Marco Pellegrini

Monte carlo approximation of form factors with error bounded a priori

The exchange of radiant energy (e.g., visible light, infrared radiation) in simple macroscopic physical models is sometimes approximated by the solution of a system of linear equations (energy transport equations). A variable in such a system represents the total energy emitted by a discrete surface element. The coefficients of these equations depend on the form factors between pairs of surface elements. A form factor is the fraction of energy leaving a surface element which directly reaches another surface element. Form factors depend only on the geometry of the physical model. Determining good approximations of form factors is the most time-consuming step in these methods, when the geometry of the model is complex due to occlusions.In this paper, we introduce a new characterization of form factors based on concepts from integral geometry. Using this characterization, we develop a new and asymptotically efficient Monte Carlo method for the simultaneous approximation of all form factors in an occluded polyhedral environment. The approximation error is bounded without recourse to special hypothesis. This algorithm is, for typical scenes, one order of magnitude faster than methods based on the hemisphere paradigm or on Monte Carlo ray-shooting.Let A be any set of convex nonintersecting polygons in R3 with a total of n edges and vertices. Let ε be the error parameter and let δ be the confidence parameter. We compute an approximation of each nonzero form factor such that with probability at least 1 - δ the absolute approximation error is less than ε. The expected running time of the algorithm is O((ε-2 log δ-1 )(n log2n + K log n)), where K is the expected number of regular intersections for a random projection of A. The number of regular intersections can range from 0 to quadratic in n, but for typical applications it is much smaller than quadratic. The expectation is with respect to the random choices of the algorithm and the result holds for any input.

On lines missing polyhedral sets in 3-space

AbstractWe show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:(1)Ann% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9vqFf0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVe0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaacI% cacaWGUbWaaWbaaSqabeaacaaIZaaaaOGaaGOmamaaCaaaleqabaGa% am4yamaakaaabaGaciiBaiaac+gacaGGNbGaamOBaaadbeaaaaGcca% GGPaaaaa!42F1!nn

Incidence and nearest-neighbor problems for lines in 3-space

O(n^3 2^{csqrt {log n} } )

On Collision-Free Placements of Simplices and the Closest Pair of Lines in 3-Space

n upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.(2)AnO(n1.5+ε) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ε>0 is an arbitrary small but fixed constant.(3)Ann% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9vqFf0x% c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVe0xe9Fve9% Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaacI% cacaWGUbWaaWbaaSqabeaacaaIZaaaaOGaaGOmamaaCaaaleqabaGa% am4yamaakaaabaGaciiBaiaac+gacaGGNbGaamOBaaadbeaaaaGcca% GGPaaaaa!42F1!nn

Repetitive Hidden Surface Removal for Polyhedra

O(n^3 2^{csqrt {log n} } )

Ray shooting and lines in space

n upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.(4)An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n4/3+ε+k1/3n1+ε+klog2n), wherek is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5].nThe tools used to obtain these results include Plücker coordinates of lines, random sampling, and polarity transformations in 3-space.