The application of Fourier Series to cartography and GIS, in particular to shape analysis, line simplification and enhancement, and the measurement of cartographic scale, is explored. Vector representations of the Australian coastline are manipulated using Fourier techniques and the output appraised in terms of its ability to encapsulate the shape of the original coastline, or as a simplified or enhanced version of the original. Using Fourier techniques on vector representations of parts of the Australian coastline at six different cartographic scales, a relationship between cartographic scale and line complexity is found to exist, and an empirical expression of that relationship developed. The intent of developing an empirical relationship is to bring greater consistency to the process of assigning cartographic scale to data, and thereby to the process of map production, data simplification and GIS analysis. Fourier techniques are found to be a poor encapsulator of cartographic shape, a good method of line simplification and enhancement, and a promising method of measuring cartographic scale. As background, some revision of the history and mathematics of Fourier Series, and previous research into the application of Fourier Series to cartography, is given. [ABSTRACT FROM AUTHOR]