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Difference between revisions of "Surface of screw motion"

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''helical surface''
 
''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
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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
  
<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>
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$$r=\{u\cos v,u\sin v,f(u)+hv\},\quad h=\text{const},$$
  
 
and its line element is
 
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>
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$$ds^2=(1+f'^2)du^2+2hf'dudv+(u^2+h^2)dv^2.$$
 
 
A surface of screw motion can be deformed into a rotation surface so that the generating helical lines are parallel (Boor's theorem). If <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.
 
 
 
 
 
 
 
====Comments====
 
  
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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"> 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>
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<table>
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<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>
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley  (1961)</TD></TR>
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<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>
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</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
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article