= Cones generated by rotation and dilation characteristics
{title2=$\dot r=r-k\times r$}
For a fixed real unit <vector> $k$, the linear <ordinary differential equation> $\dot r=r-k\times r$ combines <dilation> with <rotation> about the axis $\mathbb Rk$. Its solution from a line $s\ell$, with $|\ell|=1$, is
$$
r(s,t)=se^t\bigl(\kappa k+\cos t\,v-\sin t\,(k\times v)\bigr),\qquad\kappa=k\cdot\ell,\quad v=\ell-\kappa k.
$$
Since $k\cdot r=se^t\kappa$ and $|r|=|s|e^t$, the union of these <characteristic curves> is given by $|k\cdot r|=|\kappa||r|$. For $0<|\kappa|<1$ it is a circular <cone> of semi-angle $\arccos|\kappa|$: varying $t$ supplies every azimuth and varying $s$ supplies every radius on both halves. For $\kappa=0$ it is a <plane>; for $|\kappa|=1$ it is only the axis. In the last case the line is itself a <characteristic curve>, so the <initial conditions>[initial data] do not generate a two-dimensional <surface>. The cone is a <smooth surface> away from its vertex, and it is a <graph of a function> only on suitable patches. Invariance of a graph patch under <characteristic curves> holds only while the curve stays in its domain; global flow invariance requires continuation of that patch.
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