Geometry Package Research Articles

Interoperable Interactive Geometry for Europe: an introduction

Interactive or dynamic geometry is one of the three major software pillars of technology-based mathematics education, next to computer algebra systems (CAS) and spreadsheets. We can trace back the roots of this great tool for visualization and interactive manipulation to the work of Sutherland (1963), a system called Sketchpad which laid ground in computer science and human computer interaction. Since the introduction of Cabri-Geometre (Baulac, Bellemain and Laborde, 1988) and Geometers’ Sketchpad (Jackiw 1991), we could see several generations of such tools applied in mathematics education. During the last 20 years, interactive geometry software could establish a settled position in the mathematics education community, and for many researchers it is no longer a question whether to use these tools or not, but how to use them. However, what has convinced many in the academic field of mathematics education has still not become a standard in mathematics teaching in the classrooms (Intergeo Consortium 2008). While the lack of availability of computers in some schools or the cost of licensing might be reasons for the apparent underrepresentation, this cannot be the only reason. In particular, in Europe, most schools can offer access to networked computers, and there are several interactive geometry packages available free of cost. Also, there are thousands of web pages containing sketches, examples, activities and exercises—subsumed as ‘‘resources’’—using dynamic geometry. An initiative of Christian Mercat back in 2006 started what became the Intergeo project (Kortenkamp et al. 2009), funded through the eContentplus program of the European Union between October 2007 and 2010. This project intended to bring together teachers at all school levels from K-12 to university teaching and the wealth of resources available on the internet. By assembling the consortium and associate partners of the project from several major interactive geometry software producers from Europe (among them Cabri, Cinderella, GeoGebra, GEONExT, Geoplan-Geospace, TracenPoche, WIRIS and Z.u.L/C.a.R.), it was possible to include not only more than 3,000 resources contributing to the project, but the project could also strive for a common exchange format, a lingua franca for interactive geometry. A necessary condition for a true re-use of a resource lies in the possibility of using the resource in the environment familiar to the user regardless of the particular system used to create the resource. In addition, it was not clear how teachers can find exactly the content needed for his or her particular classroom situation. Helping people in this search for a specific content usually means adding to the resource itself metadata providing information about the resource. Then search engines can identify material based on keyword searches. The task is not trivial for mathematics and in particular geometry, as information cannot be extracted from a formula or graphical representation as easily as from a text. U. Kortenkamp (&) CERMAT, University of Education Karlsruhe, Postfach 11 10 62, 76060 Karlsruhe, Germany e-mail: kortenkamp@ph-karlsruhe.de; kortenkamp@cinderella.de

Mapping Pedagogical Opportunities Provided by Mathematics Analysis Software

This paper proposes a taxonomy of the pedagogical opportunities that are offered by mathematics analysis software such as computer algebra systems, graphics calculators, dynamic geometry or statistical packages. Mathematics analysis software is software for purposes such as calculating, drawing graphs and making accurate diagrams. However, its availability in classrooms also provides opportunities for positive changes to teaching and learning. Very many examples are documented in the professional and research literature, and in this paper we organize them into 10 types. These are displayed in the form of a ‘pedagogical map’, which further classifies them according to whether the opportunity arises from new opportunities for the mathematical tasks used, change to interpersonal aspects of the classroom or change to the point of view on mathematics as a subject. The map can be used in teacher professional development to draw attention to possibilities for lessons or as a catalyst for professional discussion. For research on teaching, it can be used to map current practice, or to track professional growth. The intention of the map is to summarise the potential benefits of teaching with technology in a form that may be useful for both teachers and researchers.

SU‐FF‐T‐433: Reducing the Simulation Time of a Prostate Cancer Clinical Setup by Two Orders of Magnitude

Purpose: Demonstrate the feasibility of utilizing the Fast Dose Calculator (FDC), a track repeating algorithm, for the simulation of a clinical proton radiotherapy setup including: beam nozzle with range shifter, aperture, range compensator and patient geometry. Methods and Materials: A database of discretized proton trajectories in water was generated with GEANT4. Information about the energy deposited, length and direction is stored for each trajectory step. Proton trajectories in any other material are extrapolated from the database by scaling step lengths and angles. Scaling parameters for various materials are obtained by comparing GEANT4 doses in the various materials to those obtained from FDC by scaling the trajectories in water. FDC allows for arbitrary geometry descriptions based on the geometry package of ROOT. The voxelized geometry from a CT‐scan of a patient being treated for prostate cancer was used to simulate the dose deposited by a passively scattered proton beam. A beam defining aperture and a range compensator were included in the simulation. This setup was used to calculate the dose distribution produced by a proton beam defined by the phase space at the front‐end of the aperture. In a second set of simulations a simplified version of a nozzle was included. In both cases the results were compared with a GEANT4 full simulation. Results: Doses calculated with FDC and GEANT4 were compared using the Gamma function. The comparison showed good agreement with less than 1% of the voxels with values above unity. FDC was found to be more than one hundred times faster than GEANT4. Conclusions: A fast track repeating algorithm has been tested in complex and realistic geometries for proton therapy. The algorithm preserves adequate accuracy and reduces calculation times by more than two orders of magnitude. Research Sponsored by the Gulf Coast Center for Computational Cancer Research.

TU-D-BRB-06: A Monte-Carlo Investigation of the Beam Quality Conversion Factor

Purpose: Clinical reference dosimetry based on absorbed dose standards requires the use of a beam quality conversion factor, kQ. This parameter is ionization chamber-specific and varies with beam quality. Current dosimetry protocols provide values of kQ for commonly used chambers (AAPM TG-51 (1999), IAEA TRS-398 (2000)). However, additional chambers are being used clinically for which values are not available. The aim is to obtain values of kQ for various ionization chambers over a range of photon beam quality using Monte Carle (MC) simulations. There is particular interest in obtaining values for those chambers which cannot be calculated using the TG-51 approach because of missing data, specifically for chambers with a relatively small collecting volume. Method and Materials: Direct MC calculations were made of the absorbed dose to water and the absorbed dose to gas in an ionization chamber at the reference depth in a water phantom. The simulations were performed using the EGSnrc user-codes cavity and egs_chamber to model the ionization chambers using the extended geometry package. The source input used a collimated point source from tabulated spectra for Cobalt-60 beams and various clinical linear accelerators. The chamber geometries were modeled using specifications from the various manufacturers' manuals. Values for seven of the eleven chambers that were simulated in this study are available in TG-51. Results: The calculations of values which are available in TG-51 had a maximum difference of 0.4% from the TG-51 values. The maximum relative uncertainty in these calculations was 0.2%. In cases where experimental results were available the calculations were compared with measurements. Values were obtained for the chambers not provided in TG-51. Conclusion: Direct calculations of kQ show agreement with those provided in TG-51. The simulations are being extended to cylindrical chambers in electron beams and to plane-parallel chambers in photon beams.

The Pierre Auger Observatory offline software

The Pierre Auger Observatory aims to discover the nature and origins of the highest energy cosmic rays. The large number of physicists involved in the project and the diversity of simulation and reconstruction tasks pose a challenge for the offline analysis software, not unlike the challenges confronting software for very large high energy physics experiments. Previously we have reported on the design and implementation of a general purpose but relatively lightweight framework which allows collaborators to contribute algorithms and sequencing instructions to build up the variety of applications they require. In this report, we update the status of this work and describe some of the successes and difficulties encountered over the last few years of use. We explain the machinery used to manage user contributions, to organize the abundance of configuration files, to facilitate multi-format file handling, and to provide access to event and time-dependent detector information residing in various data sources. We also describe the testing procedures used to help maintain stability of the code in the face of a large number of contributions. Foundation classes will also be discussed, including a novel geometry package which allows manipulation of abstract geometrical objects independent of coordinate system choice.

Geometry simulation and physics with the CMS forward pixel detector

The Forward Pixel Detector of CMS is an integral part of the Tracking system, which will play a key role in addressing the full physics potential of the collected data. It has a very complex geometry that encompasses multilayer structure of its detector modules. This presentation describes the development of geometry simulation for the Forward Pixel Detector. A new geometry package has been developed, which uses the detector description database (DDD) interface for the XML (eXtensive Markup Language) to GEANT simulation. This is necessary for digitization and GEANT4 reconstruction software for tracking. The expected physics performance is also discussed.

SU‐GG‐T‐320: Monte Carlo Generation of Phase Spaces for Dose Computation in TomoTherapy

Purpose: Generate phase spaces using Monte Carlo simulations for TomoTherapy® dose computation, measure accuracy of phase spaces, and present properties of phase spaces. Method and Materials: The PENELOPE 2006 Monte Carlo code system is used to generate phase spaces for the TomoTherapy® treatment device. The PENELOPE geometry package is used to model the head of the treatment device for each one of the three possible field sizes: 1 × 40 cm2, 2.5 × 40 cm2 and 5 × 40 cm2, and a phase‐space plane is positioned downstream from the collimating jaws of the device. Incident electrons are projected onto the linear accelerator's target assembly and Monte Carlo transport is performed through the head of the linear accelerator until particles cross the phase‐space plane. The incident electron energy distribution and the focal spot size are adjusted until the resulting dose profiles computed with a phase space match the TomoTherapy® open‐field gold standard profiles. Fluence maps, dose profiles and off‐axis spectra are presented. Results: A 5.5 MeV monoenergetic incident electron source produced optimal results. Computed dose profiles match the TomoTherapy® open‐field gold standard profiles within 2% / 1mm. The energy spectra, fluence maps and dose profiles exhibit the shape produced by a treatment device with no flattening filter, such as the TomoTherapy® device: minimal beam hardening along the central axis, minimal spectral shape change as a function of off‐axis angle, about 99.5% of the particles in the phase spaces are photons, and cone shape fluence maps and dose profiles in the lateral direction. Conclusion: Phase spaces for the three possible field sizes were generated. Dose computed from these phase spaces match TomoTherapy® gold standard dose profiles within 2% / 1mm and can be used for static or helical delivery dose computations.

Monte Carlo simulation of a realistic anatomical phantom described by triangle meshes: Application to prostate brachytherapy imaging

Purpose Monte Carlo codes can simulate the transport of radiation within matter with high accuracy and can be used to study medical applications of ionising radiations. The aim of our work was to develop a Monte Carlo code capable of generating projection images of the human body. In order to obtain clinically realistic images a detailed anthropomorphic phantom was prepared. These two simulation tools are intended to study the multiple applications of imaging in radiotherapy, from image guided treatments to portal imaging. Methods We adapted the general-purpose code PENELOPE 2006 to simulate a radiation source, an ideal digital detector, and a realistic model of the patient anatomy. The anthropomorphic phantom was developed using computer-aided design tools, and is based on the NCAT phantom. The surface of each organ is modelled using a closed triangle mesh, and the full phantom contains 330 organs and more than 5 million triangles. A novel object-oriented geometry package, which includes an octree structure to sort the triangles, has been developed to use this complex geometry with PENELOPE. Results As an example of the capabilities of the new code, projection images of the human pelvis region were simulated. Radioactive seeds were included inside the phantom’s prostate. Therefore, the resulting simulated images resemble what would be obtained in a clinical procedure to assess the positioning of the seeds in a prostate brachytherapy treatment. Conclusions The new code can produce projection images of the human body that are comparable to those obtained by a real imaging system (within the limitations of the anatomical phantom and the detector model). The simulated images can be used to study and optimise an imaging task (i.e., maximise the object detectability, minimise the delivered dose, find the optimum beam energy, etc.). Since PENELOPE can simulate radiation from 50 eV to 1 GeV, the code can also be used to simulate radiotherapy treatments and portal imaging. Using the octree data structure, the new geometry model does not significantly increase the computing time when compared to the simulation of a much simpler quadric geometry. In conclusion, we have shown that it is feasible to use PENELOPE and a complex triangle mesh geometry to simulate real medical physics applications.

The offline software framework of the Pierre Auger Observatory

The Pierre Auger Observatory is designed to unveil the nature and the origins of the highest energy cosmic rays. The large and geographically dispersed collaboration of physicists and the wide-ranging collection of simulation and reconstruction tasks pose some special challenges for the offline analysis software. We have designed and implemented a general purpose framework which allows collaborators to contribute algorithms and sequencing instructions to build up the variety of applications they require. The framework includes machinery to manage these user codes, to organize the abundance of user-contributed configuration files, to facilitate multi-format file handling, and to provide access to event and time-dependent detector information which can reside in various data sources. A number of utilities are also provided, including a novel geometry package which allows manipulation of abstract geometrical objects independent of coordinate system choice. The framework is implemented in C ++ , and takes advantage of object oriented design and common open source tools, while keeping the user side simple enough for C ++ novices to learn in a reasonable time. The distribution system incorporates unit and acceptance testing in order to support rapid development of both the core framework and contributed user code.

SU-FF-T-302: Monte Carlo Dose Verification and Application of Complex IOHDR Geometries

Purpose: Dose computation algorithm may lead to serious dose errors in intra-operative high dose rate (IOHDR) brachytherapy due to the lack of scatter from one side of the applicator and/or complex geometries of treated areas. Our group has reported the inaccuracy in clinical dose delivery for general cases through several experimental approaches. However, more accurate Monte Carlo calculations are needed to quantify dose errors in IOHDR delivered with more complex geometries and scattering environment. Method and Materials: Dose calculations were done using the EGSnrc Monte Carlo code system and the setup geometry including source structure was designed with the geometry package using EGSnrc C++ library class. User-code was developed from an existing EGSnrc user-code CAVRZnrc. To benchmark the code, a few steps were carried out to ensure that the accuracy of the cavity dose for a given simple geometry option and the ratio of cavity doses with the varying distance from the source to the center of cavity for different geometries. Based on accurate results of benchmarking geometries, test input files were developed to verify the accuracy of delivered doses for the measured known results that we had already reported the with linear array ion-chamber. Results: For the 1.0 cm prescription distance, Monte Carlo results showed about 14 % higher for added tissue equivalent materials of 15 cm. Also, calculated doses were 1.5 to 2 % higher than measured doses for added materials of above 5 cm. However, Monte Carlo showed an excellent agreement for 1.5 cm prescription distance and simulated uncertainties of 0.01 % to 0.1 %. Conclusion: Computed doses obtained using EGSnrc system for known geometries validate the results of ion-chamber measurement. This Monte Carlo calculation will improve the approach of dose calculations for complex geometries and the efficiency of dose delivery in clinical cases.

MO‐D‐M100J‐04: Exploiting Dosimetric Margins to Reduce IMRT Treated Volumes

Purpose: To develop an IMRT optimization strategy in which CTV‐to‐PTV margins are iteratively reduced until the dosimetric margin distribution (DMD) meets a 90% population coverage criterion (PCC), with coverage being the percent of patients for which the CTV minimum dose is 100% or more of the planned PTV minimum dose. Method and Materials: The DMD is the 3D margin distribution between the CTV and the treated volume (TV). Consistent with ICRU guidelines, the TV is the volume enclosed by the PTV minimum dose isodose surface. Optimization utilizes an iterative approach which converges on a DMD that exactly meets a 90% PCC. In each iteration, the DMD is obtained by exporting the CTV and TV as meshes, and calculating the margins between them via a computational geometry package. Coverage with respect to the PCC is computed based upon the DMD. In the next iteration, the PTV is re‐created using a CTV‐to‐PTV margin determined via linear interpolation from prior coverage estimates to move the plan towards the desired PCC. Intensities are then reoptimized, and coverage is re‐evaluated. Iterations continue until the 90% PCC is met. Results: The proposed algorithm was run for a prostate IMRT plan. For 2mm simulated random and systematic setup errors, the algorithm yielded 3mm CTV‐to‐PTV margins, less than the 6.4mm margin prescribed by the van Herk margin formula (VHMF). While the DMD maintained the desired 90% coverage, the TV was reduced by 63% relative to a VHMF CTV‐to‐PTV margin plan. Conclusion: To accommodate patient setup errors in a self‐consistent manner, it is feasible and desirable to generate IMRT plans in which the DMD meets a population coverage criterion. DMDs assure the desired PCC is achieved and can result in lower treated volumes compared with traditional margin approaches. (Work supported by NIH R01CA98524).

A brief review of GeoGebra: dynamic mathematics

There have always been two strands to elementary mathematics: algebra and geometry. Therefore it is perhaps not surprising that many computer packages have specifically targeted themselves at one strand. Computer algebra systems (CAS) are focused on manipulation of expressions and dynamic geometry packages concentrate on relationships between points, lines circles and so on. Perhaps the best known examples of dynamic geometry are The Geometer’s Sketchpad (http://www.dynamicgeometry. com/) and Cabri Geometry. (Cabri also has a rather splendid three dimensional version Cabri3D, and I have found version 2 invaluable for demonstrations in lectures.)

In search of more triangle centres. A source of classroom projects in Euclidean geometry

A point E inside a triangle ABC can be coordinatized by the areas of the triangles EBC, ECA, and EAB. These are called the barycentric coordinates of E. It can also be coordinatized using the six segments into which the cevians through E divide the sides of ABC, or the six angles into which the cevians through E divide the angles of ABC, or the six triangles into which the cevians through E divide ABC, etc. This article introduces several coordinate systems of these types, and investigates those centres of ABC whose coordinates, relative to a given coordinate system, are linear (or quasi-linear) with respect to appropriate elements of ABC, such as its side-lengths, its angles, etc. This results in grouping known centres into new families, and in discovering new centres. It also leads to unifying several results that are scattered in the literature, and creates several open questions that may be suitable for classroom discussions and team projects in which algebra and geometry packages are expected to be useful. These questions may also be used for Mathematical Olympiad training and may serve as supplementary material for students taking a course in Euclidean geometry.

Monte Carlo simulation of X‐ray emission using the general‐purpose code PENELOPE

AbstractPENELOPE is a general‐purpose Monte Carlo code system for the simulation of coupled electron and photon transport. It is applicable to all materials and covers the energy range from 50 eV up to 1 GeV. The code implements the most reliable cross sections and relaxation data available for the elements. Cross sections for molecules are generated by adding the atomic cross sections of the constituent atoms. The simulation of electron and positron histories is performed by using a mixed, class II, scheme; hard interactions are simulated individually, whereas the effect of soft interactions is described using multiple‐scattering approximations. The code includes a flexible geometry package for automatic tracking of particles in complex geometries consisting of homogeneous bodies limited by quadric surfaces. In this article, we describe briefly the interaction models and transport algorithms implemented in PENELOPE and present results of simulations of X‐ray generation by electron beams. Copyright © 2005 John Wiley & Sons, Ltd.

Evaluation of the doses to aircrew members taking into consideration the aircraft structures

The radiation field originated by the interactions of primary galactic cosmic rays in the atmosphere has been investigated using the FLUKA code. From the calculated differential fluence distributions in particle energy and direction, the values of effective and ambient dose equivalent rates have been derived as a function of vertical cut-off, altitude and deceleration potential. A mathematical model of an aircraft has been developed using the combinatorial geometry package of FLUKA. The irradiation of the mathematical model in the cosmic ray environment has been simulated at 10.7 km of altitude using the calculated fluence distributions at the polar region for a deceleration potential of 465 MV. Effective dose rate and ambient dose equivalent rate have been determined inside the aircraft at several locations along the fuselage. A significant shielding effect from the aircraft structures has been observed in the middle section of the passenger cabin.

Evaluation of the influence of aircraft shielding on the aircrew exposure through an aircraft mathematical model.

In order to investigate the influence of aircraft shielding on the galactic component of cosmic rays, an aircraft mathematical model has been developed by the combinatorial geometry package of the Monte-Carlo transport code FLUKA. The isotropic irradiation of the aircraft in the cosmic ray environment has been simulated. Effective dose and ambient dose equivalent rates have been determined inside the aircraft at several locations along the fuselage, at a typical civil aviation altitude (10 580 m), for vertical cut-off rigidity of 0.4 GV (poles) and 17.6 GV (equator) and deceleration potential of 465 MV. The values of both quantities were generally lower than those in the free atmosphere. They depend, in an intricate manner, on the location within the aircraft, quantity of fuel, number of passengers, etc. The position onboard of crew members should be taken into account when assessing individual doses. Likewise due consideration must be taken when positioning detectors which are used to measure H*(10). Care would be needed to avoid ambiguity when comparing the results of calculation with the experimental data.

Validation of the EGS usercode DOSE3D for internal beta dose calculation at the cellular and tissue levels

Internal radiotherapy is currently focusing on beta emitters such as 90Y or 131I because of their high-energy emissions. However, conventional dosimetric methods (MIRD) are known to be limited for such applications. They are unable to take into account microscopic radionuclide distribution because standardized anthropomorphic phantoms are used, and absorbed dose is calculated at the organ level. New tools are therefore required for dose assessment at cellular and tissue level (10–100 µm). The purpose of this study was to validate, at this scale, a Monte Carlo usercode (DOSE3D), based on the MORSE combinatorial geometry package and the EGS code system. Dose point-kernel calculations in water were compared to those published by Cross et al and Simpkin and Mackie. They confirm that DOSE3D is a reliable tool for cellular dosimetry in various geometric configurations.

The analog linear interpolation approach for Monte Carlo simulation of PGNAA: The CEARPGA code

The analog linear interpolation approach (ALI) has been developed and implemented to eliminate the big weight problem in the Monte Carlo simulation code CEARPGA. The CEARPGA code was previously developed to generate elemental library spectra for using the Monte Carlo – library least-squares (MCLLS) approach in prompt gamma-ray neutron activation analysis (PGNAA). In addition, some other improvements to this code have been introduced, including (1) adopting the latest photon cross-section data, (2) using an improved detector response function, (3) adding the neutron activation backgrounds, (4) generating the individual natural background libraries, (5) adding the tracking of annihilation photons from pair production interactions outside of the detector and (6) adopting a general geometry package. The simulated result from the new CEARPGA code is compared with those calculated from the previous CEARPGA code and the MCNP code and experimental data. The new CEARPGA code is found to give the best result.

The ROOT geometry package

Abstract The new ROOT (in: Proceeding of AIHENP’96, Lausanne, September 1996; http://root.cern.ch/ ) geometry package is a tool designed for building, browsing, tracking and visualizing a detector geometry. The code works standalone with respect to any tracking Monte-Carlo engine; therefore, it does not contain any constraints related to physics. However, the package defines a number of hooks in order to ease-up tracking, allowing user-defined objects to be attached to the basic architectural structures or superimposed on top of them. The modeling functionality was designed to optimize particle transport through complex geometries and it is accessible in an easy and transparent way. The package is currently under development and new features will be implemented, but the basic functionality has already been extensively tested on several detector geometries. The final goal is to be able to use the same geometry for several purposes, such as tracking, reconstruction or visualization, taking advantage of the ROOT features related to bookkeeping, I/O, histograming, browsing and graphical interfaces.

Development of SCINFUL-CG Code to Calculate Response Function of Hybrid Neutron Detectors Using Scintillators

A computer code, SCINFUL-CG, has been developed to calculate the response functions of a hybrid scintillation detector for neutrons. SCINFUL is a code used to calculate the response function of organic scintillator in simple geometry for neutrons of 0.1 to 80MeV. To expand the capability of this code, development of SCINFUL-CG was performed by introducing the geometry package MARS-CG, and the neutron transport cross-section data of silicon, oxygen and aluminum taken from the LA150 library. Validity of SCINFUL-CG was confirmed by comparing the calculated results with those by SCINFUL and MCNP-4B, and with the response functions measured in some neutron fields. From the results, it was concluded that SCINFUL-CG can be used for calculating the response functions and neutron spectra for the hybrid neutron detector.