| Summary: | Computer animation often requires that complex objects need to be modeled by fitting them with simpler shapes such as splines. It is necessary that these simpler shapes respond continuously as the object changes shape. This response can become complex if the topology of the error function is changing. The topological events which can occur when a spline is fit to a changing object are classified and analyzed, with a view towards defining conditions which make them possible. The Least Squares Orthogonal Distance Fitting (ODF) method is used to fit a continuous parametric function whose shape is changing. The fit is performed using a closed-shape Beta2-spline. Three types of topological events can occur: self-annihilation of two solutions, simultaneous merge of three solutions into one, and crossover of two solutions in which a local minimum and saddle point interchange roles. The events can be detected and classified by inspecting the determinants of two second-order response matrices. The last two types of event occur only within sub-manifolds defined by additional constraints not normally present in the ODF solution. The nature of these constraints is defined. The constrained events are interpreted in the context of catastrophe theory, which describes dynamical systems that can exist in multiple states with the possibility of discontinuous transitions between them. Within this theory they appear as narrowly avoided cusp catastrophes.
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