I've seen a few questions & answers here for special cases on finding the parametric equations for a given curvature. E.g; Find the parametric equation for a curve with given curvature. However I'm afraid I don't understand the general process. Could someone guide me through the process?
I care about parametric equations of the form
$$\gamma(s)=(x(s),y(s))$$
Hence having signed curvature
$$\kappa=\frac{x'y''-y'x''}{(x'^2+y'^2)^\frac{3}{2}}$$
My question is
Given the equation for $\kappa(s)$, how do you find the family of solutions for $\gamma(s)$?
I assume there is a unique curve which satisfies $\kappa(s)$, though the final solution will have three constants, $x_0$, $y_0$, and $\theta$, which will encode an arbitrary translation and rotation (or some equivalents) of such curve, as, intuitively, curvature does not care about translation or rotation of the entire curve.
As a final note, I'm simply an overoptimistic undergrad, and as such I've only academically dealt with first-order differential equations and have only self-taught curvature. Regardless, I do conceptually understand each. As such, I'd appreciate an answer roughly on my level of understanding.
Best Answer
Not only is there an arbitrary rotation and translation, but also a reflection and parametrisation of the curve. So, first of all, take the standard arclength parametrisation in which the definition of the curvature becomes $$\mathbf{t}'(s)=\kappa(s)\mathbf{n}(s)$$ where $\mathbf{t}(s)=(x'(s),y'(s))$ is the tangent vector and $\mathbf{n}(s)=(-y'(s),x'(s))$ is 'the' normal vector. The latter is only defined up to a sign, so one has to choose one of them arbitrarily. This fixes the handedness of the curve, i.e. the reflection.
Hence the differential equation to solve is $$\begin{pmatrix}x''(s)\cr y''(s)\end{pmatrix}=\kappa(s)\begin{pmatrix}-y'(s)\cr x'(s))\end{pmatrix}$$ As a second order equation, this ought to give four constants of integration, but there is the arclength constraint $(x')^2+(y')^2=1$, so in fact only three constants remain: two for translations, and one for rotation.