Section: New Results

Geometric foundations

Detecting the Intersection of Two Convex Shapes by Searching on the 2-sphere

Participant: Samuel Hornus.

We take a look at the problem of deciding whether two convex shapes intersect or not. We do so through the well known lens of Minkowski sums and with a bias towards applications in computer graphics and robotics. We describe a new technique that works explicitly on the unit sphere, interpreted as the sphere of directions. In extensive benchmarks against various well-known techniques, ours is found to be slightly more efficient, much more robust and comparatively easy to implement. In particular, our technique is compared favourably to the ubiquitous algorithm of Gilbert, Johnson and Keerthi (GJK), and its decision variant by Gilbert and Foo. We provide an in-depth geometrical understanding of the differences between GJK and our technique and conclude that our technique is probably a good drop-in replacement when one is not interested in the actual distance between two non-intersecting shapes.

The work was published in the journal Computer-Aided Design (special issue for the Proceedings of Solid and Physical Modeling: SPM’17) [9]. The paper has received a best paper award (2nd place) at the SPM conference.

Decomposition of a Hexahedron into a Set of Tetrahedra

Participant: Laurent Alonso.

This year was marked by some works on the combinatorial decomposition of a generic hexahedron in a set of nonintersecting tetrahedra up to symmetries: it is well known that there are only six decompositions of a cube into tetrahedra ; we show that there are at most 1360 potential different decompositions of a hexahedron and at least 1357 are geometrically valid. Additional work is in progress in order to show that the last 3 remaining decompositions do not correspond to valid geometrical solutions.

Hash-based CSG Evaluation on GPU

Participants: Cédric Zanni, Sylvain Lefebvre.

We have developed a new evaluation scheme for Constructive Solid Geometry (CSG) modeling that is well adapted to modern GPU. The approach falls into the category of screen space techniques and can handle a large range of geometric representation. The proposed method relies on the idea of hashing in order to reduce the memory footprint for the processing of a given ray in the scene (e.g. for discovering which part of the space is within or outside the object) while allowing the evaluation of the CSG in amortized constant time. This memory reduction in turn allows to subdivide the space in order to apply progressively the rendering algorithm, ensuring that required data fit in the graphic memory. This improvement over previous approach allows to handle objects of higher complexity during both modeling and slicing for additive manufacturing.