An edge equation is simply a discriminating function like those we used in our curve and line-drawing algorithms.

An edge equation segments a planar region into three parts, a boundary, and two half-spaces. The boundary is identified by points where the edge equation is equal to zero. The half-spaces are distiguished by differences in the edge equation's sign. We can choose which half-space gives a positive sign by multiplication by -1.
We can scale all three edges so that their negative halfspaces are on the triangle's exterior.