page_title: Generalized helicoid
section_title: On closed ruled generalized helicoids
hierarchical_section_title: Generalized helicoid / Ruled generalized helicoids / On closed ruled generalized helicoids
context_page_description: In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the profile curve, along a line, its axis. Any point of the given curve is the starting point of a circular helix. If the profile curve is contained in a plane through the axis, it is called the meridian of the generalized helicoid. Simple examples of generalized helicoids are the helicoids. The meridian of a helicoid is a line which intersects the axis orthogonally.
  Essential types of generalized helicoids are
  ruled generalized helicoids. Their profile curves are lines and the surfaces are ruled surfaces.
  circular generalized helicoids. Their profile curves are circles.
  In mathematics helicoids play an essential role as minimal surfaces.
  In the technical area generalized helicoids are used for staircases, slides, screws, and pipes.
context_section_description: A closed ruled generalized helicoid has a profile line that intersects the axis. If the profile line is described by    one gets the following parametric representation 
  If  (common helicoid) the surface does not intersect itself. 
  
  If  (oblique type) the surface intersects itself and the curves (on the surface)
   with 
  consist of double points. There exist infinite double curves. The smaller   the greater are the distances between the double curves.
caption: on the selfintersection of closed ruled generalized helicoids