Helix

Mathematical description

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

${\displaystyle \begin{array}{cc}x(t)& =\cos (t),\\ y(t)& =\sin (t),\\ z(t)& =t.\end{array}}$

As the parameter t increases, the point ${\displaystyle (x(t),y(t),z(t))}$ traces a right-handed helix of pitch 2π (or slope 1) and radius 1 about the z-axis, in a right-handed coordinate system.

In cylindrical coordinates (r, θ, h), the same helix is parametrised by:

${\displaystyle \begin{array}{cc}r(t)& =1,\\ \theta (t)& =t,\\ h(t)& =t.\end{array}}$

A circular helix of radius a and slope ⁠a/b⁠ (or pitch 2πb) is described by the following parametrisation:

${\displaystyle \begin{array}{cc}x(t)& =a\cos (t),\\ y(t)& =a\sin (t),\\ z(t)& =bt.\end{array}}$

Another way of mathematically constructing a helix is to plot the complex-valued function e^xi as a function of the real number x (see Euler's formula). The value of x 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 x, y or z components.

Arc length, curvature and torsion

A circular helix of radius ${\displaystyle a>0}$ and slope ⁠a/b⁠ (or pitch 2πb) expressed in Cartesian coordinates as the parametric equation

${\displaystyle t\mapsto (a\cos t,a\sin t,bt),t\in [0,T]}$

has an arc length of

${\displaystyle A=T\cdot \sqrt{a^2+b^2},}$

a curvature of

${\displaystyle \frac{a}{a^2+b^2},}$

and a torsion of

${\displaystyle \frac{b}{a^2+b^2}.}$

Arc length per revolution (${\displaystyle T=2\pi }$): ${\displaystyle A=\sqrt{(2\pi a{)}^2+(2\pi b{)}^2}}$ or ${\displaystyle A=\sqrt{(2\pi a{)}^2+p^2}}$ where ${\displaystyle p=}$pitch.

Twisted length per unit straight length (axial length): ${\displaystyle A=\frac{\sqrt{(2\pi a{)}^2+p^2}}{p}}$

A helix has constant non-zero curvature and torsion.

A helix is the vector-valued function

$${\displaystyle \begin{array}{cc}\mathbf{r}& =a\cos t\mathbf{i}+a\sin t\mathbf{j}+bt\mathbf{k}\\ \mathbf{v}& =-a\sin t\mathbf{i}+a\cos t\mathbf{j}+b\mathbf{k}\\ \mathbf{a}& =-a\cos t\mathbf{i}-a\sin t\mathbf{j}+0\mathbf{k}\\ |\mathbf{v}|& =\sqrt{(-a\sin t{)}^2+(a\cos t{)}^2+b^2}=\sqrt{a^2+b^2}\\ |\mathbf{a}|& =\sqrt{(-a\sin t{)}^2+(a\cos t{)}^2}=a\\ s(t)& ={\int }_0^t\sqrt{a^2+b^2}d\tau =\sqrt{a^2+b^2}t\end{array}}$$

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

$${\displaystyle \mathbf{r}(s)=a\cos \frac{s}{\sqrt{a^2+b^2}}\mathbf{i}+a\sin \frac{s}{\sqrt{a^2+b^2}}\mathbf{j}+\frac{bs}{\sqrt{a^2+b^2}}\mathbf{k}}$$

The unit tangent vector is

$${\displaystyle \frac{d\mathbf{r}}{ds}=\mathbf{T}=\frac{-a}{\sqrt{a^2+b^2}}\sin \frac{s}{\sqrt{a^2+b^2}}\mathbf{i}+\frac{a}{\sqrt{a^2+b^2}}\cos \frac{s}{\sqrt{a^2+b^2}}\mathbf{j}+\frac{b}{\sqrt{a^2+b^2}}\mathbf{k}}$$

The normal vector is

$${\displaystyle \frac{d\mathbf{T}}{ds}=\kappa \mathbf{N}=\frac{-a}{a^2+b^2}\cos \frac{s}{\sqrt{a^2+b^2}}\mathbf{i}+\frac{-a}{a^2+b^2}\sin \frac{s}{\sqrt{a^2+b^2}}\mathbf{j}+0\mathbf{k}}$$

Its curvature is

$${\displaystyle \kappa =\left|\frac{d\mathbf{T}}{ds}\right|=\frac{a}{a^2+b^2}}$$.

The unit normal vector is

$${\displaystyle \mathbf{N}=-\cos \frac{s}{\sqrt{a^2+b^2}}\mathbf{i}-\sin \frac{s}{\sqrt{a^2+b^2}}\mathbf{j}+0\mathbf{k}}$$

The binormal vector is

$${\displaystyle \begin{array}{cc}\mathbf{B}=\mathbf{T}\times \mathbf{N}& =\frac{1}{\sqrt{a^2+b^2}}\left(b\sin \frac{s}{\sqrt{a^2+b^2}}\mathbf{i}-b\cos \frac{s}{\sqrt{a^2+b^2}}\mathbf{j}+a\mathbf{k}\right)\\ \frac{d\mathbf{B}}{ds}& =\frac{1}{a^2+b^2}\left(b\cos \frac{s}{\sqrt{a^2+b^2}}\mathbf{i}+b\sin \frac{s}{\sqrt{a^2+b^2}}\mathbf{j}+0\mathbf{k}\right)\end{array}}$$

Its torsion is $${\displaystyle \tau =\left|\frac{d\mathbf{B}}{ds}\right|=\frac{b}{a^2+b^2}.}$$

Examples

An example of a double helix in molecular biology is the nucleic acid double helix.

An example of a conic helix is the Corkscrew roller coaster at Cedar Point amusement park.

Some curves found in nature consist of multiple helices of different handedness joined together by transitions known as tendril perversions.

Most hardware screw threads are right-handed helices. The alpha helix in biology as well as the A and B forms of DNA are also right-handed helices. The Z form of DNA is left-handed.

In music, pitch space is often modeled with helices or double helices, most often extending out of a circle such as the circle of fifths, so as to represent octave equivalency.

In aviation, geometric pitch is the distance an element of an airplane propeller would advance in one revolution if it were moving along a helix having an angle equal to that between the chord of the element and a plane perpendicular to the propeller axis; see also: pitch angle (aviation).