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Context. As primary anchors of the distance scale, Cepheid stars play a crucial role in our understanding of the distance scale of the Universe because of their period-luminosity relation. Determining precise and consistent parameters (radius, temperature, color excess, and projection factor) of Cepheid pulsating stars is therefore very important. Aims. With the high-precision parallaxes delivered by the early third Gaia data release (EDR3), we aim to derive various parameters of Cepheid stars in order to calibrate the period-luminosity and period-radius relations and to investigate the relation of period to p -factor. Methods. We applied an implementation of the parallax-of-pulsation method through the algorithm called spectro-photo-interferometry of pulsating stars (SPIPS), which combines all types of available data for a variable star (multiband and multicolor photometry, radial velocity, effective temperature, and interferometry measurements) in a global modeling of its pulsation. Results. We present the SPIPS modeling of a sample of 63 Galactic Cepheids. Adopting Gaia EDR3 parallaxes as an input associated with the best available dataset, we derive consistent values of parameters for these stars such as the radius, multiband apparent magnitudes, effective temperatures, color excesses, period changes, Fourier parameters, and the projection factor. Conclusions. Using the best set of data and the most precise distances for Milky Way Cepheids, we derive new calibrations of the period-luminosity and period-radius relations: M K S = −5.529 ±0.015   −  3.141 ±0.050 (log P   −  0.9) and log R = 1.763 ±0.003   +  0.653 ±0.012 (log P   −  0.9). After investigating the dependences of the projection factor on the parameters of the stars, we find a high dispersion of its values and no evidence of its correlation with the period or with any other parameters such as radial velocity, temperature, or metallicity. Statistically, the p -factor has an average value of p  = 1.26 ± 0.07, but with an unsatisfactory agreement ( σ  = 0.15). In absence of any clear correlation between the p -factor and other quantities, the best agreement is obtained under the assumption that the p -factor can take any value in a band with a width of 0.15. This result highlights the need for a further examination of the physics behind the p -factor. 

Context. The surface brightness – color relationship (SBCR) is a poweful tool for determining the angular diameter of stars from photometry. It was for instance used to derive the distance of eclipsing binaries in the Large Magellanic Cloud (LMC), which led to its distance determination with an accuracy of 1%. Aims. We calibrate the SBCR for red giant stars in the 2.1 ≤ V − K ≤ 2.5 color range using homogeneous VEGA/CHARA interferometric data secured in the visible domain, and compare it to the relation based on infrared interferometric observations, which were used to derive the distance to the LMC. Methods. Observations of eight G–K giants were obtained with the VEGA/CHARA instrument. The derived limb-darkened angular diameters were combined with a homogeneous set of infrared magnitudes in order to constrain the SBCR. Results. The average precision we obtain on the limb-darkened angular diameters of the eight stars in our sample is 2.4%. For the four stars in common observed by both VEGA/CHARA and PIONIER/VLTI, we find a 1 σ agreement for the angular diameters. The SBCR we obtain in the visible has a dispersion of 0.04 magnitude and is consistent with the one derived in the infrared (0.018 magnitude). Conclusions. The consistency of the infrared and visible angular diameters and SBCR reinforces the result of 1% precision and accuracy recently achieved on the distance of the LMC using the eclipsing-binary technique. It also indicates that it is possible to combine interferometric observations at different wavelengths when the SBCR is calibrated.