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Ribbon (mathematics)

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In differential geometry, a ribbon (or strip) is the combination of a smooth space curve and its corresponding normal vector. More formally, a ribbon denoted by ( X , U ) {\displaystyle (X,U)} includes a curve X {\displaystyle X} given by a three-dimensional vector X ( s ) {\displaystyle X(s)} , depending continuously on the curve arc-length s {\displaystyle s} ( a ≤ s ≤ b {\displaystyle a\leq s\leq b} ), and a unit vector U ( s ) {\displaystyle U(s)} perpendicular to ∂ X ∂ s ( s ) {\displaystyle {\partial X \over \partial s}(s)} at each point. Ribbons have seen particular application as regards DNA.

Properties and implications

The ribbon ( X , U ) {\displaystyle (X,U)} is called simple if X {\displaystyle X} is a simple curve (i.e. without self-intersections) and closed and if U {\displaystyle U} and all its derivatives agree at a {\displaystyle a} and b {\displaystyle b} . For any simple closed ribbon the curves X + ε U {\displaystyle X+\varepsilon U} given parametrically by X ( s ) + ε U ( s ) {\displaystyle X(s)+\varepsilon U(s)} are, for all sufficiently small positive ε {\displaystyle \varepsilon } , simple closed curves disjoint from X {\displaystyle X} .

The ribbon concept plays an important role in the Călugăreanu formula, that states that

L k = W r + T w , {\displaystyle Lk=Wr+Tw,}

where L k {\displaystyle Lk} is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis; W r {\displaystyle Wr} denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and T w {\displaystyle Tw} is the total twist number (or simply twist), the rate of rotation of the ribbon around its axis.

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.

See also

References

  1. Blaschke, W. (1950) Einführung in die Differentialgeometrie. Springer-Verlag. ISBN 9783817115495
  2. Vologodskiǐ, Aleksandr Vadimovich (1992). Topology and Physics of Circular DNA (First ed.). Boca Raton, FL. p. 49. ISBN 978-1138105058. OCLC 1014356603.{{cite book}}: CS1 maint: location missing publisher (link)
  3. Fuller, F. Brock (1971). "The writhing number of a space curve" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522.
  4. Dennis, M. R.; Hannay, J. H. (26 August 2005). "Geometry of Călugăreanu's theorem". Proceedings of the Royal Society A. 461 (2062): 3245–3254.

Bibliography

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