Generalization of Superposition of Rectangles based on Direction Relations Fadoua Ghourabi Ochanomizu University, Japan ghourabi.fadoua@ocha.ac.jp Kazuko Takahashi Kwansei Gakuin University, Japan ktaka@kwansei.ac.jp Abstract We present a method of superposing rectangles. The su- perposition is under the condition that some of the re- gions should be visible. We first define a qualitative spa- tial representation of the rectangles. In particular, direc- tion relations are used to express the positions of the must-be-visible regions. The representation is extend- able to accommodate higher degree of granularity, and therefore to cover any arrangement of regions. Proper- ties of success and effectiveness are defined to evaluate the superposition. 1 Introduction Qualitative spatial representation emerged as an area of knowledge representation. The foundation in qualitative spa- tial representation is to treat objects of the space qualita- tively, i.e. what matters is how objects are related. Posi- tions of objects in the space is one of the relevant prob- lems that is addressed by the field of qualitative spatial rep- resentation. The direction relations describe where an ob- ject is positioned w.r.t. a reference. We distinguish two cat- egories of direction relations. The direction relations are es- tablished either w.r.t. a relative reference, in that case we use relative direction relations (Frank 1991), or w.r.t. an ab- solute reference, in that case we use cardinal direction re- lations (Clementini, Felice, and Hern´ andes 1997). The latter type of relations is used in the context of geographical space. They are the 9 classic relations north, south, east, west, north east, north west, south east, south west and origin. The relative direction relations are used in a local context and they are the 9 relations up, bottom, right, left, up right, up left, bottom right, bottom left and same. In this paper, we use relative direction relations to not only represent positions of objects but also to treat their super- positions. We illustrate with a practically oriented problem, namely superposing rectangular structures. A tangible ex- ample is the arrangement of rectangular GUIs. When inter- acting with software, we often work with several GUIs at the same time. They are arranged randomly (e.g. Fig. 1(a)) or using tiling window manager without overlapping. How- ever, the way the GUIs are arranged should depend on the Copyright c  2015, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. (a) Random arrangement (b) Manual arrangement where relevant content is visible Figure 1: Arrangements of GUI windows with superposition content. We therefore often manually resize, drag and super- pose the interfaces for better visibility (e.g. Fig. 1(b)). Our goal is to find solutions of superposing GUI rectangles while keeping important content visible. The rest of this paper is organized as follows. In Sect. 2, we summarize the original superposition method and its lim- itations. We describe the qualitative representation in Sect. 3. Then, in Sect. 4, we discuss the expressiveness, and we present the generalization of superposition and its properties in Sect. 5. In Sect. 6, we conclude with remarks on future directions of research.