A Probabilistic Analysis of the Power of Arithmetic Filters

The assumption of real-number arithmetic, which is at the basis of conventional geometric algorithms, has been seriously challenged in recent years, since digital computers do not exhibit such capability. A geometric predicate usually consists of evaluating the sign of some algebraic expression. In most cases, rounded computations yield a reliable result, but sometimes rounded arithmetic introduces errors which may invalidate the algorithms. The rounded arithmetic may produce an incorrect result only if the exact absolute value of the algebraic expression is smaller than some (small) varepsilon, which represents the largest error that may arise in the evaluation of the expression. The threshold varepsilon depends on the structure of the expression and on the adopted computer arithmetic, assuming that the input operands are error-free. A pair (arithmetic engine,threshold) is an "arithmetic filter". In this paper we develop a general technique for assessing the efficacy of an arithmetic filter. The analysis consists of evaluating both the threshold and the probability of failure of the filter. To exemplify the approach, under the assumption that the input points be chosen randomly in a unit ball or unit cube with uniform density, we analyze the two important predicates "which-side" and "insphere". We show that the probability that the absolute values of the corresponding determinants be no larger than some positive value V, with emphasis on small V, is Theta(V) for the which-side predicate, while for the insphere predicate it is Theta(V^(2/3)) in dimension 1, O(sqrt(V)) in dimension 2, and O(sqrt(V) ln(1/V)) in higher dimensions. Constants are small, and are given in the paper.Comment: 22 pages 7 figures Results for in sphere test inproved in cs.CG/990702

Further Results on Arithmetic Filters for Geometric Predicates

An efficient technique to solve precision problems consists in using exact computations. For geometric predicates, using systematically expensive exact computations can be avoided by the use of filters. The predicate is first evaluated using rounding computations, and an error estimation gives a certificate of the validity of the result. In this note, we studies the statistical efficiency of filters for cosphericity predicate with an assumption of regular distribution of the points. We prove that the expected value of the polynomial corresponding to the in sphere test is greater than epsilon with probability O(epsilon log 1/epsilon) improving the results of a previous paper by the same authors.Comment: 7 pages 2 figures presented at the 15th European Workshop Comput. Geom., 113--116, 1999 improve previous results (in other paper

On the heating of source of the Orion KL hot core

We present images of the J=10-9 rotational lines of HC3N in the vibrationally excited levels 1v7, 1v6 and 1v5 of the hot core (HC) in Orion KL. The images show that the spatial distribution and the size emission from the 1v7 and 1v5 levels are different. While the J=10-9 1v7 line has a size of 4''x 6'' and peaks 1.1'' NE of the 3 mm continuum peak, the J=10--9 1v5 line emission is unresolved (<3'') and peaks 1.3'' south of the 3 mm peak. This is a clear indication that the HC is composed of condensations with very different temperatures (170 K for the 1v7 peak and $>230$ K for the 1v5 peak). The temperature derived from the 1v7 and 1v5 lines increases with the projected distance to the suspected main heating source I. Projection effects along the line of sight could explain the temperature gradient as produced by source I. However, the large luminosity required for source I, >5 10^5 Lsolar, to explain the 1v5 line suggests that external heating by this source may not dominate the heating of the HC. Simple model calculations of the vibrationally excited emission indicate that the HC can be internally heated by a source with a luminosity of 10^5 Lsolar, located 1.2'' SW of the 1v5 line peak (1.8'' south of source I). We also report the first detection of high-velocity gas from vibrationally excited HC3N emission. Based on excitation arguments we conclude that the main heating source is also driving the molecular outflow. We speculate that all the data presented in this letter and the IR images are consistent with a young massive protostar embedded in an edge-on disk.Comment: 13 pages, 3 figures, To be published in Ap.J. Letter

First results with the Gas Discharge Visualisation (GDV) method (Kirlian photography) to assess the inner Quality of apples

GDV technology can provide useful information for distinguishing healthy and stressed plants and, in some cases, it can provide useful information for distinguishing different varieties and rootstocks. It can also provide information for distinguishing fruits grown using different fertilization treatments. However, in our cases with fruit of very similar standard quality, we were not able to find complementary information to distinguish organically from conventionally grown apple fruit. Currently we are carrying out two larger and refined studies on just that topic with 5 conventional/ organic comparison pairs from harvests of two years