Section: New Results

Effective 3D global visibility, theory and applications

Participants : Guillaume Batog, Hazel Everett, Xavier Goaoc, Sylvain Lazard, Sylvain Petitjean, Linqiao Zhang.

In recent years, our activity in the area of 3D visibility focused on three main directions: (i) the computation and complexity analysis of the 3D visibility complex, (ii) the computation and complexity analysis of the boundary of shadows cast by area light sources, and (iii) the study of some fundamental questions in geometric transversal theory.

The 3D visibility complex is a partition of the space of rays according to visibility. The questions we consider are two-fold. On one hand, we study its size, both experimentally and theoretically. This size is reflected in the number of unobstructed line segments tangent to four objects of the scene; while this number can be (n⁴) in unstructured scenes, its behavior for more realistic input is still not understood. On the other hand, we work on the computation of the 3D Visibility skeleton, a substructure of the visibility complex. A few years ago we presented a worst-case near-optimal sweep-plane algorithm for computing this structure in a restrictive setting [3] . Over the last years, we have been working on an efficient and reliable implementation of that algorithm, and on the development of an algorithm in a more general setting.

Shadows play a central role in human perception and a wide variety of approaches have been considered for simulating and rendering them. Unfortunately, computing realistic shadows efficiently is a very difficult problem, particularly in the case of non-point light sources, due to the complicated internal structure that such shadows may have. The only current solution for that problem goes through a discretization of area light sources by many point light sources; shadows cannot be certified with this approach and their structure is lost. As a first approximation, one can compute the main shadow boundaries, that is the boundary between the regions in full-light, penumbra, and umbra. Specifically, a point is in the umbra if it does not see any part of any light source; it is in full light if it sees entirely all the light sources; otherwise, it is in the penumbra. These boundaries are traced out by one-parameter families of segments that belong to the 3D Visibility skeleton. On one hand, we try to estimate the complexity of these shadow boundaries, both in theory and in practice. On the other hand, we work on applying our methods for computing the visibility skeleton to this particular problem.

The study of fundamental properties of sets of lines, that is, line geometry, underlies many algorithmic questions including 3D visibility computations. A few years ago, we generalized Helly's classical theorem to sets of line transversals to disjoint unit balls in  [5] . This theorem builds on two types of results: a local Helly-type theorem, also called a pinning theorem, and a bound on the topological complexity of the space of line transversals, obtained by counting so-called geometric permutations . Our activity focuses on extending both types of results.

Visibility complex and skeleton

This year, our PhD student Linqiao Zhang, co-supervised with S. Whitesides at McGill University, defended her Ph.D. thesis entitled On the three-dimensional visibility skeleton: implementation and analysis  [11] . She developed a first implementation of our sweep-plane algorithm that computes robustly the vertices of the visibility skeleton of a set of convex polyhedra in generic position. This implementation is a key element of a prototype we developed for computing limits of umbra [24] (see below); it also allowed us to study experimentally the size of the 3D visibility skeleton in a random setting and, in particular, to show that the constant involved in the asymptotic complexity is small [11] [66] . Related to this work, we studied various predicates, arising in three-dimensional visibility, concerning line transversals to lines and segments in 3D. In particular, we computed the degrees of standard methods of evaluating these predicates. We showed that the degrees of some of these methods are surprisingly high (up to 168), which may explain why computing line transversals with finite-precision floating-point arithmetic is prone to error [11] ; this work also published this year in the journal Computational Geometry, Theory and Applications  [16] .

We also obtained a new result on the complexity of the set free lines and free lines segments tangent to balls in 3D. In particular, we proved that, in the presence of n disjoint unit balls, the set of free line segment, that is the so-called visibility complex, has complexity (n⁴) . This result is very surprising because it was natural to conjecture that the visibility complex of fat objects of similar size had a lower worst-case complexity than that for thin triangles. Our result settles negatively this conjecture, and shows exactly the opposite, that is, that fatness and similarity, alone, do not reduce the worst-case complexity of that structure. This result was submitted to the Symposium on Computational Geometry in December 2009 [35] .

In the context of the PhD thesis of Guillaume Batog, we are working on an invariant-based method for designing efficient evaluation strategies for predicates. In a nutshell, the idea is to find a group action on the input space of a predicate with two properties: it has few orbits, and any two inputs in the same orbit are equivalent for the predicate. The invariants of that action are known, under certain conditions, to distinguish the various orbits, and can then be used to decide the predicate. This year, we unfolded this approach for predicates on line systems that arise in visibility computations, for instance deciding the number of lines intersecting four given lines. A communication is in preparation.

Shadow computations

We published this year two results on the umbra and penumbra cast by non-trivial light sources.

First, we published the first non-trivial bounds on the size of the umbra region. These result show that the umbra can be surprisingly complicated, even in the presence of disjoint fat obstacles; for instance, we proved that the umbra cast on a plane by a segment light source in the presence of two fat polytopes of size n , can have up to (n) connected components in the worst case. These results were published in the journal Computational Geometry, Theory and Applications  [13] .

We also developed, over the last years, a new method for the exact computation of shadow boundaries. The associated prototype software is, up to our knowledge, the first implementation that computes exactly such shadow boundaries of polygonal light sources in the presence of polyhedral obstacles for nontrivial scenes (e.g., 50 polytopes with 1 500 vertices). Using this implementation, we observed experimentally a significant gap between the size of the visibility skeleton and the, much smaller, complexity of the shadow boundaries. We presented these results at the European Workshop on Computational Geometry  [24] .

Pinning and line transversals

In the last few years, we strived to understand the combinatorial properties of so-called isolated line transversals, also called pinnings : these lines meet every member in a family of objects, but they lose this property when subject to any arbitrarily small perturbation. We obtained a basis theorem when the objects are disjoint balls in in 2006 [5] : any pinning of a line by disjoint balls in contains a pinning of size at most 2d-1 . We strengthened this result in three ways. First, we proved that this constant, 2d-1, is best possible for all dimension, a result obtained in 2008. In 2009, we presented this result at the Eurocomb conference [21] and submitted a complete version for journal publication. Then, we extended this result to pinnings of a line by possibly intersecting smooth convex sets [37] . Last, in 2008 we extended this result to pinnings by polyhedra in . This year, we added to this extension a complete description of minimal pinning configurations; these results were submitted to the Symposium on Computational Geometry in December 2009 [29] .

An order in which a line can intersect a collection of geometric objects is called a geometric permutation of that collection. When the objects are disjoint balls, the geometric permutations count the connected components of the space of line transversals ; more generally, it gives a lower bound on its topological complexity. Substantial effort has been devoted to understanding how the geometry of the objects constrains the number of geometric permutations; the asymptotic behavior of the maximum number of geometric permutations of n disjoint convex sets in has remained an open problem for two decades. We have been working on a new approach to this question. The idea is to analyze how the knowledge of a bound for constant-size problems (say, knowing that the maximum number of geometric permutations for 10 objects is at most 24) helps bounding the asymptotic behavior for n objects. This year, we made a first step in this direction: we delineated the main growth rates (constant, polynomial, exponential) through a generalization of Sauer's Lemma for shatter functions of set systems.