Symmetry is important in the development and performance of the visual system. Most of the current research findings on perception and computational modeling of symmetry focus on the biological significance of bilateral symmetry in the visual system of primates and psychophysics of reflection symmetry in human vision. There seems to be little previous work done on modeling detection of translation symmetry in the human visual system. In this paper, we study the problem of detection and characterization of translation symmetry in plane surfaces. We also explore their generalization to the case of a planar surface situated in the 3-space with a slant, tilt, or both. Besides contribution to computational modeling of human visual perception of symmetry, this research provides algorithms for characterization of texture and geometric properties of planar surfaces in computer vision. On the theoretical side, we discuss the reduction of a complete solution to perception of arbitrary planar symmetries to the special cases of characterization of translations and reflections. On the computational side, we provide an algorithm based on geometric and statistical considerations to detect and determine the fundamental domain for translation symmetries in planar surfaces parallel to the viewer's eye (or the robot camera's image plane). We have included a brief review and critique of results from mathematics of tiling, psychophysics, and neurobiology of the visual systems pertaining to detection of bilateral symmetry. Finally, we discuss the limitations and successes of our algorithms, and the issues of effectiveness and performance of our computation methods.