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''surface strip (in the narrow sense)''
 
''surface strip (in the narrow sense)''
  
A one-parameter family of planes tangent to a surface. In the general sense, a strip is the union of a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904901.png" /> and a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904902.png" /> orthogonal to the tangent vector of the curve at each of its points. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904903.png" /> is given in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904904.png" /> by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904905.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904906.png" /> is the natural parameter of the curve and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904907.png" /> is the position vector of the points of the curve. Along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904908.png" /> one has a vector-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s0904909.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049010.png" /> is a unit vector orthogonal to the tangent vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049011.png" /> at the corresponding points of the curve. One then says that a surface strip <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049012.png" /> with normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049013.png" /> is defined along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049014.png" />. The vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049015.png" /> is called the geodesic normal vector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049016.png" />; together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049018.png" />, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049019.png" /> forms the Frénet frame for the strip. Given the moving Frénet frame for a strip, one has the Frénet derivation formulas:
+
A one-parameter family of planes tangent to a surface. In the general sense, a strip is the union of a curve $  l $
 +
and a vector $  \mathbf m $
 +
orthogonal to the tangent vector of the curve at each of its points. Suppose that $  l $
 +
is given in the space $  \mathbf R  ^ {3} $
 +
by an equation $  \mathbf r = \mathbf r ( s) $,  
 +
where s $
 +
is the natural parameter of the curve and $  \mathbf r ( s) $
 +
is the position vector of the points of the curve. Along $  l $
 +
one has a vector-function $  \mathbf m = \mathbf m ( s) $,  
 +
where $  \mathbf m ( s) $
 +
is a unit vector orthogonal to the tangent vector $  \mathbf t = d \mathbf r / d s $
 +
at the corresponding points of the curve. One then says that a surface strip $  \Phi = \{ l , \mathbf m \} $
 +
with normal $  \mathbf m ( s) $
 +
is defined along $  l $.  
 +
The vector $  \pmb\tau = [ \mathbf m , \mathbf t ] $
 +
is called the geodesic normal vector of $  \Phi $;  
 +
together with $  \mathbf t $
 +
and $  \mathbf m $,  
 +
the vector $  \pmb\tau $
 +
forms the Frénet frame for the strip. Given the moving Frénet frame for a strip, one has the Frénet derivation formulas:
 +
 
 +
$$
 +
 
 +
\frac{d \mathbf t }{ds}
 +
  =  k _ {g} \pmb\tau + k _ {n} \mathbf m ; \
 +
 
 +
\frac{d \pmb\tau }{ds}
 +
  =  - k _ {g} \mathbf t + \kappa _ {g} \mathbf m ;
 +
$$
  
<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/s090/s090490/s09049020.png" /></td> </tr></table>
+
$$
  
<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/s090/s090490/s09049021.png" /></td> </tr></table>
+
\frac{d \mathbf m }{ds}
 +
  = - k _ {n} \mathbf t + \kappa _ {g} \pmb\tau ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049022.png" /> denotes the geodesic curvature of the strip, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049023.png" /> denotes its normal curvature and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049024.png" /> denotes its geodesic torsion, which are scalar functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049025.png" />.
+
where $  k _ {g} $
 +
denotes the geodesic curvature of the strip, $  k _ {n} ( s) $
 +
denotes its normal curvature and $  \kappa _ {g} ( s) $
 +
denotes its geodesic torsion, which are scalar functions of s $.
  
If the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049026.png" /> is collinear with the [[Principal normal|principal normal]] at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049027.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049028.png" /> and the strip is then called a geodesic strip. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049029.png" /> is collinear with the [[Binormal|binormal]] of the curve at each point, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090490/s09049030.png" /> and the strip is called an asymptotic strip.
+
If the vector $  \mathbf m $
 +
is collinear with the [[Principal normal|principal normal]] at each point of $  l $,  
 +
then $  k _ {g} = 0 $
 +
and the strip is then called a geodesic strip. If $  \mathbf m $
 +
is collinear with the [[Binormal|binormal]] of the curve at each point, one has $  k _ {n} = 0 $
 +
and the strip is called an asymptotic strip.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,  "Einführung in die Differentialgeometrie" , Springer  (1950)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,  "Einführung in die Differentialgeometrie" , Springer  (1950)</TD></TR></table>

Latest revision as of 08:23, 6 June 2020


surface strip (in the narrow sense)

A one-parameter family of planes tangent to a surface. In the general sense, a strip is the union of a curve $ l $ and a vector $ \mathbf m $ orthogonal to the tangent vector of the curve at each of its points. Suppose that $ l $ is given in the space $ \mathbf R ^ {3} $ by an equation $ \mathbf r = \mathbf r ( s) $, where $ s $ is the natural parameter of the curve and $ \mathbf r ( s) $ is the position vector of the points of the curve. Along $ l $ one has a vector-function $ \mathbf m = \mathbf m ( s) $, where $ \mathbf m ( s) $ is a unit vector orthogonal to the tangent vector $ \mathbf t = d \mathbf r / d s $ at the corresponding points of the curve. One then says that a surface strip $ \Phi = \{ l , \mathbf m \} $ with normal $ \mathbf m ( s) $ is defined along $ l $. The vector $ \pmb\tau = [ \mathbf m , \mathbf t ] $ is called the geodesic normal vector of $ \Phi $; together with $ \mathbf t $ and $ \mathbf m $, the vector $ \pmb\tau $ forms the Frénet frame for the strip. Given the moving Frénet frame for a strip, one has the Frénet derivation formulas:

$$ \frac{d \mathbf t }{ds} = k _ {g} \pmb\tau + k _ {n} \mathbf m ; \ \frac{d \pmb\tau }{ds} = - k _ {g} \mathbf t + \kappa _ {g} \mathbf m ; $$

$$ \frac{d \mathbf m }{ds} = - k _ {n} \mathbf t + \kappa _ {g} \pmb\tau , $$

where $ k _ {g} $ denotes the geodesic curvature of the strip, $ k _ {n} ( s) $ denotes its normal curvature and $ \kappa _ {g} ( s) $ denotes its geodesic torsion, which are scalar functions of $ s $.

If the vector $ \mathbf m $ is collinear with the principal normal at each point of $ l $, then $ k _ {g} = 0 $ and the strip is then called a geodesic strip. If $ \mathbf m $ is collinear with the binormal of the curve at each point, one has $ k _ {n} = 0 $ and the strip is called an asymptotic strip.

References

[1] W. Blaschke, "Einführung in die Differentialgeometrie" , Springer (1950)
How to Cite This Entry:
Strip (generalized). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strip_(generalized)&oldid=18146
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article