  Transmission spectroscopy is one of the premier methods used to probe the
temperature, composition, and cloud properties of exoplanet atmospheres. Recent
studies have demonstrated that the multidimensional nature of exoplanet
atmospheres -- due to non-uniformities across the day-night transition and
between the morning and evening terminators -- can strongly influence
transmission spectra. However, the computational demands of 3D radiative
transfer techniques have precluded their usage within atmospheric retrievals.
Here we introduce TRIDENT, a new 3D radiative transfer model which rapidly
computes transmission spectra of exoplanet atmospheres with day-night,
morning-evening, and vertical variations in temperature, chemical abundances,
and cloud properties. We also derive a general equation for transmission
spectra, accounting for 3D atmospheres, refraction, multiple scattering,
ingress/egress, grazing transits, stellar heterogeneities, and nightside
thermal emission. After introducing TRIDENT's linear algebra-based approach to
3D radiative transfer, we propose new parametric prescriptions for 3D
temperature and abundance profiles and 3D clouds. We show that multidimensional
transmission spectra exhibit two significant observational signatures: (i)
day-night composition gradients alter the relative amplitudes of absorption
features; and (ii) morning-evening composition gradients distort the
peak-to-wing contrast of absorption features. Finally, we demonstrate that
these signatures of multidimensional atmospheres incur residuals > 100 ppm
compared to 1D models, rendering them potentially detectable with JWST.
TRIDENT's rapid radiative transfer, coupled with parametric multidimensional
atmospheres, unlocks the final barrier to 3D atmospheric retrievals.
