A Simple Approach for Boundary Improvement of Euler Diagrams

Abstract

Abstract-General methods for drawing Euler diagrams tend to generate irregular polygons. Yet, empirical evidence indicates that smoother contours make these diagrams easier to read. In this paper, we present a simple method to smooth the boundaries of any Euler diagram drawing. When refining the diagram, the method must ensure that set elements remain inside their appropriate boundaries and that no region is removed or created in the diagram. Our approach uses a force system that improves the diagram while at the same time ensuring its topological structure does not change. We demonstrate the effectiveness of the approach through case studies and quantitative evaluations.

Drawing Dynamic Graphs Without Timeslices

“…Our force-directed algorithm is based on earlier variants by Simonetto et al [46,47]. As in most force-directed algorithms, after an initialization phase, the algorithm iteratively improves the current layout of the drawing in 3D for a given number of iterations in the following way:…”

“…max g ∈ I : ∃n ∈ s u,i , ∃q ∈ s v,j , g = n[2] = q [2] where x [2] is the time coordinate of the point x in the space-time cube. We compute the attractive force between the points m and p, and n and q, using EdgeContraction(δ) [47] (see Fig. 3c).…”

“…This force smooths node trajectories, helping with node movements over time. For trajectory bends, we use the CurveSmoothing [47] force, which pulls a bend b to the centroid of the triangle ∆abc formed by using the previous and next bends or endpoints a and c. A trajectory endpoint a has no such triangle. Therefore, it is pulled in 2D toward the midpoint of the segment formed with the closest bend or endpoint b.…”

“…This allows large movements at the beginning of the computation, and smaller refinements towards the end. This constraint is similar to DecreasingMaxMovement(δ) [47].…”

A toolkit for knot diagram sketching, encoding and re-generation

“…By clicking on the OK button, a new dialog It is also possible to export a drawing in svg format (optionally as an html file), whilst the code can be exported in txt or html format. This facilitates the use of the tool for the production of diagrams that occur in research papers, student submissions, or to include in web-pages; • open the Options window in order to alter tool settings, such as the size of the gap used in the visualisation of the crossings; • perform knot beautification using EulerSmooth [30].…”