#Umbra vs penumbra full#

Nous exposons en premier lieu des résultats sur la taille de la silhouette d'un objet vu d'un point, c'est à dire sur la complexite de l'ensemble des droites ou segments tangents à l'objet et passant par le point.

#Umbra vs penumbra full#

These results prove that the umbra, which is bounded by arcs of conics,is intrinsically much more intricate than the full light/penumbra boundary whichis bounded by linesegments and whose worst-case complexity is in Ω(nα(k) +km +k2) and O(nα(k) + kmα(k) +k2), where m is the complexity of the polygonallight source.Ĭette thèse présente principalement des résultats sur la combinatoire des droites et segments qui apparaissent naturellement dans l'étude des problèmes de visibilité en trois dimensions. We also prove that, in the presence of kdisjoint polytopes of total complexity n, some of which being light sources,the umbra cast on a plane may have Ω(n2k3 +nk5) connected components and has complexity O(n3k3).These are the first bounds on the size of the umbra in terms of both k and n. In a scene consisting of a segment light source and kdisjoint polytopes of total complexity n, we prove an Ω(nk2+k4)lower bound on the maximum number of connected components of the umbra and a O(nk3) upper bound on its complexity. In particular, we show that a single segment light source may cast on a plane, in thepresence of two triangles, four connected components of umbra and that two fatconvex obstacles of total complexity n can engender Ω(n) connectedcomponents of umbra.

In this paper we prove various bounds on the complexity of the umbra andthe penumbra cast by a segment or polygonal light source on a plane in the presence ofpolygon or polytope obstacles.

While the common boundary of the penumbraand the full light is well understood, less is known about the boundary of theumbra. A point is in the umbra if it does not see any part of anylight source it is in full light if it sees entirely all the light sources otherwise, it is in the penumbra. Computing shadow boundaries is a difficult problem in the case of non-pointlight sources.