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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650601.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650602.png" /> 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|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
  
<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/m/m065/m065060/m0650603.png" /></td> </tr></table>
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$$r=\rho(u)+\eta(v)p(u)+\zeta(v)q(u),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650606.png" /> is the principal normal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650607.png" /> is the binormal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650608.png" /> is the torsion of the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m0650609.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m06506010.png" />. Its line element is given by:
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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:
  
<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/m/m065/m065060/m06506011.png" /></td> </tr></table>
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$$ds^2=[1+k(\zeta\sin\theta-\eta\cos\theta)]^2du^2+(\eta'^2+\zeta'^2)dv^2,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m06506012.png" /> are the equations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m06506013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m06506014.png" /> is the curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065060/m06506015.png" />.
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where $\eta(v),\zeta(v)$ are the equations of $L$ and $k$ is the curvature of $\Gamma$.
  
  

Revision as of 13:16, 4 October 2014

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 $\Gamma$.


Comments

References

[a1] G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Chelsea, reprint (1972) pp. Sects. 85–87
How to Cite This Entry:
Moulding surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Moulding_surface&oldid=33483
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article