Visualization of Geometric Spanner Algorithms

It is easier to understand an algorithm when it can be seen in interactive mode. The current study implemented four algorithms to construct geometric spanners; the path-greedy, gap-greedy, Theta-graph and Yao-graph algorithms. The data structure visualization framework (http://www.cs.usfca.edu/~galles/visualization/) developed by David Galles was used. Two features were added to allow its use in spanner algorithm visualization: support point-based algorithms and export of the output to Ipe drawing software format. The interactive animations in the framework make steps of visualization beautiful and media controls are available to manage the animations. Visualization does not require extensions to be installed on the web browser. It is available at http://cs.yazd.ac.ir/cgalg/AlgsVis/

Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Choosing Colors for Geometric Graphs via Color Space Embeddings

Graph drawing research traditionally focuses on producing geometric embeddings of graphs satisfying various aesthetic constraints. After the geometric embedding is specified, there is an additional step that is often overlooked or ignored: assigning display colors to the graph's vertices. We study the additional aesthetic criterion of assigning distinct colors to vertices of a geometric graph so that the colors assigned to adjacent vertices are as different from one another as possible. We formulate this as a problem involving perceptual metrics in color space and we develop algorithms for solving this problem by embedding the graph in color space. We also present an application of this work to a distributed load-balancing visualization problem.Comment: 12 pages, 4 figures. To appear at 14th Int. Symp. Graph Drawing, 200

Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

A geometric graph is angle-monotone if every pair of vertices has a path between them that---after some rotation---is $x$- and $y$-monotone. Angle-monotone graphs are $\sqrt 2$-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone---specifically, we prove that the half-$\theta_6$-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex $s$ to any vertex $t$ whose length is within $1 + \sqrt 2$ times the Euclidean distance from $s$ to $t$. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Experimental analysis of the accessibility of drawings with few segments

The visual complexity of a graph drawing is defined as the number of geometric objects needed to represent all its edges. In particular, one object may represent multiple edges, e.g., one needs only one line segment to draw two collinear incident edges. We study the question if drawings with few segments have a better aesthetic appeal and help the user to asses the underlying graph. We design an experiment that investigates two different graph types (trees and sparse graphs), three different layout algorithms for trees, and two different layout algorithms for sparse graphs. We asked the users to give an aesthetic ranking on the layouts and to perform a furthest-pair or shortest-path task on the drawings.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Computer Aspects of Solid Freeform Fabrication: Geometry, Process Control, and Design

Solid Freefonn Fabrication (SFF) is a class of manufacturing technologies aimed at the production of mechanical components without part-specific tooling or process planning. Originally used for creating modelsfor visualization, many industrial users of SFF technologies are realizing the greater potentialofSFF as legitimate manufacturing processes for producing patterns and, in some cases, functional.parts. Thus, SFF is becoming an important aspect of the product realization process in these industries. Solid Freefonn Fabrication arose from the dream of "push-button" prototyping, in which solid reproductions of three-dimensional geometric models are created automatically under computer control. Perhaps more than any other class of manufacturing technologies, computer software development has been an integral part of the emergence of SFF. As SFF technologies evolve toward the ability to create functional parts, computer issues gain more importance. This paper discusses three aspects of software design for SFF: processing of geometric data, global and local control of SFF processes, and computer-based analysis and design for SFF manufacturing. The discussion of geometric processing issues focuses on accuracy and completeness of input models, and the algorithms required to process such models. The interplay between the physics of SFF processing and the desired output geometry is discussed in terms of the development of model-based control algorithms for SFF. These two areas, geometric processing and control, are necessary for the practical implementation of any SFF technology. However, for SFF to realize its potential as an alternative for manufacturing functional parts, engineers must be provided with analysis and design tools for predicting mechanical properties, ensuring dimensional accuracy, choosing appropriate materials, selecting process parameter values, etc. For each of these three different but related areas of software design, the state-of-theart is assessed, contemporary research is summarized, and future needs are outlined.Mechanical Engineerin