Detection and Classification of Critical Points for Linear Metamorphosis

Abstract

We apply topological analysis to functionally based shape metamorphosis. Functionally based methods have two problems: shape interpolation is applied without defining the topological information and the time moments of topological changes are not known. Thus, it is difficult to identify the time intervals for key frames of shape metamorphosis animation. Moreover, information on the types of the topological changes is missing. We present a method of the critical points detection based on the Morse theory and classification using the Hessian matrix for solving these problems. The defining function of the linear metamorphosis is treated as a height function. By analyzing how the critical points are changing at a particular height level, we detect the critical points of the metamorphosis process. The critical points can be used for ease in/ ease out effects in animation. In addition, we classify the detected critical points into maximum point, minimum point, and saddle point types. Using the type of the critical points and the sign of the function time derivative at the critical points, we can define the topological information for the shape metamorphosis. We illustrate these methods using shape metamorphosis in 2D and 3D spaces.