Diane L. Souvaine

Planar minimally rigid graphs and pseudo-triangulations

Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than p). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide---to the best of our knowledge---the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.

On compatible triangulations of simple polygons

Abstract It is well known that, given two simple n -sided polygons, it may not be possible to triangulate the two polygons in a compatible fashion, if ones choice of triangulation vertices is restricted to polygon corners. Is it always possible to produce compatible triangulations if additional vertices inside the polygon are allowed? We give a positive answer and construct a pair of such triangulations with O( n 2 ) new triangulation vertices. Moreover, we show that there exists a ‘universal’ way of triangulating an n -sided polygon with O( n 2 ) extra triangulation vertices. Finally, we also show that creating compatible triangulations requires a quadratic number of extra vertices in the worst case.

Computational Geometry in a Curved World

We extend the results of straight-edged computational geometry into the curved world by defining a pair of new geometric objects, thesplinegon and thesplinehedron, as curved generalizations of the polygon and polyhedron. We identify three distinct techniques for extending polygon algorithms to splinegons: the carrier polygon approach, the bounding polygon approach, and the direct approach. By these methods, large groups of algorithms for polygons can be extended as a class to encompass these new objects. In general, if the original polygon algorithm has time complexityO(f(n)), the comparable splinegon algorithm has time complexity at worstO(Kf(n)) whereK represents a constant number of calls to members of a set of primitive procedures on individual curved edges. These techniques also apply to splinehedra. In addition to presenting the general methods, we state and prove a series of specific theorems. Problem areas include convex hull computation, diameter computation, intersection detection and computation, kernel computation, monotonicity testing, and monotone decomposition, among others.

Computing least median of squares regression lines and guided topological sweep

Abstract Given a set of data points pi = (xi, yi ) for 1 ≤ i ≤ n, the least median of squares regression line is a line y = ax + b for which the median of the squared residuals is a minimum over all choices of a and b. An algorithm is described that computes such a line in O(n 2) time and O(n) memory space, thus improving previous upper bounds on the problem. This algorithm is an application of a general method built on top of the topological sweep of line arrangements.

Time- and Space-Efficient Algorithms for Least Median of Squares Regression

Abstract The least median of squared residuals regression line (or LMS line) is that line y = ax + b for which the median of the residuals |yi - axi - b |2 is minimized over all choices of a and b. If we rephrase the traditional ordinary least squares (OLS) problem as finding the a and b that minimize the mean of | yi - axi - b |2, one can see that in a formal sense LMS just replaces a “mean” by a “median.” This way of describing LMS regression does not do justice to the remarkable properties of LMS. In fact, LMS regression behaves in ways that distinguish it greatly from OLS as well as from many other methods for robustifying OLS (see, e.g., Rousseeuw 1984). As illustrations given here show, the LMS regression line should provide a valuable tool for studying those data sets in which the usual linear model assumptions are violated by the presence of some (not too small) groups of data values that behave distinctly from the bulk of the data. This feature of LMS regression is illustrated by the fit given in...

An efficient algorithm for guard placement in polygons with holes

In this paper we consider the problem of placing guards to supervise an art gallery with holes. No gallery withn vertices andh holes requires more than [(n+h)/3] guards. For some galleries this number of guards is necessary. We present an algorithm which places the [(n+h)/3] guards inO(n2) time.

Efficient computation of location depth contours by methods of computational geometry

The concept of location depth was introduced as a way to extend the univariate notion of ranking to a bivariate configuration of data points. It has been used successfully for robust estimation, hypothesis testing, and graphical display. The depth contours form a collection of nested polygons, and the center of the deepest contour is called the Tukey median. The only available implemented algorithms for the depth contours and the Tukey median are slow, which limits their usefulness. In this paper we describe an optimal algorithm which computes all bivariate depth contours in O(n2) time and space, using topological sweep of the dual arrangement of lines. Once these contours are known, the location depth of any point can be computed in O(log2n) time with no additional preprocessing or in O(log n) time after O(n2) preprocessing. We provide fast implementations of these algorithms to allow their use in everyday statistical practice.

The Floodlight Problem

Given three angles summing to 2π, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of specified angles not exceeding π can be placed at n fixed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2π. We give O(nlog n) algorithms for both these problems.

An intuitive approach to measuring protein surface curvature.

A natural way to measure protein surface curvature is to generate the least squares fitted (LSF) sphere to a surface patch and use the radius as the curvature measure. While the concept is simple, the sphere‐fitting problem is not trivial and known means of protein surface curvature measurement use alternative schemes that are arguably less straightforward to interpret. We have developed an approach to solve the LSF sphere problem by turning the sphere‐fitting problem into a solvable plane‐fitting problem using a transformation known as geometric inversion. The approach works on any arbitrary surface patch, and returns a radius of curvature that has direct physical interpretation. Additionally, it is flexible in its ability to find the curvature of an arbitrary surface patch, and the “resolution” can be adjusted to highlight atomic features or larger features such as peptide binding sites. We include examples of applying the method to visualization of peptide recognition pockets and protein conformational change, as well as a comparison with a commonly used solid‐angle curvature method showing that the LSF method produces more pronounced curvature results. Proteins 2005.

Decomposition and intersection of simple splinegons

A splinegon is a polygon whose edges have been replaced by “well-behaved” curves. We show how to decompose a simple splinegon into a union of monotone pieces and into a union of differences of unions of convex pieces. We also show how to use a fast triangulation algorithm to test whether two given simple splinegons intersect. We conclude with examples of splinegons that make the extension of algorithms from polygons to splinegons difficult.