UNIVERSAL PRINCIPAL AXES - AN EASY-TO-CONSTRUCT TOOL USEFUL IN DEFINING SHAPE ORIENTATIONS FOR ALMOST EVERY KIND OF SHAPE

Abstract

A convenient tool for defining shape orientations called universal principal axes is introduced. These axes are formed of half lines starting from the shape centroid, with their directional angles expressed as functions of the polar angle of the first non-zero complex number encountered in the sequence integral integral(x + iy)l dA with l = 2, 3, 4,..., provided that the integration domain is taken to be the given two-dimensional (2D) shape and the origin is assumed to be the shape centroid. Universal principal axes are shown to be independent of the translation, scaling, and rotation of the coordinate system used, and they are therefore qualified in defining shape orientations. A major benefit of using the proposed universal principal axes is that they exist for almost every kind of shape. Another benefit is that there is no need to judge in advance whether a given shape is mirror-symmetric, rotationally symmetric, irregular, etc. Moreover, in the case of rotationally symmetric shapes, the universal principal axes make the preprocessing procedure for detecting the number of folds contained in the given shape unnecessary although such procedure is essential for many existing tools designed to define orientations of rotationally symmetric shapes. Defining shape orientations by universal principal axes is therefore quite convenient. An algorithm describing the construction of universal principal axes and several examples showing the detected universal principal axes for some shapes of distinct types are given. The relationship between universal and conventional principal axes is also discussed.