Difference between revisions of "Surface of screw motion"
From Encyclopedia of Mathematics
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''helical surface'' | ''helical surface'' | ||
| − | A surface described by a plane curve | + | A surface described by a plane curve $L$ which, while rotating around an axis at a uniform rate, also advances along that axis at a uniform rate. If $L$ is located in the plane of the axis of rotation $z$ and is defined by the equation $z=f(u)$, the position vector of the surface of screw motion is |
| − | + | $$r=\{u\cos v,u\sin v,f(u)+hv\},\quad h=\text{const},$$ | |
and its line element is | and its line element is | ||
| − | + | $$ds^2=(1+f'^2)du^2+2hf'dudv+(u^2+h^2)dv^2.$$ | |
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| + | A surface of screw motion can be deformed into a rotation surface so that the generating helical lines are parallel (Boor's theorem). If $f=\text{const}$, one has a [[helicoid]]; if $h=0$, one has a [[rotation surface]], or surface of revolution. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> | ||
Latest revision as of 20:09, 5 March 2024
helical surface
A surface described by a plane curve $L$ which, while rotating around an axis at a uniform rate, also advances along that axis at a uniform rate. If $L$ is located in the plane of the axis of rotation $z$ and is defined by the equation $z=f(u)$, the position vector of the surface of screw motion is
$$r=\{u\cos v,u\sin v,f(u)+hv\},\quad h=\text{const},$$
and its line element is
$$ds^2=(1+f'^2)du^2+2hf'dudv+(u^2+h^2)dv^2.$$
A surface of screw motion can be deformed into a rotation surface so that the generating helical lines are parallel (Boor's theorem). If $f=\text{const}$, one has a helicoid; if $h=0$, one has a rotation surface, or surface of revolution.
References
| [a1] | M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French) |
| [a2] | H.S.M. Coxeter, "Introduction to geometry" , Wiley (1961) |
| [a3] | M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145 |
How to Cite This Entry:
Surface of screw motion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surface_of_screw_motion&oldid=17025
Surface of screw motion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Surface_of_screw_motion&oldid=17025
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article