Article history: Received 18 August 2010 Accepted 18 March 2011 Available online xxxx Communicated by T.M. Chan

\‘c consider some problems in computattonal geometry when every one of the input points i\ movmg in a prescribed manner.

Article history: Received 21 January 2006 Received in revised form 1 January 2007 Accepted 2 February 2007 Available online 12 March 2007 Communicated by P.G. Spirakis

Article history: Received 11 February 2007 Received in revised form 12 February 2008 Accepted 20 May 2008 Communicated by F. Hurtado

We present a method for extracting contours from digital images using techniques from computational geometry. Our approach is different from traditional pixelbased methods in image processing. Instead of working directly with pixels, we extract a set of oriented feature points from the input digital images, then apply classical geometric techniques such as clustering, linking, and simplificatio...

We consider the notion of property testing as applied to computational geometry. We aim at developing efficient algorithms which determine whether a given (geometrical) object has a predetermined property Q or is “far” from any object having the property. We show that many basic geometric properties have very efficient testing algorithms, whose running time is significantly smaller than the obj...

We solve algorithmic geometrical problems in real 3-space or the real plane arising from applications in the area of cad, computer vision, and motion planning. The problems include parallel and central projection problems, shade and cast shadow problems, reconstruction of objects from images, oosets of objects, Voronoi diagrams of a nite families of objects, and collision of moving objects. Our...

In matrix computations, such as in factoring matrices, Hermitian and, preferably, positive definite elements are occasionally required. Related problems can often be cast as that of existence of respective elements in a matrix subspace. For two dimensional matrix subspaces, first results in this regard are due to Finsler. To assess positive definiteness in larger dimensional cases, the task bec...

We show how persistence can be used to solve a number of geometric problems where prepro-cessing is required to facilitate query answering. EEcient solutions for most of the problems discussed already exist in the literature; however, persistence provides an eecient and conceptually simpler alternative to existing solutions.

In this thesis we study a variety of problems in combinatorial and computational geometry, which deal with various aspects of arrangements of geometric objects, in the plane and in higher dimensions. Some of these problems have algorithmic applications, while others provide combinatorial bounds for various structures in such arrangements. The thesis involves two main themes: (i) Counting Crossi...