2

The Euler spiral is a planar curve, whose curvature increases linearly with arc length from the origin. This Wikipedia article features a derivation of its equation based on the fact, that the curvature is in fact the rate of change of the angle of the tangent vector. I was wondering what an alternative derivation would look like, that would start with an unknown parametrization:

$$\binom{x(t)}{y(t)}$$

and based on the formulas for curvature:

$$k(t) = \frac{x'(t)y''(t) - y'(t)x''(t)}{(x'(t)^2+y'(t)^2)^{3/2}}$$

and arc length:

$$L(t) = \int _0 ^t \sqrt{x'(\tau)^2+y'(\tau)^2} \ d\tau$$

would get to something like:

$$const. = \frac{(x'(t)^2+y'(t)^2)^{3/2}}{x'y'' - y'x''} \cdot \int _0 ^t \sqrt{x'(\tau)^2+y'(\tau)^2} \ d\tau$$

Here I am stuck, one should probably simplify this by using a parametrization by the arc length..? And where do the sine and cosine in the resulting parametric equations come from?

Community
  • 1
FliegenderZirkus
  • 123

2 Answers2

2

Here, we derive an equation for the Euler (or Cornu spiral). The natural form of a curve is one that is expressed in term of its arc length, s; it is independent of any coordinate system. The natural form of the Euler spiral is correctly identified here as $\rho s=s/\kappa=\text{constant}$, where $\rho$ is the radius of curvature and $\kappa$ is the curvature. If you are interested in the tangent angle, $\theta$, then you have to know that

$$\theta=\int \kappa(s)ds$$

If you want to express the spiral in a coordinate system (here, I choose complex variables) then you need

$$z=\int e^{i\int \kappa(s)ds}ds =\int e^{i \theta(s)} ds$$

Generally, I use the canonical form for the Euler spiral, to wit,

$$z(u)=\int_0^u e^{i \pi s^2/2} ds$$

where the factor $\pi/2$ puts the terminus of the spiral at $z=(1+i)/2$.

The Euler spiral is frequently expressed in Cartesian cordinates in terms of the Fresnel integrals (see here: http://dlmf.nist.gov/search/search?q=fresnel). However, we have shown (as probably many others have) that the integral can be expressed in closed-form as follows

$$z(u)=\frac{1+i}{2} erf\left(\frac{1-i}{2} \sqrt{\pi} \cdot u\right)$$

This equation will allow you to calculate both negative and positive values of $u$ for a two-sided Euler spiral. Here, you can see that as $u\rightarrow\pm\infty$, $z\rightarrow\pm \frac{1+i}{2}$.

Full disclosure, this answer is very similar the one that I have previously given here: Is this Cornu spiral positively oriented or not?.

Community
  • 1
Cye Waldman
  • 7,524
0

Ok I'm on a phone so bear with me.

The angle of the derivative of a plane curve with respect to X axis is:

Atan (Dx/dy)

Now, the curvature func k is the derivative of angle. we can thus get angle with integral calculus.

now that we have angle, we can form a system of two equations:

1) dx/dy = tan (angle)

2) Dxdx + dydy = 1

EQ 2 constraints tangent to be unit length, producing an arc length parameterization.

Solve for dx, dy and you get trig identifies that reduce to:

Dx = sin (angle) Do = cos (angle)

In case of clothoid, integrating 2s produces s^2 angle func, and we get the fresnel s.

Joe
  • 204
  • 1
    This is a very good question. Fresnels are actually a terrible curve formulation, you get a better one by integrating k = k1 + (k2 - k1)*s, whinch defines curvature as a linear interpolation of two constants. – Joe Mar 05 '15 at 05:49
  • Eh looks like I didn't read the whole question. To answer again, curvature is also the magnitude of the second derivative of a curve parameterized by arc length. – Joe Mar 05 '15 at 05:55
  • Thanks for your help. In the end I found a derivation in a book on differential geometry that I referenced to and gave up on my own efforts;) – FliegenderZirkus Mar 12 '15 at 11:38