$c_1,c_2,c_3$ are referred to as the [*characters*]{} of the world-line.]{}

\[remarkuniq\][Conversely, if $k_1,k_2,k_3:I\to \R$ are smooth functions satisfying (\[eqmrk4\]), by solving the linear system (\[mcc\]) with initial condition $\mathrm{B}(u_0)\in \mathrm{A}^{\uparrow}_+(2,4)$, we get a smooth map $\mathrm{B}:I\to \mathrm{A}^{\uparrow}_+(2,4)$. Then, $\gamma=[B_0]$ is a linearly full conformal world-line with conformal curvatures $k_1,k_2,k_3$ and canonical conformal frame $B$. Any other world-line $\widetilde{\gamma}$ with the same curvatures is congruent to $\gamma$ with respect to the restricted conformal group, ie there exist $\mathbf{X}\in \mathrm{A}^{\uparrow}_+(2,4)$ such that $\widetilde{\gamma}=\mathbf{X}\cdot \gamma$.]{}

[The sign ambiguity of the third curvature can be removed with the following reasoning : if the third conformal curvature of $\gamma$ is $k_3$ and if $\mathbf{X}$ is an orientation-preserving and time-reversing conformal transformation, then $\mathbf{X}\cdot \gamma(-u)$ is a conformal world-line with conformal curvatures $k_1(-u),k_2(-u)$ and $-k_3(-u)$. Therefore, up to a time-reversing conformal transformation, the curvature $k_3$ is positive.]{}

[The sign of the third curvature is said the [*conformal helicity*]{} of the world-line. By the previous remark we see that is not restrictive to consider conformal world-lines with positive helicity. From now on this additional assumption is implicitly assumed.]{}

[If $k_1,k_2$ and $k_3$ are constant, then the variational equations imply $k_1=(k_2^2+k_3^2)/2$. In this case $\mathcal{K}$ is a fixed 