Editing Helix (section)

===Arc length, curvature and torsion{{anchor|Arc length|Curvature|Torsion}}===

A circular helix of radius <math>a>0</math> and slope {{math|{{sfrac|''a''|''b''}}}} (or pitch {{math|2''πb''}}) expressed in Cartesian coordinates as the [[parametric equation]]
:<math>t\mapsto (a\cos t, a\sin t, bt), t\in [0,T]</math>
has an [[arc length]] of
:<math>A = T\cdot \sqrt{a^2+b^2},</math>
a [[Curvature#One dimension in three dimensions: Curvature of space curves|curvature]] of
:<math>\frac{a}{a^2+b^2},</math>
and a [[Torsion of a curve|torsion]] of
:<math>\frac{b}{a^2+b^2}.</math>

A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

<math display="block">\begin{align}
\mathbf{r}&=a\cos t \mathbf{i}+a\sin t \mathbf{j}+ b t\mathbf{k}\\[6px]

\mathbf{v}&=-a\sin t \mathbf{i}+a\cos t \mathbf{j}+ b \mathbf{k}\\[6px]

\mathbf{a}&=-a\cos t \mathbf{i}-a\sin t \mathbf{j}+ 0\mathbf{k}\\[6px]

|\mathbf{v}|&=\sqrt{(-a\sin t )^2 +(a\cos t)^2 + b^2}=\sqrt{a^2 +b^2}\\[6px]

|\mathbf{a}| &= \sqrt{(-a\sin t )^2 +(a\cos t)^2 } = a\\[6px]

s(t) &= \int_{0}^{t}\sqrt{a^2 +b^2}d\tau = \sqrt{a^2 +b^2} t
\end{align}</math>

So a helix can be reparameterized as a function of {{mvar|s}}, which must be unit-speed:

<math display="block">\mathbf{r}(s) = a\cos \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i}+a\sin \frac{s}{\sqrt{a^2 +b^2}} \mathbf{j}+ \frac{bs}{\sqrt{a^2 +b^2}} \mathbf{k}</math>

The unit tangent vector is

<math display="block">\frac{d \mathbf{r}}{d s} = \mathbf{T} = \frac{-a}{\sqrt{a^2 +b^2} }\sin \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i}+\frac{a}{\sqrt{a^2 +b^2} }\cos \frac{s}{\sqrt{a^2 +b^2} }\mathbf{j}+ \frac{b}{\sqrt{a^2 +b^2}} \mathbf{k}</math>

The normal vector is

<math display="block">\frac{d \mathbf{T}}{d s} = \kappa \mathbf{N} = \frac{-a}{a^2 +b^2 }\cos \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i}+\frac{-a}{a^2 +b^2} \sin \frac{s}{\sqrt{a^2 +b^2} }\mathbf{j}+ 0 \mathbf{k}</math>

Its curvature is

<math display="block">\kappa = \left|\frac{d\mathbf{T}}{ds}\right|= \frac{a}{a^2 +b^2 }</math>.

The unit normal vector is

<math display="block">\mathbf{N}=-\cos \frac{s}{\sqrt{a^2 +b^2} } \mathbf{i} - \sin \frac{s}{\sqrt{a^2 +b^2} } \mathbf{j} + 0 \mathbf{k}</math>

The binormal vector is

<math display="block"> \begin{align}
\mathbf{B}=\mathbf{T}\times\mathbf{N} &= \frac{1}{\sqrt{a^2 +b^2 }} \left( b\sin \frac{s}{\sqrt{a^2 +b^2}}\mathbf{i} - b\cos \frac{s}{\sqrt{a^2 +b^2}}\mathbf{j}+ a \mathbf{k}\right)\\[12px]
\frac{d\mathbf{B}}{ds} &= \frac{1}{a^2 +b^2} \left( b\cos \frac{s}{\sqrt{a^2 +b^2}} \mathbf{i} + b\sin \frac{s}{\sqrt{a^2 +b^2}}\mathbf{j}+ 0 \mathbf{k} \right)
\end{align}</math>

Its torsion is
<math display="block">\tau = \left| \frac{d\mathbf{B}}{ds} \right| = \frac{b}{a^2 +b^2}.</math>