View source for Surface of screw motion

''helical surface''

A surface described by a plane curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913801.png" /> which, while rotating around an axis at a uniform rate, also advances along that axis at a uniform rate. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913802.png" /> is located in the plane of the axis of rotation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913803.png" /> and is defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913804.png" />, the position vector of the surface of screw motion is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913805.png" /></td> </tr></table>

and its line element is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913806.png" /></td> </tr></table>

A surface of screw motion can be deformed into a rotation surface so that the generating helical lines are parallel (Boor's theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913807.png" />, one has a [[Helicoid|helicoid]]; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091380/s0913808.png" />, one has a [[Rotation surface|rotation surface]], or surface of revolution.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,   "Introduction to geometry" , Wiley  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.P. Do Carmo,   "Differential geometry of curves and surfaces" , Prentice-Hall  (1976)  pp. 145</TD></TR></table>