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We extend two results about the ordinary continued fraction expansion to best simultaneous Diophantine approximations of vectors or matrices. The first is Levy-Khintchin Theorem about the almost sure growth rate of the denominators of the convergents. The second is a Theorem of Bosma, Hendrik and Wiedijk about the almost sure limit distribution of the sequence of products q_n d(q_n θ,Z) where the qn's are the denominators of the convergents associated with the real number θ by the ordinary continued fraction algorithm. Beside these two main results, we show that when d≥2, for almost all vectors θ∈R^d, lim inf_{n→∞} q_{n+k} d(q_n θ,Z^d)=0 for all positive integers k, where (q_n)_{n∈N} is the sequence of best approximation denominators of θ.

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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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