= Domain determined by a finite segment under a radial characteristic flow
{title2=$\dot x=x-a,\ \dot y=y,\ x(0)=b<a,\ 0\leq y(0)\leq c\implies x<a,\ 0\leq y\leq c(a-x)/(a-b)$}
For a first-order <partial differential equation> with the displayed projected <characteristic> field, label the initial segment by $r=y(0)\in[0,c]$. The <characteristic curves> are $x=a-(a-b)e^s$, $y=re^s$. They are the branches of straight lines through $(a,0)$ on the side $x<a$. The inverse coordinates are
$$
e^s=\frac{a-x}{a-b},\qquad r=\frac{(a-b)y}{a-x}.
$$
They are unique exactly in the displayed sector. The <Jacobian determinant> is $-(a-b)e^{2s}\ne0$, so the initial segment is noncharacteristic and its transported data determine a unique classical solution on the sector's interior whenever the characteristic equation for the dependent variable has a unique solution. The bounding rays inherit data from the two endpoints. The point $(a,0)$ is approached only at infinite parameter time, and characteristics on the opposite side cannot reach the initial segment.
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