In mathematics (differential geometry) by a ribbon (or strip)
(
X
,
U
)
is meant a smooth space curve
X
given by a three-dimensional vector
X
(
s
)
, depending continuously on the curve arc-length
s
(
a
≤
s
≤
b
), together with a smoothly varying unit vector
U
(
s
)
perpendicular to
X
at each point (Blaschke 1950).
The ribbon
(
X
,
U
)
is called simple and closed if
X
is simple (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
+
ε
U
given parametrically by
X
(
s
)
+
ε
U
(
s
)
are, for all sufficiently small positive
ε
, simple closed curves disjoint from
X
.
The ribbon concept plays an important role in the Cǎlugǎreǎnu-White-Fuller formula (Fuller 1971), that states that
L
k
=
W
r
+
T
w
,
where
L
k
is the asymptotic (Gauss) linking number (a topological quantity),
W
r
denotes the total writhing number (or simply writhe) and
T
w
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.
