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The complex polar oxide heterojunction of ${\mathrm{SrTiO}}_{3}/{\mathrm{LaAlO}}_{3}$ (STO/LAO) is of great interest due to the emergent physical phenomena observed at the interface. STO and LAO separately are wide band-gap insulators. However, upon joining them at the 001, 110, and 111 crystallographic planes, the interface undergoes a transition to a conductive state. Although first-principles modeling of the 001 plane interface has been widely studied, there is a lack of reports regarding the 110 and 111. This paper expands the theoretical model of the STO/LAO heterointerface to the three crystallographic planes (001, 110, and 111) where the conductivity has been experimentally reported. The calculations showed that whereas at the 001 interface the conductivity appears at a critical thickness of 4 monolayers of LAO, the 110 and 111 planes have no clear critical thickness; these two interfaces were always conductive. Nevertheless, the number of conductive electrons per unit cell increases with the thickness of the LAO layer in the 110 and 111 interfaces. This is related to the energy levels downshifting due to the electrostatic potential buildup (which was in the opposite direction respect to the 001 interface), increasing the number of conductive sates below the Fermi level. Given the absence of a critical thickness and the fact that chemical intermixing and oxygen vacancies at the interface were not considered, the main mechanism responsible for the conductivity in the 110 and 111 planes was attributed to the large structure reconstruction that locally changes the energy levels at the interface causing charge transfer and accumulation at the layers close to the interface.