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 and a vector orthogonal to the tangent vector of the curve at each of its points. Suppose that is given in the space by an equation , where is the natural parameter of the curve and is the position vector of the points of the curve. Along one has a vector-function , where is a unit vector orthogonal to the tangent vector at the corresponding points of the curve. One then says that a surface strip with normal is defined along . The vector is called the geodesic normal vector of ; together with and , the vector forms the Frénet frame for the strip. Given the moving Frénet frame for a strip, one has the Frénet derivation formulas:

where denotes the geodesic curvature of the strip, denotes its normal curvature and denotes its geodesic torsion, which are scalar functions of .

If the vector is collinear with the principal normal at each point of , then and the strip is then called a geodesic strip. If is collinear with the binormal of the curve at each point, one has and the strip is called an asymptotic strip.

References

[1]	W. Blaschke, "Einführung in die Differentialgeometrie" , Springer (1950)