The union of Unit Balls has Quadratic Complexity, even if They all Contain the Origin

We provide a lower bound construction showing that the union of unit balls in R3 has quadratic complexity, even if they all contain the origin. This settles a conjecture of Sharir

On the number of line tangents to four triangles in three-dimensional space

Colloque avec actes et comité de lecture. internationale.International audienceWe establish upper and lower bounds on the number of connected components of lines tangent to four triangles in $\mathbb{R}^3$. We show that four triangles in $\mathbb{R}^3$ may admit at least 88 tangent lines, and at most 216 isolated tangent lines, or an infinity (this may happen if the lines supporting the sides of the triangles are not in general position). In the latter case, the tangent lines may form up to 216 connected components, at most 54 of which can be infinite. The bounds are likely to be too large, but we can strengthen them with additional hypotheses: for instance, if no four lines, each supporting an edge of a different triangle, lie on a common ruled quadric (possibly degenerate to a plane), then the number of tangents is always finite and at most 162; if the four triangles are disjoint, then this number is at most 210; and if both conditions are true, then the number of tangents is at most 156 (the lower bound 88 still applies)

From white to green : Snow cover loss and increased vegetation productivity in the European Alps

Mountains are hotspots of biodiversity and ecosystem services, but they are warming about twice as fast as the global average. Climate change may reduce alpine snow cover and increase vegetation productivity, as in the Arctic. Here, we demonstrate that 77% of the European Alps above the tree line experienced greening (productivity gain) andPeer reviewe

On the Number of Maximal Free Line Segments Tangent to Arbitrary Three-dimensional Convex Polyhedra

We prove that the lines tangent to four possibly intersecting convex polyhedra in $^3$ with $n$ edges in total form $\Theta(n^2)$ connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrary degenerate scenes. More generally, we show that a set of $k$ possibly intersecting convex polyhedra with a total of $n$ edges admits, in the worst case, $\Theta(n^2k^2)$ connected components of maximal free line segments tangent to any four of the polytopes. This bound also holds for the number of connected components of possibly occluded lines tangent to any four of the polytopes

Lines tangent to four triangles in three-dimensional space

International audienceWe investigate the lines tangent to four triangles in $\mathbb{R}^3$. By a construction, there can be as many as 62 tangents. We show that there are at most 162 connected components of tangents, and at most 156 if the triangles are disjoint. In addition, if the triangles are in (algebraic) general position, then the number of tangents is finite and it is always even

Intraspecific differentiation: Implications for niche and distribution modelling

Aim: Mounting evidence suggests that failure of species distribution models to integrate local adaptation hinders our ability to predict distribution ranges, raising the question of whether modelling should be performed at the level of species (clade models) or intraspecific lineages (subclade models), characterized by the restricted availability of occurrence points. While Ensembles of Small Models (ESMs) offer an attractive framework for small datasets, their evaluation remains critical. We address these issues in the case of very small datasets inherent to subclade models and discuss which modelling strategy should be applied based on niche overlap among lineages. Location: Sweden. Taxon: Mosses. Methods: Ensembles of Small Models were evaluated by null models built from randomly sampled presence points. We compared the extent of suitable area predicted by the projections of clade and subclade models. Niche overlap was quantified using Schoener's D and Hellinger'sImetrics, and the significance of these metrics in terms of niche conservatism or divergence was assessed by similarity tests. Results: We introduced a simple procedure for evaluating ESMs based on the pooling of the statistics used to assess model accuracy from the replicates. Despite fairly high AUC and TSS values, 2 of the 23 subclade models did not perform better than null models and should be discarded. Combined predictions from subclade models contributed, on average, five times more than clade models to the total suitable area predicted by the combination of subclade and clade models. The D and I metrics averaged 0.45 and 0.71, with evidence for niche conservatism in half of the species and no signal for niche divergence. Main conclusions: In addition to the assessment of ESM accuracy based on the simple procedure described here, we recommend that ESMs should be systematically evaluated against null models. Lumping or splitting occurrence data at the intraspecific level substantially impacted model projections. Given the poor performance of models based on small datasets, even when employing ESMs, we pragmatically suggest that, in the absence of evidence for niche divergence during diversification of closely related intraspecific lineages, SDMs should be based on all available occurrence data at the species level

From white to green: Snow cover loss and increased vegetation productivity in the European Alps

Mountains are hotspots of biodiversity and ecosystem services, but they are warming about twice as fast as the global average. Climate change may reduce alpine snow cover and increase vegetation productivity, as in the Arctic. Here, we demonstrate that 77% of the European Alps above the tree line experienced greening (productivity gain) an

Lines and free line segments tangent to arbitrary three-dimensional convex polyhedra

SUE WHITESIDES∗ ∗ Abstract. Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in R 3 with a total of n edges consists of Θ(n 2) connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of k possibly intersecting convex polyhedra with a total of n edges admits, in the worst case, Θ(n 2 k 2) connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an O(n 2 k 2 log n) time and O(nk 2) space algorithm that, given a scene of k possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines. Key words. computational geometry, 3D visibility, visibility complex, visual event