Find all flat curves with constant curvature

Question

I suspect that the answer is "straight line and circle". I understand how to show that these two objects have constant curvature, but no idea how to show that there are no other curves with constant curvature.

Answer 1

Just a thought. For a curve expressible as $y = f(x)$, the curvature $\kappa$ appears in the following ODE, $$ \vert y'' \vert = \kappa (1 + y'^2)^{\frac{3}{2}}$$ Locally any curve can be expressed in the form $y = f(x)$, by suitable axes transformations.

So if $\kappa = 0$ then the second derivative is zero, so locally we have a straight line

If $\kappa \neq 0$, you end up with an ODE $$ w' = f(w, x)$$ with $f$ continuous locally but not globally Lipschitz continuous) which at least locally i believe has a unique solution (Picard-Lindehof Theorem).

As we know an arc of circumference will satisfy the given ODE, the arc and the solution must coincide.

Some care must be taken with the absolute sign and "signed" curvature, or otherwise.

One could consider an arc of a circumference, take a point on it and dividing it in two portions, and reflect the one portion with respect to the tangent line. Then there would be infinite lines of constant curvature.