Geometrical and topological features of isotrigon textures

Abstract

Isorigon textures are considered from the perspectives of geometry and topology. The textures considered here consist of finite size pixels. Thus we should use discrete mathematics. We use the Gaussian and mean curvatures to characterize textures by introducing the discrete scheme of curvature calculation from our previous work (Mem. Cntr. lnjor. Sci. Kokushikan U 30 (2009) 1-13). We also consider Minkowski functionals using the morphological analysis of images, one of which comes from the application of the Gauss-Bonnet theorem on closed oriented surfaces. This idea is an intrinsic property of surface so that the application of Gauss-Bonnet theorem leads to the topology of surfaces. In the topology of surfaces integrated Gaussian curvature plays a key role. Thus the integrated Gaussian curvature of surfaces consisting of finite size elements is calculated for isotrigon textures. The integrated Gaussian curvature has large values for isotrigon textures and we discuss why this occurs.