Ribbon (mathematics)
In mathematics (differential geometry) by a ribbon (or strip) is meant a smooth space curve
given by a three-dimensional vector
, depending continuously on the curve arc-length
(
), together with a smoothly varying unit vector
perpendicular to
at each point (Blaschke 1950).
The ribbon is called simple and closed if
is simple (i.e. without self-intersections) and closed and if
and all its derivatives agree at
and
.
For any simple closed ribbon the curves
given parametrically by
are, for all sufficiently small positive
, simple closed curves disjoint from
.
The ribbon concept plays an important role in the Cǎlugǎreǎnu-White-Fuller formula (Fuller 1971), that states that
where is the asymptotic (Gauss) linking number (a topological quantity),
denotes the total writhing number (or simply writhe) and
is the total twist number (or simply twist).
Ribbon theory investigates geometric and topological aspects of a mathematical reference ribbon associated with physical and biological properties, such as those arising in topological fluid dynamics, DNA modeling and in material science.
References
- Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
- Fuller, F.B. (1971) The writhing number of a space curve. PNAS USA 68, 815-819.