  Fast rotation is responsible for important changes in the structure and
evolution of stars. Optical long baseline interferometry now permits the study
of its effects on the stellar surface, mainly gravity darkening and flattening.
We aim to determine the fundamental parameters of the fast-rotating star
Altair, in particular its evolutionary stage, mass, and differential rotation,
using state-of-the-art stellar interior and atmosphere models together with
interferometric, spectroscopic, and asteroseismic observations. We use ESTER 2D
stellar models to produce the relevant surface parameters needed to create
intensity maps from atmosphere models. Interferometric and spectroscopic
observables are computed from these intensity maps and several stellar
parameters are then adjusted using the MCMC algorithm Emcee. We determined
Altair's equatorial radius to be 2.008 +/- 0.006 Rsun, the position angle 301.1
+/- 0.3 degrees, the inclination 50.7 +/- 1.2 degrees, and the equatorial
angular velocity 0.74 +/- 0.01 times the Keplerian angular velocity. This
angular velocity leads to a flattening of 0.220 +/- 0.003. We also deduce from
the spectroscopically derived vsini ~ 243 km/s, a true equatorial velocity of
~314 km/s corresponding to a rotation period of 7h46m (~3 c/d). The data also
impose a strong correlation between mass, metallicity, hydrogen abundance, and
core evolution. Thanks to asteroseismic data, we constrain the mass of Altair
to 1.86 +/- 0.03 Msun and further deduce its metallicity Z = 0.019 and its core
hydrogen mass fraction Xc = 0.71, assuming an initial solar hydrogen mass
fraction X = 0.739. These values suggest that Altair is ~100 Myrs old. Finally,
the 2D ESTER model also gives the internal differential rotation of Altair,
showing that its core rotates approximately 50% faster than the envelope, while
the surface differential rotation does not exceed 6%.
