Grid Geometries Which Preserve Properties of Euclidean Geometry: A Study of Graphics Line Drawing Algorithms

Abstract

One of the most frequent displays on a graphics screen is a combination of straight line segments. However, the screen is not really the Euclidean plane, but is instead a rectangular array of pixels, each of which can be turned on or off. Thus, the theoretically nice but imaginary Euclidean geometry is approximated by a not so theoretically nice but implementable grid geometry (see Figure 0). We isolate several key properties of straight lines in the Euclidean plane and define analogous properties with respect to paths drawn in a grid. We present a theoretical and practical study of a new grid geometry, called a smooth geometry, which has combinations of these properties not found in previously proposed algorithms.