New Period-Luminosity and Period-Color Relations of Classical Cepheids: I. Cepheids in the Galaxy

321 Galactic fundamental-mode Cepheids with good B,V,I photometry by Berdnikov et al. 2000 and with homogenized color excesses E(B-V) based on Fernie et al. 1995 are used to determine their P-C relation in the range 0.4<log P<1.6. Distances of 25 Cepheids in open clusters (Feast 1999) and of 28 Cepheids with BBW distances (Gieren et al. 1998) are used in a first step to determine the absorption coefficients R_B=4.17, R_V=3.17, and R_I=1.89. The two sets define two independent P-L relations which agree very well in slope. They are therefore combined into a single mean Galactic P-L relation. Galactic Cepheids are redder in (B-V) than those in LMC and SMC as shown by the over 1000 Cloud Cepheids with good standard B,V,I photometry by Udalski et al 1999. Also the (B-V), (V-I) two-color diagrams differ between Cepheids in the Galaxy and the Clouds, attributed both to the effects of metallicity differences, on the spectral energy distributions of the Cepheids and to a shift in the effective temperature of the middle of the instability strip for LMC and SMC relative to the Galaxy by about Delta log Te ~ 0.02 at M_V=-4.0 mag, hotter for both LMC and SMC. Differences in the P-C relations between the Galaxy and the Clouds show that there cannot be a universal P-L relation from galaxy-to-galaxy in any given color. The inferred non-uniqueness of the slope of the P-L relations in the Galaxy, LMC, and SMC is born out by the observations. The Cloud Cepheids follow a shallower overall slope of the P-L relations in B,V,I than the Galactic ones. LMC and SMC Cepheids are brighter in V than in the Galaxy by up to 0.5 mag at short periods and fainter at long periods.