We analyse interferometric data obtained for Regulus with AMBER (Astronomical Multi- BEam combineR) at high spectral resolution ($lambda/deltalambda approx 12000$) across the Br$gamma$ spectral line. The study of the photocentre displacement allows us to constrain a large number of stellar parameters -- equatorial radius $R_{rm eq}$, equatorial velocity $V_{rm eq}$, inclination $i$, rotation-axis position angle $PA_{rm rot}$, and flattening -- with an estimation of gravity-darkening coefficient $beta$ using previously published theoretical results. We use the Simulation Code of Interferometric-observations for ROtators and CirCumstellar Objects (SCIROCCO), a semi-analytical algorithm dedicated to fast rotators. We chose Regulus because it is a very well-known edge-on star, for which an alternative approach is needed to check the previously published results. Our analysis showed that a significant degeneracy of solution is present. By confronting the results obtained by differential interferometry with those obtained by conventional long-base interferometry, we obtain similar results (within the uncertainties), thereby validating our approach, where $V_{eq}$ and $i$ are found separately. From the photocentre displacement, we can independently deduce $PA_{rot}$. We use two minimization methods to restrict observed stellar parameters via a fast rotator model: a non-stochastic method ($chi^2$ fit) and a stochastic one (Markov Chain Monte Carlo method), in order to check whether the correct global minimum is achieved particularly with respect to the degeneracies of the gravity darkening parameter $beta$, where we demonstrate, using a quantitative analysis of parameters, that the estimate of $beta$ is easier for stars with an inclination angle of around $45^circ$.
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