Computing envelopes in dynamic geometry environments

We review the behavior of standard dynamic geometry software when computing envelopes, relating these approaches with the various definitions of envelope. Special attention is given to the recently released version of GeoGebra 5.0, that implements a recent parametric polynomial solving algorithm, allowing sound computations of envelopes of families of plane curves. Specific details on this novel approach are provided in this paper

A Parametric Approach to 3D Dynamic Geometry

Dynamic geometry systems are computer applications allowing the exact on-screen drawing of geometric diagrams and their interactive manipulation by mouse dragging. Whereas there exists an extensive list of 2D dynamic geometry environments, the number of 3D systems is reduced. Most of them, both in 2D and 3D, share a common approach, using numerical data to manage geometric knowledge and elementary methods to compute derived objects. This paper deals with a parametric approach for automatic management of 3D Euclidean constructions. An open source library, implementing the core functions in a 3D dynamic geometry system, is described here. The library deals with constructions by using symbolic parameters, thus enabling a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof. Basic functions are defined for symbolically checking the truth of statements. Using recent results from the theory of parametric polynomial systems solving, the bottleneck in the automatic determination of geometric loci and envelopes is solved. As far as we know, there is no comparable library in the 3D case, except the paramGeo3D library (designed for computing equations of simple 3D geometric objects, which, however, lacks specific functions for finding loci and envelopes)

Evaluation of the energy impact of PCM tiles in an Airport Terminal Departure hall

Copyright @ 2013 CIBSEIn most past studies, passive PCM (phase change materials) systems have been tested for relatively small office spaces where the airflow is of minimal consequence in the overall energy consumption of the space. This paper on the other hand, reports on the application of PCM tiles on the floor of an Airport terminal space, similar to London Heathrow Terminal 5 departure hall, where in such large open spaces, the influence of airflow is crucial for the evaluation of the energy performance of AC units. In this paper, the evaluation of the energy performance of PCM tiles is obtained through a coupled simulation of TRNSYS and CFD. TRNSYS simulates the AC unit and PID control systems, while CFD is used to simulate the airflow and radiation inside the terminal space. The phase change process is simulated in CFD using an in-house developed model which considers hysteresis effects and the non-linear enthalpy-temperature relationship of PCMs. Although, a displacement ventilation (DV) system is actually employed at Heathrow Terminal 5, this study also compares the performance of the PCM tiles for a mixed ventilation (MV) system. Due to large computing times associated with CFD, discrete time-dependent scenarios under different UK weather conditions are used. The yearly energy demand is then determined through the heating/cooling degree day concept using base temperatures of 18 and 23 °C for HDD and CDD, respectively, similar to the comfort temperature range in the Terminal. The results show that the use of PCM tiles on the floor of the Terminal departure hall can lead to annual energy savings of around 3% for the DV system and 6% for the MV system, corresponding to 174 MWh/year and 379 MWh/year for the Terminal building.This work was funded by the UK Engineering and Physical Sciences Research Council (EPSRC), Grant No: EP/H004181/1

A Singular web service for geometric computations

Outsourcing algebraic computations in dynamic geometry is a possible strategy used when software distribution constraints apply. Either if the target user machine has hardware limitations, or if the computer algebra system cannot be easily (or legally) packaged inside the geometric software, this approach can solve current shortcomings in dynamic environments. We report the design and implementation of a web service using Singular, a program specialized in ideal theory and commutative algebra. Besides its canonical address, a virtual appliance and a port to a low-cost ARM based computer are also provided. Any interactive geometric environment can then outsource computations where Singular is used, and incorporate their results into the system. In particular, we illustrate the capabilities of the web service by extending current abilities of GeoGebra to deal with algebraic loci and envelopes by means of a recent algorithm for studying parametric polynomial systems

Some issues on the automatic computation of plane envelopes in interactive environments

This paper addresses some concerns, and describes some proposals, on the elusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Gröbner Cover algorithm to study envelopes of families of algebraic curves, and we give a protocol toward its implementation in dynamic geometry environments. The proposal is illustrated through some examples. A beta version of GeoGebra is used to highlight new envelope abilities in interactive environments, and limitations of our approach are discussed, since the computations are performed in an algebraically closed field

Separation-Sensitive Collision Detection for Convex Objects

We develop a class of new kinetic data structures for collision detection between moving convex polytopes; the performance of these structures is sensitive to the separation of the polytopes during their motion. For two convex polygons in the plane, let $D$ be the maximum diameter of the polygons, and let $s$ be the minimum distance between them during their motion. Our separation certificate changes $O(\log(D/s))$ times when the relative motion of the two polygons is a translation along a straight line or convex curve, $O(\sqrt{D/s})$ for translation along an algebraic trajectory, and $O(D/s)$ for algebraic rigid motion (translation and rotation). Each certificate update is performed in $O(\log(D/s))$ time. Variants of these data structures are also shown that exhibit \emph{hysteresis}---after a separation certificate fails, the new certificate cannot fail again until the objects have moved by some constant fraction of their current separation. We can then bound the number of events by the combinatorial size of a certain cover of the motion path by balls.Comment: 10 pages, 8 figures; to appear in Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 1999; see also http://www.uiuc.edu/ph/www/jeffe/pubs/kollide.html ; v2 replaces submission with camera-ready versio