Abstract This paper extends a recently proposed robust computational framework for constructing the boundary representation (brep) of the volume swept by a given smooth solid moving along a one parameter family h of rigid motions. Our extension allows the input solid to have sharp features, and thus it is a significant and useful generalization of that work. This naturally requires a precise description of the geometry of the surface generated by the sweep of a sharp edge supported by two intersecting smooth faces. We uncover the geometry along with the related issues like parametrization and singularities via a novel mathematical analysis. Correct trimming of such a surface is achieved by an analysis of the interplay between the cone of normals at a sharp point and its trajectory under h. The overall topology is explained by a key lifting theorem which allows us to compute the adjacency relations amongst entities in the swept volume by relating them to corresponding adjacencies in the input solid. Moreover, global issues related to body‐check such as orientation, singularities and self‐intersections are efficiently resolved. Examples from a pilot implementation illustrate the efficiency and effectiveness of our framework.

Learning to rank is an important area at the interface of machine learning, information retrieval and Web search. The central challenge in optimizing various measures of ranking loss is that the objectives tend to be non-convex and discontinuous. To make such functions amenable to gradient based optimization procedures one needs to design clever bounds. In recent years, boosting, neural networks, support vector machines, and many other techniques have been applied. However, there is little work on directly modeling a conditional probability Pr(y|xq) where y is a permutation of the documents to be ranked and xqrepresents their feature vectors with respect to a query q. A major reason is that the space of y is huge: n! if n documents must be ranked. We first propose an intuitive and appealing expected loss minimization objective, and give an efficient shortcut to evaluate it despite the huge space of ys. Unfortunately, the optimization is non-convex, so we propose a convex approximation. We give a new, efficient Monte Carlo sampling method to compute the objective and gradient of this approximation, which can then be used in a quasi-Newton optimizer like LBFGS. Extensive experiments with the widely-used LETOR dataset show large ranking accuracy improvements beyond recent and competitive algorithms.