Composition, doping, size, and strain distribution within quantum dots, and at the dots-substrate interfaces, determine the confinement potential of electrons and holes creating a complex band structure. We use ultra-high vacuum Kelvin probe force microscopy to obtain the two-dimensional confinement potential in and around InAs and InSb dots epitaxially grown on GaAs. It is found that the potential manifests rich features governed by the strain and composition variations in the vicinity of the individual quantum dots. The results can adjust or confirm theoretical predictions for many epitaxial dots systems.