  Exoplanet transmission spectra, which measure the absorption of light passing
through a planet's atmosphere during transit, are most often assessed globally,
resulting in a single spectrum per planetary atmosphere. However, the inherent
three-dimensional nature of planetary atmospheres, via thermal, chemical, and
dynamical processes, can imprint inhomogeneous structure and properties in the
observables. In this work, we devise a technique for spatially mapping the
atmospheres of exoplanets in transmission. Our approach relaxes the assumption
that transit light curves are created from circular stars occulted by circular
planets, and instead we allow for flexibility in the planet's sky-projected
shape. We define the planet's radius to be a single-valued function of angle
around its limb, and we refer to this mathematical object as a transmission
string. These transmission strings are parameterised in terms of Fourier
series, a choice motivated by these series having adjustable complexity,
generating physically practical shapes, while being reducible to the classical
circular case. The utility of our technique is primarily intended for
high-precision multi-wavelength light curves, from which inferences of
transmission spectra can be made as a function of angle around a planet's
terminator, enabling analysis of the multidimensional physics at play in
exoplanet atmospheres. More generally, the technique can be applied to any
transit light curve to derive the shape of the transiting body. The algorithm
we develop is available as an open-source package, called harmonica.
