Overlapped-triangle analysis with hierarchical ranking of dominance

Plane geometric figures (PGFs) are essential diagrams that regularly appear in mathematics documents. To understand PGFs profoundly and intuitively, it is necessary to decompose them into visual elements rather than traditional line segments. In this paper, we present a method for detection and analysis of overlapped triangles on the basis of dominance rank. Overlapped triangles lead to a large quantity of redundant sub-triangles or incident triangles. The most important and representative triangles must be detected and adopted to reduce redundancy. Diverse types of relationships among potential triangles present at least three obstacles: (1) how to precisely detect all potential triangles, (2) how to select predominant triangles, and (3) how to verify that the combination of these predominant triangles can reconstruct the original figure. Based on Gestalt theory, we propose a method for selecting predominant triangles, including selection of the main convex triangle and composition of interior auxiliary triangles. Experiments show that our algorithm can detect most of the potential triangles and can present reasonable decomposition solutions using only few predominant elements.