Using scanning tunneling microscopy, we reproducibly obtained electrical conductance spectra of superconductor- superconductor atomic point contacts formed on various atomic sites of the substrate. From the analysis of the subharmonic features due to the multiple Andreev reflection, spatial distributions of the number of the conduction channels and their transmission probabilities were obtained. It is found that the number of channels and the transmission probabilities strongly depend on an atomic site where the contact is formed. We also revealed how the conduction channels and their transmission probability evolve from the tunneling to contact regimes. The Josephson current also exhibits the atomic site dependence, which is explained by the site-dependent transmission probabilities. Our results demonstrate a crucial role of the atomic geometry in the conduction channels of the ultimately confined conductor for the first time experimentally.