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Most 3D laser scanners are based on 3D optical triangulation algorithms, where the location of each 3D point is estimated as the intersection of a camera ray and a plane of light projected by a laser line generator. Since a physical laser line generator projects a sheet of light of finite thickness, inaccurate measurement and errors result from assuming that the plane of light is infinitesimally thin. We propose a new mathematical formulation for 3D optical triangulation based on interval arithmetic, where 3D points are only determined within certain bounds along the camera rays, and multiple measurements are used to tighten these bounds. We propose the Line Segment Cloud as an alternative surface representation to visualize the measurement errors within the proposed framework. We introduce the Iterative Line Segment Tightening algorithm to convert line segment clouds to point clouds, as a preprocessing step prior to surface reconstruction. We describe how to construct a low cost laser line 3D scanner, where the camera is fixed with respect to the object and the laser line generator is mounted on a high resolution motion platform. We describe a GPU-based implementation where the large number of captured images are processed in real time. Finally, we present some experimental results.