In this paper we describe a hierarchical face clustering algorithm for triangle meshes based on fitting primitives belonging to an arbitrary set. The method proposed is completely automatic, and generates a binary tree of clusters, each of which fitted by one of the primitives employed. Initially, each triangle represents a single cluster; at every iteration, all the pairs of adjacent clusters are considered, and the one that can be better approximated by one of the primitives forms a new single cluster. The approximation error is evaluated using the same metric for all the primitives, so that it makes sense to choose which is the most suitable primitive to approximate the set of triangles in a cluster. Based on this approach, we implemented a prototype which uses planes, spheres and cylinders, and have experimented that for meshes made of 100k faces, the whole binary tree of clusters can be built in about 8 seconds on a standard PC. The framework here described has natural application in reverse engineering processes, but it has been also tested for surface de-nosing, feature recovery and character skinning.

Summary.Remeshing is a key component of many geometric algorithms, including modeling, editing, animation and simulation. As such, the rapidly developing field of geometry processing has produced a profusion of new remeshing techniques over the past few years. In this paper we survey recent developments in remeshing of surfaces, focusing mainly on graphics applications. We classify the techniques into five categories based on their end goal: structured, compatible, high quality, feature and error-driven remeshing. We limit our description to the main ideas and intuition behind each technique, and a brief comparison between some of the techniques. We also list some open questions and directions for future research.

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Due to the wide diffusion of 3D printing technologies, geometric algorithms for Additive Manufacturing are being invented at an impressive speed. Each single step, in particular along the Process Planning pipeline, can now count on dozens of methods that prepare the 3D model for fabrication, while analysing and optimizing geometry and machine instructions for various objectives. This report provides a classification of this huge state of the art, and elicits the relation between each single algorithm and a list of desirable objectives during Process Planning. The objectives themselves are listed and discussed, along with possible needs for tradeoffs. Additive Manufacturing technologies are broadly categorized to explicitly relate classes of devices and supported features. Finally, this report offers an analysis of the state of the art while discussing open and challenging problems from both an academic and an industrial perspective.

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We introduce a novel algorithm to transform any generic set of triangles in 3D space into a well-formed simplicial complex. Intersecting elements in the input are correctly identified, subdivided, and connected to arrange a valid configuration, leading to a topologically sound partition of the space into piece-wise linear cells. Our approach does not require the exact coordinates of intersection points to calculate the resulting complex. We represent any intersection point as an unevaluated combination of input vertices. We then extend the recently introduced concept of indirect predicates [Attene 2020] to define all the necessary geometric tests that, by construction, are both exact and efficient since they fully exploit the floating-point hardware. This design makes our method robust and guaranteed correct, while being virtually as fast as non-robust floating-point based implementations. Compared with existing robust methods, our algorithm offers a number of advantages: it is much faster, has a better memory layout, scales well on extremely challenging models, and allows fully exploiting modern multi-core hardware with a parallel implementation. We thoroughly tested our method on thousands of meshes, concluding that it consistently outperforms prior art. We also demonstrate its usefulness in various applications, such as computing efficient mesh booleans, Minkowski sums, and volume meshes.