Learn Frenet Equations in 2-D Which Result in the Cornu Spiral

This Insights article is intended as an introduction to the Frenet-Serret equations by showing an interesting application that results from a case in two-dimensional motion. The two-dimensional result for this case=the Cornu Spiral has some interesting associated mathematics that also arises in solutions of certain diffraction equations in optics.

Frenet-Serret equations:

The Frenet-Serret equations are vector equations that are used to describe the motion of an object w.r.t. path length. The equations use a unit vector $\hat{T}$ tangential to the path,  plus a unit vector $\hat{N}$ perpendicular to the path.  The vector $\hat{N}$ lies in the instantaneous plane of the trajectory consisting of $\hat{T}$ and $\hat{N}$ because $\hat{N}$ is determined by $\frac{d \hat{T}}{ds}=\kappa \hat{N}$ where $\kappa$ is the curvature of the path ($\frac{1}{\kappa}$ is the instantaneous radius of curvature.) A third unit vector $\hat{B}=\hat{T} \times \hat{N}$ defines the system of coordinates at any instant.  The three Frenet-Serret equations are

1) $\frac{d \hat{T}}{ds}=\kappa \hat{N}$ where $\kappa$ is a proportionality constant called the curvature.

2) $\frac{d \hat{N}}{ds}=-\kappa \hat{T}+\tau \hat{B}$

3) $\frac{d \hat{B}}{ds}=-\tau \hat{N}$ where $\tau$ is a proportionality constant called the torsion.

Two-dimensional application of Frenet-Serret equations:

The equations become somewhat simplified for motion in the x-y plane,  where $\hat{B}$ then becomes $\hat{z}$. The acceleration vector can be computed in two dimensions and the result is components tangential and perpendicular to the path.   It was recognized that they could be useful in solving the problem of what occurs in two dimensions when a force is applied normal to the path that increases linearly with time (resulting in an acceleration normal to the path that increases linearly with time.  Note: When there is a constant acceleration normal to the path,  the result is circular motion). This two-dimensional application should also prove useful in understanding the Frenet-Serret equations in three dimensions.  Two questions arise with such an acceleration that increases linearly with time:  Is there any increase in speed and what is the shape of the path?

Computing the components of the acceleration:

We can begin with $\vec{v}=(\frac{ds}{dt}) \hat{T}$ and need to find $\vec{a}=\frac{d \vec{v}}{dt}$.

Using the chain rule, we can write the acceleration as $\vec{a}=(\frac{d^2 s}{dt^2}) \hat{T}+(\frac{ds}{dt}) \frac{d \hat{T}}{dt}$.

It is useful to write $\hat{T}=cos(\phi) \hat{i} +sin(\phi) \hat{j}$.

Then $\frac{d \hat{T}}{dt}=(\frac{d \hat{T}}{d \phi})( \frac{d \phi}{ds})( \frac{ds}{dt})=\hat{N} \kappa (\frac{ds}{dt})$ where $\kappa=\frac{ d \phi}{ds}=\frac{1}{r}$ and where $r$ is the instantaneous radius of curvature. Here we have used the first Frenet-Serret equation,  which follows very simply from a little vector calculus.  This gives the result that the acceleration

$\vec{a}=(\frac{d^2 s}{dt^2})\hat{T}+\kappa (\frac{ds}{dt})^2 \hat{N}$.

Applying the result to the case where the acceleration perpendicular to the path increases linearly with time:

Upon inspection of this acceleration equation,  if there is no component of $\vec{a}$ tangent to the path,  then $\frac{d^2 s}{dt^2}=0$.  This means that the speed $\frac{ds}{dt}$ must be constant.  We will call this speed  $v_o$. Now let the normal component of the acceleration increase linearly with time so that $\vec{a}=(\alpha t )\hat{N}$ where $\alpha$ is a constant. We have $\alpha t=(\frac{d \phi}{ds}) v_o^2$,  and we can write $t=\frac{s}{v_o}$,  so that,  upon integrating, $\phi=\frac{1}{2} s^2 \frac{\alpha}{v_o^3}$,  which is of the form  $\phi=\beta s^2$.  $\\$ Now the path of the object can be determined by finding $\vec{r}=\int \vec{v} \, dt =\int \hat{T} ds=\int (cos(\beta s^2) \hat{i}+sin(\beta s^2) \hat{j}) \, ds$.  This last integral is one that appears in diffraction theory and has been tabulated and is known as the Fresnel integrals.   The result is also often given in a graphical form which is known as the Cornu Spiral.(see figure 1)  For the problem at hand,  the path of the object will be some scaled version of the Cornu Spiral,  depending upon the value for $\beta$,  etc.  For the Cornu Spiral shown in the figure,   $\beta=\frac{\pi}{4}$.  $\\$

                                                                                                                              Figure 1.  The Cornu Spiral with $\beta=\frac{\pi}{4}$.

Summary:

An application of the Frenet-Serret equations in two dimensions shows that when a (centripetal) force that increases linearly with time is applied perpendicular to the path of an object,  the path will be that of a Cornu Spiral (or some scaled version) and the speed of the object will remain constant.

Read my next article: An Integral Result from Parseval’s Theorem