Generalized helicoid
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. Analytical representation Screw motion of a point Moving a point on a screwtype curve means, the point is rotated and displaced along a line (axis) such that the displacement is proportional to the rotation-angle. The result is a circular helix. If the axis is the z-axis, the motion of a point can be described parametrically by is called slant, the angle , measured in radian, is called the screw angle and the pitch (green). The trace of the point is a circular helix (red). It is contained in the surface of a right circular cylinder. Its radius is the distance of point to the z-axis. In case of , the helix is called right handed; otherwise, it is said to be left handed. (In case of the motion is a rotation around the z-axis.) Screw motion of a curve The screw motion of curve yields a generalized helicoid with the parametric representation The curves are circular helices. The curves are copies of the given profile curve. Example: For the first picture above, the meridian is a parabola. Ruled generalized helicoids Types If the profile curve is a line one gets a ruled generalized helicoid. There are four types: (1) The line intersects the axis orthogonally. One gets a helicoid (closed right ruled generalized helicoid). (2) The line intersects the axis, but not orthogonally. One gets an oblique closed type. If the given line and the axis are skew lines one gets an open type and the axis is not part of the surface (s. picture). (3) If the given line and the axis are skew lines and the line is contained in a plane orthogonally to the axis one gets a right open type or shortly open helicoid. (4) If the line and the axis are skew and the line is not contained in ... (s. 3) one gets an oblique open type. Oblique types do intersect themselves (s. picture), right types (helicoids) do not. One gets an interesting case, if the line is skew to the axis and the product of its distance to the axis and its slope is exactly . In this case the surface is a tangent developable surface and is