  We report high-resolution angular magnetoresistance (AMR) experiments
performed on crystals of ReO$_3$ with high mobility (90,000 cm$^2$/Vs at 2 K)
and extremely low residual resistivity (5-8 n$\Omega$cm). The Fermi surface,
comprised of intersecting cylinders, supports open orbits. The resistivity
$\rho_{xx}$ in a magnetic field $B$ = 9 T displays a singular pattern of
behavior. With $\bf E\parallel \hat{x}$ and $\bf B$ initially
$\parallel\bf\hat{z}$, tilting $\bf B$ in the longitudinal $k_z$-$k_x$ plane
leads to a steep decrease in $\rho_{xx}$ by a factor of 40. However, if $\bf B$
is tilted in the transverse $k_y$-$k_z$ plane, $\rho_{xx}$ increases steeply by
a factor of 8. Using the Shockley tube integral approach, we show that, in
ReO$_3$, the singular behavior results from the rapid conversion of closed to
open orbits, resulting in opposite signs for AMR in orthogonal planes. The
floor values of $\rho_{xx}$ in both AMR scans are identified with specific sets
of open and closed orbits. Also, the "completion angle" $\gamma_c$ detected in
the AMR is shown to be an intrinsic geometric feature that provides a new way
to measure the Fermi radius $k_F$. However, additional sharp resonant features
which appear at very small tilt angles in the longitudinal AMR scans are not
explained by the tube integral approach.
