Twist (mathematics) | Wikipedia audio article
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SUMMARY
=======
In mathematics (differential geometry) twist is the rate of rotation of a smooth ribbon around the space curve
X
=
X
(
s
)
{\displaystyle X=X(s)}
, where
s
{\displaystyle s}
is the arc length of
X
{\displaystyle X}
and
U
=
U
(
s
)
{\displaystyle U=U(s)}
a unit vector perpendicular at each point to
X
{\displaystyle X}
. Since the ribbon
(
X
,
U
)
{\displaystyle (X,U)}
has edges
X
{\displaystyle X}
and
X
′
=
X
+
ε
U
{\displaystyle X'=X+\varepsilon U}
the twist (or total twist number)
T
w
{\displaystyle Tw}
measures the average winding of the curve
X
′
{\displaystyle X'}
around
and along the curve
X
{\displaystyle X}
. According to Love (1944) twist is defined by
T
w
=
1
2
π
∫
(
d
U
d
s
×
U
)
⋅
d
X
d
s
d
s
,
{\displaystyle Tw={\dfrac {1}{2\pi }}\int \left({\dfrac {dU}{ds}}\times U\right)\cdot {\dfrac {dX}{ds}}ds\;,}
where
d
X
/
d
s
{\displaystyle dX/ds}
is the unit tangent vector to
X
{\displaystyle X}
.
The total twist number
T
w
{\displaystyle Tw}
can be decomposed (Moffatt & Ricca 1992) into normalized total torsion
T
∈
[
0
,
1
)
{\displaystyle T\in [0,1)}
and intrinsic twist
N
∈
Z
{\displaystyle N\in \mathbb {Z} }
as
T
w
=
1
2
π
∫
τ
d
s
+
[
Θ
]
X
2
π
=
T
+
N
,
{\displaystyle Tw={\dfrac {1}{2\pi }}\int \tau \;ds+{\dfrac {\left[\Theta \right]_{X}}{2\pi }}=T+N\;,}
...
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