Namespaces
Variants
Actions

Strip (generalized)

From Encyclopedia of Mathematics
Revision as of 17:24, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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)
How to Cite This Entry:
Strip (generalized). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Strip_(generalized)&oldid=48872
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article