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Accurate measurements of the physical structure of protoplanetary discs arecritical inputs for planet formation models. These constraints aretraditionally established via complex modelling of continuum and lineobservations. Instead, we present an empirical framework to locate the COisotopologue emitting surfaces from high spectral and spatial resolution ALMAobservations. We apply this framework to the disc surrounding IM Lupi, where wereport the first direct, i.e. model independent, measurements of the radial andvertical gradients of temperature and velocity in a protoplanetary disc. Themeasured disc structure is consistent with an irradiated self-similar discstructure, where the temperature increases and the velocity decreases towardsthe disc surface. We also directly map the vertical CO snow line, which islocated at about one gas scale height at radii between 150 and 300 au, with aCO freeze-out temperature of $21\pm2$ K. In the outer disc ($> 300$ au), wherethe gas surface density transitions from a power law to an exponential taper,the velocity rotation field becomes significantly sub-Keplerian, in agreementwith the expected steeper pressure gradient. The sub-Keplerian velocitiesshould result in a very efficient inward migration of large dust grains,explaining the lack of millimetre continuum emission outside of 300 au. Thesub-Keplerian motions may also be the signature of the base of an externallyirradiated photo-evaporative wind. In the same outer region, the measured COtemperature above the snow line decreases to $\approx$ 15 K because of thereduced gas density, which can result in a lower CO freeze-out temperature,photo-desorption, or deviations from local thermodynamic equilibrium.

Direct mapping of the temperature and velocity gradients in discs. Imaging the vertical CO snow line around IM Lupi