We consider the Stokes conjecture concerning the shape of extreme 2-dimensional water waves. By new geometric methods including a non-linear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.

Section (1) is devoted to a discussion of the model-fitting problem, which finds its explicit solution in equation (1.13). In section (2) the maximum likelihood, (ML), estimates of the model parameters are investigated, and for the class of series considered shown to possess the same optimum properties as in the case of independent series. Next, the covariance matrix of the parameter estimates is expressed in terms of the spectral function of the generating process (eq. 3.7). The last section is concerned with certain working approximations to the ML statistics.

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Man-made objects usually exhibit descriptive curved features (i.e., curve networks). The curve network of an object conveys its high-level geometric and topological structure. We present a framework for extracting feature curve networks from unstructured point cloud data. Our framework first generates a set of initial curved segments fitting highly curved regions. We then optimize these curved segments to respect both data fitting and structural regularities. Finally, the optimized curved segments are extended and connected into curve networks using a clustering method. To facilitate effectiveness in case of severe missing data and to resolve ambiguities, we develop a user interface for completing the curve networks. Experiments on various imperfect point cloud data validate the effectiveness of our curve network extraction framework. We demonstrate the usefulness of the extracted curve networks for surface reconstruction from incomplete point clouds.

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