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
)