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''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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,   "Einführung in die Differentialgeometrie" , Springer  (1950)</TD></TR></table>