This paper presents the first design principles that optimize the visualization of sets using linear diagrams.These principles are justified through empirical studies that evaluate the impact of graphical features ontask performance. Linear diagrams represent sets using straight line segments, with line overlaps corre-sponding to set intersections. This work builds on recent empirical research which establishes that lineardiagrams can be superior to prominent set visualization techniques, namely Euler and Venn diagrams. Weaddress the problem of how to best visualize overlapping sets using linear diagrams. To solve the problem,we investigate which graphical features of linear diagrams significantly impact user task performance. Tothis end, we conducted seven crowd-sourced empirical studies involving a total of 1760 participants. Thesestudies allowed us to identify the following design principles, which significantly aid task performance: usea minimal number of line segments, use guide-lines where overlaps start and end, and draw lines that arethin as opposed to thick bars. We also evaluated the following graphical properties which did not signifi-cantly impact task performance: colour, orientation, and set-order. The results are brought to life througha freely available software implementation that automatically draws linear diagrams with user-controlledgraphical choices. An important consequence of our research is that users are now able to create effectivevisualizations of sets automatically, thus improving human-computer interaction.