In
mathematics, a helix is a
curve in 3-
dimensional space. The following
parametrisation in
Cartesian coordinates defines a particular helix; perhaps the simplest equations for one is : \begin{align} x(t) &= \cos(t),\\ y(t) &= \sin(t),\\ z(t) &= t. \end{align} As the
parameter increases, the point (x(t), y(t), z(t)) traces a right-handed helix of pitch (or slope 1) and radius 1 about the -axis, in a right-handed coordinate system. In
cylindrical coordinates , the same helix is parametrised by: : \begin{align} r(t) &= 1,\\ \theta(t) &= t,\\ h(t) &= t. \end{align} A circular helix of radius and slope (or pitch ) is described by the following parametrisation: : \begin{align} x(t) &= a\cos(t),\\ y(t) &= a\sin(t),\\ z(t) &= bt. \end{align} Another way of mathematically constructing a helix is to plot the complex-valued function as a function of the real number (see
Euler's formula). The value of and the real and imaginary parts of the function value give this plot three real dimensions. Except for
rotations,
translations, and changes of scale, all right-handed helices are equivalent to the helix defined above. The equivalent left-handed helix can be constructed in a number of ways, the simplest being to negate any one of the , or components.
Arc length, curvature and torsion A circular helix of radius a>0 and slope (or pitch ) expressed in Cartesian coordinates as the
parametric equation :t\mapsto (a\cos t, a\sin t, bt), t\in [0,T] has an
arc length of :A = T\cdot \sqrt{a^2+b^2}, a
curvature of :\frac{a}{a^2+b^2}, and a
torsion of :\frac{b}{a^2+b^2}. Arc length per revolution ( T=2 \pi ): A = \sqrt{(2\pi a)^2+(2\pi b)^2} or A = \sqrt{(2\pi a)^2+p^2} where p = pitch. Twisted length per unit straight length (axial length): A = \frac{ \sqrt{(2\pi a)^2+p^2}}{p} A helix has constant non-zero curvature and torsion. A helix is the vector-valued function \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] s(t) &= \int_{0}^{t}\sqrt{a^2 +b^2}d\tau = \sqrt{a^2 +b^2} t \end{align} So a helix can be reparameterized as a function of , which must be unit-speed: \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} The unit tangent vector is \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} The normal vector is \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} Its curvature is \kappa = \left|\frac{d\mathbf{T}}{ds}\right|= \frac{a}{a^2 +b^2 }. The unit normal vector is \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} The binormal vector is \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} Its torsion is \tau = \left| \frac{d\mathbf{B}}{ds} \right| = \frac{b}{a^2 +b^2}. ==Examples==