Abstract: This paper presents a series of new algorithms for reliably graphing two-dimensional implicit equations and inequalities. A clear standard for interpreting the graphs generated by two-dimensional graphing soft are is introduced and used to evaluate the presented algorithms. The first approach presented uses a standard interval arithmetic library. This approach is shown to be faulty; an analysis of the failure reveals a limitation of standard interval arithmetic. Subsequent algorithms are developed in parallel with improvements and extensions to the interval arithmetic used by the graphing algorithms. Graphs exhibiting a variety of mathematical and artistic phenomena are shown to be graphed correctly by the presented algorithms. A brief comparison of the final algorithm presented to other graphing algorithms is included.