Strip (generalized)

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)