Detection of rotationally symmetric shape orientation by fold principal orientation

A systematic method, called fold principal orientation (FPO), is proposed to detect the orientation of rotationally symmetric shape. A K-fold rotationally symmetric shape (K-RSS), f(r,Θ), being angularly periodic with period 2π/K, is proved to have the property: f(r,Θ) = g(r,KΘ), where g(r,Θ) is called a fold-expanded shape. With this property, the orientation of f(r,Θ) can be detected through defining the orientation of g(r,Θ). A series of complicated situations about the fold-expanded shape, g(r,Θ), are investigated in this paper. An algorithm describing the construction of FPO and several illustrative examples showing the detected principal orientation of distinct types are given. The method proposed is also compared with fold principal axes (FPA) and fold-invariant shape-specific points (FISSP).