The concept of the grid is broadly used in digital geometry and other fields of computer science; it consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this thesis, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems. The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [-1, 1], which gives many other vital properties of this coordinate system. Moreover, addition of two vectors in the new triangular coordinate system is presented and illustrated. Accordingly, in discrete and digital geometry, rotations with the composition of translations have been measured and examined carefully on the square and the hexagonal grids. The translation has never been considered individually because it obviously leads to the isometric translation on these grids. However, the triangular grid is not a point lattice, thus, it is worth to consider the translation itself. Therefore in this thesis, translations on the triangular grid are investigated and the vectors of bijective and non-bijective translations are specified. Keywords: Barycentric coordinate system, coordinate system, hexagonal grid, triangular grid, trihexagonal grid, non-traditional grids, transformations, image processing, computer graphics, discretized translations, digital geometry