Ribbon (mathematics)

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 $$(X,U)$$ includes a curve $$X$$ given by a three-dimensional vector $$X(s)$$, depending continuously on the curve arc-length $$s$$ ($$a\leq s \leq b$$), and a unit vector $$U(s)$$ perpendicular to $$X$$ at each point. Ribbons have seen particular application as regards DNA.

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

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


 * $$Lk = Wr + Tw ,$$

where $$Lk$$ is the asymptotic (Gauss) linking number, the integer number of turns of the ribbon around its axis; $$Wr$$ denotes the total writhing number (or simply writhe), a measure of non-planarity of the ribbon's axis curve; and $$Tw$$ 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.