Ribbon (mathematics)
# 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.

