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Difference between revisions of "Moulding surface"

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A surface generated by the orthogonal trajectories of a one-parameter family of planes. Moulding surfaces have one family of planar lines of curvature that are simultaneously geodesics for the moulding surface. If the family of planes is degenerated into a bundle, then the moulding surface will be a [[Surface of revolution|surface of revolution]]. The sections of a moulding surface by planes of the family are called meridians, and the orthogonal trajectories are called parallels of the moulding surface. All meridians are congruent, so that a moulding surface can be formed by the motion of a planar line $L$ (the meridian), the plane of which moves without sliding along a certain developable surface. This surface is called the directing surface of the moulding surface and is one of the sheets of its evolute. If $\rho(u)$ is the position vector of one parallel position, then the position vector of the moulding surface will be
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A surface generated by the orthogonal trajectories of a one-parameter family of planes. Moulding surfaces have one family of planar lines of curvature that are simultaneously geodesics for the moulding surface. If the family of planes is degenerated into a bundle, then the moulding surface will be a [[surface of revolution]]. The sections of a moulding surface by planes of the family are called meridians, and the orthogonal trajectories are called parallels of the moulding surface. All meridians are congruent, so that a moulding surface can be formed by the motion of a planar line $L$ (the meridian), the plane of which moves without sliding along a certain developable surface. This surface is called the directing surface of the moulding surface and is one of the sheets of its evolute. If $\rho(u)$ is the position vector of one parallel position, then the position vector of the moulding surface will be
  
 
$$r=\rho(u)+\eta(v)p(u)+\zeta(v)q(u),$$
 
$$r=\rho(u)+\eta(v)p(u)+\zeta(v)q(u),$$
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where $\eta(v),\zeta(v)$ are the equations of $L$ and $k$ is the curvature of $\Gamma$.
 
where $\eta(v),\zeta(v)$ are the equations of $L$ and $k$ is the curvature of $\Gamma$.
 
 
 
====Comments====
 
 
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , '''1''' , Chelsea, reprint  (1972)  pp. Sects. 85–87 {{ZBL|0257.53001}}</TD></TR>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Darboux,  "Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal" , '''1''' , Chelsea, reprint  (1972)  pp. Sects. 85–87 {{ZBL|0257.53001}}</TD></TR>
 
</table>
 
</table>

Latest revision as of 15:05, 10 April 2023

A surface generated by the orthogonal trajectories of a one-parameter family of planes. Moulding surfaces have one family of planar lines of curvature that are simultaneously geodesics for the moulding surface. If the family of planes is degenerated into a bundle, then the moulding surface will be a surface of revolution. The sections of a moulding surface by planes of the family are called meridians, and the orthogonal trajectories are called parallels of the moulding surface. All meridians are congruent, so that a moulding surface can be formed by the motion of a planar line $L$ (the meridian), the plane of which moves without sliding along a certain developable surface. This surface is called the directing surface of the moulding surface and is one of the sheets of its evolute. If $\rho(u)$ is the position vector of one parallel position, then the position vector of the moulding surface will be

$$r=\rho(u)+\eta(v)p(u)+\zeta(v)q(u),$$

where $p=\nu\cos\theta+\beta\sin\theta$, $q=-\nu\sin\theta+\beta\cos\theta$, $v$ is the principal normal, $\beta$ is the binormal, $x$ is the torsion of the curve $\Gamma$, and $\theta=-\int xdu$. Its line element is given by:

$$ds^2=[1+k(\zeta\sin\theta-\eta\cos\theta)]^2du^2+(\eta'^2+\zeta'^2)dv^2,$$

where $\eta(v),\zeta(v)$ are the equations of $L$ and $k$ is the curvature of $\Gamma$.

References

[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal" , 1 , Chelsea, reprint (1972) pp. Sects. 85–87 Zbl 0257.53001
How to Cite This Entry:
Moulding surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moulding_surface&oldid=53751
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article