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Concept# Rectangle

Summary

In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as .
The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle).
A crossed rectangle is a crossed (self-intersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals (therefore only two sides are parallel). It is a special case of an antiparallelogram, and its angles are not right angles and not all equal, though opposite angles are equal. Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles

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It is an old problem of Danzer and Rogers to decide whether it is possible arrange O(1/epsilon) points in the unit square so that every rectangle of area epsilon contains at least one of them. We show that the answer to this question is in the negative if we slightly relax the notion of rectangles, as follows. Let delta be a fixed small positive number. A quasi-rectangle is a region swept out by a continuously moving segment s, with no rotation, so that throughout the motion the angle between the trajectory of the center of s and its normal vector remains at most delta. We show that the smallest number of points needed to pierce all quasi-rectangles of area e is Theta (1/epsilon log 1/epsilon).

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms.

The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs defined by geometric representations and competitivity analysis of on-line algorithms. This connection became apparent after the recent construction of triangle-free geometric intersection graphs with arbitrarily large chromatic number due to Pawlik et al. We show that any on-line graph coloring problem gives rise to a class of game graphs, which in many cases have a natural representation by geometric objects. As a consequence, problems of estimating the chromatic number of graphs with geometric representations are reduced to finding on-line coloring algorithms that use few colors or proving that such algorithms do not exist. We use this framework to derive upper and lower bounds on the maximum possible chromatic number in terms of the clique number in the following classes of graphs: rectangle overlap graphs, subtree overlap graphs and interval filament graphs. These graphs generalize interval overlap graphs (also known as circle graphs)-a well-studied class of graphs with chromatic number bounded by a function of the clique number. Our bounds are absolute for interval filament graphs and asymptotic of the form (log log n)(f(w)) for rectangle and subtree overlap graphs. In particular, we provide the first construction of geometric intersection graphs with bounded clique number and with chromatic number asymptotically greater than log log n. Moreover, with some additional assumptions on the geometric representation, the bounds obtained are tight.