((-umax)) - 1)), v, 0, vmax) → borde2
  16. CurveCartesian((umax * cos(v)), (umax * sin(v)), zInc + v + ((v * k) * (cos(umax) - 1)), v, 0, vmax) → borde
  17. CurveCartesian((u * cos(0)), (u * sin(0)), zInc + 0 + ((0 * k) * (cos(u) - 1)), u, -4, 4) → borde3

AI Drawing Prompt Reference

Draw the Steinbach helix and catenoid. Start by drawing a vertical line along the z-axis and create a point at the origin. Define the parameter ranges and use parametric equations to draw the helix and catenoid. Fill the catenoid with different colors and draw the helix. The helix and catenoid revolve around the z-axis, with the helix above the catenoid.
The prompt above was compiled from the original drawing commands and is for reference only.