  I study the location of the $\gamma$-ray emission in blazar jets by creating
a Compton-scattering approximation valid for all anisotropic radiation fields
in the Thomson through Klein-Nishina regimes, which is highly accurate and can
speed up numerical calculations by up to a factor $\sim10$. I apply this
approximation to synchrotron self-Compton, and external Compton-scattering of
photons from the accretion disk, broad-line region (BLR), and dust torus. I use
a stratified BLR model and include detailed Compton-scattering calculations of
a spherical and flattened BLR. I create two dust torus models, one where the
torus is an annulus, and one where it is an extended disk. I present detailed
calculations of the photoabsorption optical depth using my detailed BLR and
dust torus models, including the full angle dependence. I apply these
calculations to the emission from a relativistically moving blob traveling
through these radiation fields. The ratio of $\gamma$-ray to optical flux
produces a predictable pattern that could help locate the $\gamma$-ray emission
region. I show that the bright flare from 3C 454.3 in 2010 November detected by
the Fermi Large Area Telescope is unlikely to originate from a single blob
inside the BLR since it moves outside the BLR in a time shorter than the flare
duration, although emission by multiple blobs inside the BLR is possible; and
$\gamma$-rays are unlikely to originate from outside the BLR from scattering of
photons from an extended dust torus, since then the cooling timescale would be
too long to explain the observed short variability.
