A Comparison of Three Curve Intersection Algorithms

An empirical comparison is made between three algorithms for computing the points of intersection of two planar Bezier curves. The algorithms compared are: the well known Bezier subdivision algorithm, which is discussed in Lane 80; a subdivision algorithm based on interval analysis due to Koparkar and Mudur; and an algorithm due to Sederberg, Anderson and Goldman which reduces the problem to one of finding the roots of a univariate polynomial. The details of these three algorithms are presented in their respective references

Heegaard diagrams and surgery descriptions for twisted face-pairing 3-manifolds

The twisted face-pairing construction of our earlier papers gives an efficient way of generating, mechanically and with little effort, myriads of relatively simple face-pairing descriptions of interesting closed 3-manifolds. The corresponding description in terms of surgery, or Dehn-filling, reveals the twist construction as a carefully organized surgery on a link. In this paper, we work out the relationship between the twisted face-pairing description of closed 3-manifolds and the more common descriptions by surgery and Heegaard diagrams. We show that all Heegaard diagrams have a natural decomposition into subdiagrams called Heegaard cylinders, each of which has a natural shape given by the ratio of two positive integers. We characterize the Heegaard diagrams arising naturally from a twisted face-pairing description as those whose Heegaard cylinders all have integral shape. This characterization allows us to use the Kirby calculus and standard tools of Heegaard theory to attack the problem of finding which closed, orientable 3-manifolds have a twisted face-pairing description.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-10.abs.htm

Droplet shapes on structured substrates and conformal invariance

We consider the finite-size scaling of equilibrium droplet shapes for fluid adsorption (at bulk two-phase co-existence) on heterogeneous substrates and also in wedge geometries in which only a finite domain $\Lambda_{A}$ of the substrate is completely wet. For three-dimensional systems with short-ranged forces we use renormalization group ideas to establish that both the shape of the droplet height and the height-height correlations can be understood from the conformal invariance of an appropriate operator. This allows us to predict the explicit scaling form of the droplet height for a number of different domain shapes. For systems with long-ranged forces, conformal invariance is not obeyed but the droplet shape is still shown to exhibit strong scaling behaviour. We argue that droplet formation in heterogeneous wedge geometries also shows a number of different scaling regimes depending on the range of the forces. The conformal invariance of the wedge droplet shape for short-ranged forces is shown explicitly.Comment: 20 pages, 7 figures. (Submitted to J.Phys.:Cond.Mat.

Annual league tables of mortality in neonatal intensive care units: longitudinal study. International Neonatal Network and the Scottish Neonatal Consultants and Nurses Collaborative Study Group.[see comment]

OBJECTIVE: To assess whether crude league tables of mortality and league tables of risk adjusted mortality accurately reflect the performance of hospitals. DESIGN: Longitudinal study of mortality occurring in hospital. SETTING: 9 neonatal intensive care units in the United Kingdom. SUBJECTS: 2671 very low birth weight or preterm infants admitted to neonatal intensive care units between 1988 and 1994. MAIN OUTCOME MEASURES: Crude hospital mortality and hospital mortality adjusted using the clinical risk index for babies (CRIB) score. RESULTS: Hospitals had wide and overlapping confidence intervals when ranked by mortality in annual league tables; this made it impossible to discriminate between hospitals reliably. In most years there was no significant difference between hospitals, only random variation. The apparent performance of individual hospitals fluctuated substantially from year to year. CONCLUSIONS: Annual league tables are not reliable indicators of performance or best practice; they do not reflect consistent differences between hospitals. Any action prompted by the annual league tables would have been equally likely to have been beneficial, detrimental, or irrelevant. Mortality should be compared between groups of hospitals using specific criteria-such as differences in the volume of patients, staffing policy, training of staff, or aspects of clinical practice-after adjusting for risk. This will produce more reliable estimates with narrower confidence intervals, and more reliable and rapid conclusions