The action of $SE(2) = SO(2) \ltimes R^2$ on smooth curves in the plane is definition $(R_{\theta},(a,b)) \cdot (x,u(x))=R_{\theta} \cdot (x,u(x))+(a,b)$
I have already shown that $\frac{u_{xx}}{(1+u_x^2)^{\frac{3}{2}}}$ is a diferential invariant for this action, which is known as the Euclidean curvature.
I want to obtain a diferential invariant which contains $u_{xxx}$.
Any suggestions?
Many thanks.
For a constant slope $ I_1=\phi$ of curve, $ \cos \phi = \frac{dx}{ds}, $ the Euclidean curvature $ k_g= \dfrac{d \phi }{ds}=0 $, as arc derivative of Euclidean rotation.
Defines a straight line when $I_1$ is constant.
The invariant $I_2$ is $$ \frac{d\phi }{ds}= \dfrac{ u_{xx } }{(1+ u_x^{2})^{3/2} },$$
Defines a Circle when $I_2$ is constant.
Next arc derivative also is an isometric differential invariant expressible from first fundamental form coefficients.
The next invariant $I_3$ is $$ \frac{d^2\phi }{ds^2} $$
when differentiation performed
$$ \dfrac{ u_{xxx} {(1+ u_x^{2}) -3 u_x u_{xx}^2 }}{(1+ u_x^{2})^{7/2} }, $$
and so on..
Defines Cornu's spiral when $I_3$ is constant.