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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.