Computational Geometry

We study the problem of computing a planar curve, restricted to lie between two given polygonal chains, such that the integral of the square of arc-length derivative of curvature along the curve is minimized. We introduce the minimum variation B-spline problem, which is a linearly constrained optimization problem over curves, defined by B-spline functions only. An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.

In this paper, we introduce the fuzzy Voronoi diagram as an extension of the Voronoi diagram. We assume Voronoi sites to be fuzzy points and then define the Voronoi diagram for this kind of sites, then we provide an algorithm for computing this diagram based on Fortune's algorithm which costs O(nlogn) time. Also we introduce the fuzzy Voronoi diagram for a set of fuzzy circles, rather than fuzzy points, of the same radius. We prove that the boundary of this diagram is formed by the intersection of some hyperbolae, and finally we provide an O(n3logn)-time algorithm to compute the boundary.

... 870 JZC Lai Image and Vision Computing 15 (1997) 867871 Table 2 The execution time of enci ading a real image (tiffany) for 3 different algorithms' Number of codewords Available encoding algorithm (seconds) EAWFC (seconds) E c AWFS with = 3 (seconds) Speedup ...

This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of the "TRIADS of Metallic Means". The Metallic Means and their TRIADS can be geometrically substantiated, in an intriguing manner, as described herein.

We consider the problems of conflict detection and resolution in air traffic management (ATM) from the perspective of computational geometry and give algo-rithms for solving these problems efficiently. For con-flict resolution, we propose a simple method that can route ...

A wireless network consisting of large number of small sensors with low-power transceivers. These devices rely on battery power so that; improvement in the energy of these networks becomes important. Wireless sensor network (WSN) require various power management protocols to reduce the power consumption. In this paper we study that different number of clustering algorithm leads to different network performance on the energy balancing, energy consumption and network lifetime. We propose a Multi-hop Hierarchical Stable Election Protocol (MHSEP) for wireless sensor networks to enhance the network lifetime and avoid the formation of energy holes. Classical clustering protocols assume that all the nodes in a system are equipped with the equal amount of energy and as a result, network cannot take full advantage of the presence of node heterogeneity in system. Our proposed protocol is a heterogeneous-aware protocol to increase the time-interval before the death of first node (we refer to a...

Geometric computing is a rapidly evolving interdisciplinary field involving computer science, engineering and mathematics. It has relationships to many other areas within computer science such as computational geometry, computer-aided design, computer graphics, computer ...

We give algorithms to find shortest paths homotopic to given disjoint paths that wind amongst n point obstacles in the plane. Our deterministic algorithm runs in time O(k log n+n√n), and the randomized version in time O(k log n+n(log n)1+ε) where k is the input plus output sizes of the paths.

We provide two algorithms for computing the volume of the convex polytope Ω : = { x ∈ ℝ n + | Ax ≤ b }, for A , ∈ ℝ m × n , b ∈ ℝ n . The computational complexity of both algorithms is essentially described by n m , which makes them especially attractive for large n and relatively small m , when the other methods with O ( m n ) complexity fail. The methodology, which differs from previous existing methods, uses a Laplace transform technique that is well suited to the half-space representation of Ω.

Triangulation of the polygon is a fundamental algorithm in computational geometry. This paper considers techniques of object-oriented analysis and design as a new tool for solving and analyzing convex polygon triangulation. The triangulation is analyzed from three aspects: forward, reverse and round-trip engineering. We give a suggestion for improving the obtained software solution of the polygon triangulation algorithm using technique that combines UML modeling and Java programming.

Motivated by an application in computational geometry, we consider a novel variant of the problem of efficiently maintaining a forest of dynamic rooted trees. This variant includes an operation that merges two tree paths. In contrast to the standard problem, in which a single operation can only add or delete one arc, one merge can add and delete up to a linear number of arcs. In spite of this, we develop three different methods that need only polylogarithmic time per operation. The first method extends a solution of Farach and Thorup [1998] for the special case of paths. Each merge takes O (log 2 n ) amortized time on an n -node forest and each standard dynamic tree operation takes O (log n ) time; the latter bound is amortized, worst case, or randomized depending on the underlying data structure. For the special case that occurs in the motivating application, in which arbitrary arc deletions (cuts) do not occur, we give a method that takes O (log n ) time per operation, including m...

The structure of a large network (graph) can often be revealed by partitioning it into smaller and possibly more dense sub-networks that are easier to handle. One of such decompositions is based on “k-cores”, proposed in 1983 by Seidman. Together with connectivity components, cores are one among few concepts that provide efficient decompositions of large graphs and networks. In this

Visually impaired people face many problems in their day to day lives. Among them, outdoor navigation is one of the major concerns. The existing solutions based on Wireless Sensor Networks(WSN) and Global Positioning System (GPS) track ZigBee units or RFID (Radio Frequency Identification) tags fixed on the navigation system. The issues pertaining to these solutions are as follows: (1) It is suitable only when the visually impaired person is commuting in a familiar environment; (2) The device provides only a one way communication; (3) Most of these instruments are heavy and sometimes costly. Preferable solution would be to make a system which is easy to carry and cheap. The objective of this paper is to break down the technological barriers, and to propose a system by developing an Android App which would help a visually impaired person while traveling via the public transport system like Bus. The proposed system uses an inbuilt feature of smart phone such as GPS location tracker to track the location of the user and Text to Speech converter. The system also integrates Google Speech to Text converter for capturing the voice input and converts them to text. This system recommends the requirement of installing a GPS module in buses for real time tracking. With minor modification, this App can also help older people for independent navigation.

In this paper, we introduce an agent-based representation of games, in order to propose a compact representation for multi-party games in game theory. Our method is inspired by concepts in process theory and process algebra. In addition, we introduce an algorithm whose input is a game in the form of process algebra (proposed in this paper) and as an output, finds the Nash equilibrium of the game in linear space complexity.

The paper details computational mapping and modeling techniques from an ongoing design research project titled Irradiated Shade, which endeavors to develop and calibrate a computational toolset to uncover, represent, and design for the unseen dangers of ultraviolet radiation, a growing yet underexplored threat to cities, buildings, and the bodies that inhabit them. While increased shade in public spaces has been advocated as a strategy for "mitigation [of] climate change" (Kapelos and Patterson 2014), it is not a panacea to the threat. Even in apparent shade, the body is still exposed to harmful, ambient, or "scattered" UVB radiation. The study region is a binational metroplex, a territory in which significant atmospheric pollution and the effects of climate change (reduced cloud cover and more "still days" of stagnant air) amplify the "scatter" of ultraviolet wavelengths and UV exposure within shade, which exacerbates urban conditions of shade as an "index of inequality" (Bloch 2019) and threatens public health. Exposure to indirect radiation correlates to the amount of sky visible from the position of an observer (Gies and Mackay 2004). The overall size of a shade structure, as well as the design of openings along its sides, can greatly impact the UV protection factor (UPF) (Turnbull and Parisi 2005). Shade, therefore, is more complex than ubiquitous urban and architectural "sun" and "shadow studies" are capable of representing, as such analyses flatten the three-dimensional nature of radiation exposure and are "blind" to the ultraviolet spectrum. "Safe shade" is contingent on the nuances of the surrounding built environment, and designers must be empowered to observe and respond to a wider context than current representational tools allow.

The work presented below is part of the results of a research project at UPM.
The objective of the research consisted in the reconstruction of a selection of architectural projects chosen for both, its importance in the historical context and its geometry; so that the development of non-uniform rational Bsplines (NURBS) could be implemented as a tool for the graphical representation of architecture.

NURBS can represent complex geometries with precision (such as the hyperbolic paraboloid), either in the form of models or virtual models. Thus one can hypothesize and even complete information gaps -in the absence or loss of documentation-, of old or not constructed architectural projects.
During the communication it is intended to show the results achieved in one of the specific cases analyzed: the non-built market for Mexico in 1956, by Félix Candela.