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| − | Diffeomorphic equi-oriented surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713902.png" /> having parallel tangent planes at corresponding points and such that the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713903.png" /> between corresponding points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713905.png" /> is constant and equal to that between the corresponding tangent planes. The position vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713906.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713907.png" /> of two parallel surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p0713909.png" /> are connected by a relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139011.png" /> is a unit normal vector that is the same for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139014.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139015.png" />. | + | {{TEX|done}} |
| | + | Diffeomorphic equi-oriented surfaces $F_1$ and $F_2$ having parallel tangent planes at corresponding points and such that the distance $h$ between corresponding points of $F_1$ and $F_2$ is constant and equal to that between the corresponding tangent planes. The position vectors $\mathbf r_1$ and $\mathbf r_2$ of two parallel surfaces $F_1$ and $F_2$ are connected by a relation $\mathbf r_2-\mathbf r_1=h\mathbf n$, where $\mathbf n$ is a unit normal vector that is the same for $F_1$ at $r_1$ and $F_2$ at $r_2$. |
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| − | Thus, one can define a one-parameter family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139016.png" /> of surfaces parallel to a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139018.png" /> is regular for sufficiently small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139019.png" /> for which | + | Thus, one can define a one-parameter family $F_h$ of surfaces parallel to a given $F=F_0$, where $F_h$ is regular for sufficiently small values of $h$ for which |
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| − | <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/p/p071/p071390/p07139020.png" /></td> </tr></table>
| + | $$w(h)=1-2Hh+Kh^2>0.$$ |
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| − | To the values of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139022.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139023.png" /> there correspond two surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139025.png" /> that are evolutes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139026.png" />, so that parallel surfaces have a common evolute (cf. [[Evolute (surface)|Evolute (surface)]]). The [[Mean curvature|mean curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139027.png" /> and the [[Gaussian curvature|Gaussian curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139028.png" /> of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139029.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139030.png" /> are connected with the corresponding quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139033.png" /> by the relations | + | To the values of the roots $h_1$ and $h_2$ of the equation $w(h)=0$ there correspond two surfaces $F_{h_1}$ and $F_{h_2}$ that are evolutes of $$, so that parallel surfaces have a common evolute (cf. [[Evolute (surface)|Evolute (surface)]]). The [[Mean curvature|mean curvature]] $H_h$ and the [[Gaussian curvature|Gaussian curvature]] $K_h$ of a surface $F_h$ parallel to $F$ are connected with the corresponding quantities $H$ and $K$ of $F$ by the relations |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139034.png" /></td> </tr></table>
| + | $$H_h=\frac{H-Kh}{w}(h),\quad K_h=\frac Kw(h);$$ |
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| | lines of curvature of parallel surfaces correspond to each other, so that between them there is a Combescour correspondence, which is a special case of a [[Peterson correspondence|Peterson correspondence]]. | | lines of curvature of parallel surfaces correspond to each other, so that between them there is a Combescour correspondence, which is a special case of a [[Peterson correspondence|Peterson correspondence]]. |
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| | ====Comments==== | | ====Comments==== |
| − | For a linear family of closed convex parallel surfaces (depending linearly on a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139035.png" />) the Steiner formula holds: The volume of the point set bounded by them is a polynomial of degree 3 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071390/p07139036.png" />. An analogous result holds for arbitrary dimensions. The Steiner formula is a special case of formulas for general polynomials in Minkowski's theory of mixed volumes, and, even more general, in the theory of valuations. | + | For a linear family of closed convex parallel surfaces (depending linearly on a parameter $\epsilon>0$) the Steiner formula holds: The volume of the point set bounded by them is a polynomial of degree 3 in $\epsilon$. An analogous result holds for arbitrary dimensions. The Steiner formula is a special case of formulas for general polynomials in Minkowski's theory of mixed volumes, and, even more general, in the theory of valuations. |
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| | For references see [[Parallel lines|Parallel lines]]. | | For references see [[Parallel lines|Parallel lines]]. |
Revision as of 17:15, 3 August 2014
Diffeomorphic equi-oriented surfaces $F_1$ and $F_2$ having parallel tangent planes at corresponding points and such that the distance $h$ between corresponding points of $F_1$ and $F_2$ is constant and equal to that between the corresponding tangent planes. The position vectors $\mathbf r_1$ and $\mathbf r_2$ of two parallel surfaces $F_1$ and $F_2$ are connected by a relation $\mathbf r_2-\mathbf r_1=h\mathbf n$, where $\mathbf n$ is a unit normal vector that is the same for $F_1$ at $r_1$ and $F_2$ at $r_2$.
Thus, one can define a one-parameter family $F_h$ of surfaces parallel to a given $F=F_0$, where $F_h$ is regular for sufficiently small values of $h$ for which
$$w(h)=1-2Hh+Kh^2>0.$$
To the values of the roots $h_1$ and $h_2$ of the equation $w(h)=0$ there correspond two surfaces $F_{h_1}$ and $F_{h_2}$ that are evolutes of $$, so that parallel surfaces have a common evolute (cf. [[Evolute (surface)|Evolute (surface)]]). The [[Mean curvature|mean curvature]] $H_h$ and the [[Gaussian curvature|Gaussian curvature]] $K_h$ of a surface $F_h$ parallel to $F$ are connected with the corresponding quantities $H$ and $K$ of $F$ by the relations
$$H_h=\frac{H-Kh}{w}(h),\quad K_h=\frac Kw(h);$$
lines of curvature of parallel surfaces correspond to each other, so that between them there is a Combescour correspondence, which is a special case of a Peterson correspondence.
For a linear family of closed convex parallel surfaces (depending linearly on a parameter $\epsilon>0$) the Steiner formula holds: The volume of the point set bounded by them is a polynomial of degree 3 in $\epsilon$. An analogous result holds for arbitrary dimensions. The Steiner formula is a special case of formulas for general polynomials in Minkowski's theory of mixed volumes, and, even more general, in the theory of valuations.
For references see Parallel lines.
How to Cite This Entry:
Parallel surfaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parallel_surfaces&oldid=32709
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article