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At least 451 records · Page 25
All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

Aircraft geometry verification with enhanced computer generated displays

A method for visual verification of aerodynamic geometries using computer generated, color shaded images is described. The mathematical models representing aircraft geometries are created for use in theoretical aerodynamic analyses and in computer aided manufacturing. The aerodynamic shapes are defined using parametric bi-cubic splined patches. This mathematical representation is then used as input to an algorithm that generates a color shaded image of the geometry. A discussion of the techniques used in the mathematical representation of the geometry and in the rendering of the color shaded display is presented. The results include examples of color shaded displays, which are contrasted with wire frame type displays. The examples also show the use of mapped surface pressures in terms of color shaded images of V/STOL fighter/attack aircraft and advanced turboprop aircraft.

Cozzolongo, J. V.↗
DOE OSTI2023

EDX Spatial: Leveraging cloud and hybrid data management resources for spatial data

In the last few years, the National Energy Technology Laboratory has started leveraging cloud-hosted services for hosting spatial data collections published on the Energy Data eXchange (EDX). Using cloud-hosted storage and compute options offers multiple benefits for visualization and tool development through utilization of spatial data resources. The transition from on-premises services to cloud-hosted services has enabled a few key features: increased accessibility of large derivative datasets, dynamic integration of external authoritative data resources directly from outside entities through representational state transfer application programming interfaces (REST API), and the ability to produce complex mapping applications, dashboards, and online maps. As a result of the shift, NETL has launched EDX Spatial, a platform that leverages on-premises resources combined with cloud compute capabilities to enable enhanced online mapping interfaces and optimize data access. In addition, a unified workflow for handling the public release of spatial data products through EDX has been developed, including standardization of data hosting practices, metadata, symbology, and application elements. This poster reviews the opportunity of leveraging cloud-hosted and hybrid data management solutions for spatial data, and discusses the benefits and lessons learned while leveraging these services through the data repository EDX.

Morkner, Paige↗
NASA NTRS1996

Neurofilament protein is differentially distributed in subpopulations of corticocortical projection neurons in the macaque monkey visual pathways

Previous studies of the primate cerebral cortex have shown that neurofilament protein is present in pyramidal neuron subpopulations displaying specific regional and laminar distribution patterns. In order to characterize further the neurochemical phenotype of the neurons furnishing feedforward and feedback pathways in the visual cortex of the macaque monkey, we performed an analysis of the distribution of neurofilament protein in corticocortical projection neurons in areas V1, V2, V3, V3A, V4, and MT. Injections of the retrogradely transported dyes Fast Blue and Diamidino Yellow were placed within areas V4 and MT, or in areas V1 and V2, in 14 adult rhesus monkeys, and the brains of these animals were processed for immunohistochemistry with an antibody to nonphosphorylated epitopes of the medium and heavy molecular weight subunits of the neurofilament protein. Overall, there was a higher proportion of neurons projecting from areas V1, V2, V3, and V3A to area MT that were neurofilament protein-immunoreactive (57-100%), than to area V4 (25-36%). In contrast, feedback projections from areas MT, V4, and V3 exhibited a more consistent proportion of neurofilament protein-containing neurons (70-80%), regardless of their target areas (V1 or V2). In addition, the vast majority of feedback neurons projecting to areas V1 and V2 were located in layers V and VI in areas V4 and MT, while they were observed in both supragranular and infragranular layers in area V3. The laminar distribution of feedforward projecting neurons was heterogeneous. In area V1, Meynert and layer IVB cells were found to project to area MT, while neurons projecting to area V4 were particularly dense in layer III within the foveal representation. In area V2, almost all neurons projecting to areas MT or V4 were located in layer III, whereas they were found in both layers II-III and V-VI in areas V3 and V3A. These results suggest that neurofilament protein identifies particular subpopulations of corticocortically projecting neurons with distinct regional and laminar distribution in the monkey visual system. It is possible that the preferential distribution of neurofilament protein within feedforward connections to area MT and all feedback projections is related to other distinctive properties of these corticocortical projection neurons.

Non-NASA Center↗
DOE OSTI2026

Memory-Aware External Facelist Calculation: A Data-Parallel Atomic Hash Counting Approach

Unstructured volumetric meshes serve as fundamental data representations in various scientific simulations and analyses. They play a crucial role in representing complex computational domains and are essential for important numerical techniques, such as finite element analysis. Whenever such a mesh is read from a file, streamed in-situ, or generated by algorithms, scientific visualization libraries rely on calculating the external surface of a geometry, named “external facelist”, to produce a polygonal mesh for rendering. Consequently, external facelist calculation has become one of the most widely used algorithms in the scientific visualization domain, necessitating optimal performance. In this paper, we explore relevant work on external facelist calculation algorithms in two common visualization libraries, VTK and Viskores, assess their performance and memory constraints, and introduce a novel memory-aware external facelist calculation algorithm employing an atomic hash counting approach. This algorithm fully leverages Viskores' data-parallel primitive operations, facilitating its execution across diverse many-core architectures. Our algorithm features the lowest memory footprint on the GPU and the second-lowest on the CPU among all evaluated methods, and it also delivers the fastest performance on both CPU and GPU. It has been made available under an open-source license in the VTK and Viskores visualization systems.

Tsalikis, Spiros [Kitware] (ORCID:0000000151137195↗
DOE OSTI2022

Unsupervised Machine Learning for Exploratory Data Analysis of Exoplanet Transmission Spectra

Abstract Transit spectroscopy is a powerful tool for decoding the chemical compositions of the atmospheres of extrasolar planets. In this paper, we focus on unsupervised techniques for analyzing spectral data from transiting exoplanets. After cleaning and validating the data, we demonstrate methods for: (i) initial exploratory data analysis, based on summary statistics (estimates of location and variability); (ii) exploring and quantifying the existing correlations in the data; (iii) preprocessing and linearly transforming the data to its principal components; (iv) dimensionality reduction and manifold learning; (v) clustering and anomaly detection; and (vi) visualization and interpretation of the data. To illustrate the proposed unsupervised methodology, we use a well-known public benchmark data set of synthetic transit spectra. We show that there is a high degree of correlation in the spectral data, which calls for appropriate low-dimensional representations. We explore a number of different techniques for such dimensionality reduction and identify several suitable options in terms of summary statistics, principal components, etc. We uncover interesting structures in the principal component basis, namely well-defined branches corresponding to different chemical regimes of the underlying atmospheres. We demonstrate that those branches can be successfully recovered with a K-means clustering algorithm in a fully unsupervised fashion. We advocate for lower-dimensional representations of the spectroscopic data in terms of the main principal components, in order to reveal the existing structure in the data and quickly characterize the chemical class of a planet.

Matchev, Konstantin T. (ORCID:0000000341829096)↗
NASA NTRS1977

Full color hybrid display for aircraft simulators

A full spectrum color monitor, connected to the camera and lens system of a television camera supported by a gantry frame over a terrain model simulating an aircraft landing zone, projects the monitor image onto a lens or screen visually accessible to a trainee in the simulator. A digital computer produces a pattern corresponding to the lights associated with the landing strip onto a monochromatic display, and an optical system projects the calligraphic image onto the same lens so that it is superposed on the video representation of the landing field. The optical system includes a four-color wheel which is rotated between the calligraphic display and the lens, and an apparatus for synchronizing the generation of a calligraphic pattern with the color segments on the color wheel. A servo feedback system responsive to the servo motors on the gantry frame produces an input to the computer so that the calligraphically generated signal corresponds in shape, size and location to the video signal.

Chase, W. D.↗
NASA NTRS2019

Orbital Debris Ontology, Terminology, and Knowledge Modeling

The looming threat orbital debris poses to assets in orbit demands solutions. As the orbital population grows, so does this hazard, but so does the sea of data. The problem is also an opportunity for interdisciplinary innovation and cooperation. This paper focuses on the data and information management aspect of developing solutions for a sustainable and safe orbital space environment. The corresponding author’s in-progress work to develop an orbital debris domain ontology is summarized in order to discuss knowledge modeling for this domain. Methodological approaches of this effort can also contribute to standards efforts and address terminological and policy questions. Leveraging the growing volumes of orbital debris and space situational awareness (SSA) data will create a more complete picture of the orbital space environment. Part of the solution will be: consistent and correct data interpretation, sharing orbital debris and SSA data in one form or another, terminology development & harmonization, and knowledge or domain modeling. To facilitate this, [Rovetto, 2015/16] discussed ontology development for the orbital debris domain. This paper lists concepts from that paper, and subsequently developed concepts [2-9]. Ontology engineering is an interdisciplinary field related to knowledge representation and reasoning in artificial intelligence, semantic technologies and the so-called semantic web. An ontology is effectively a computable and semantically rich terminology that presents a knowledge or domain model for a topic area. Expressions of knowledge or assertions are stored using formally defined term. This knowledge base is reasoned over to yield answers to queries, among other things. Ontologies have been developed in knowledge-based projects across various disciplines, and used for such things as search engines, chatbots, enterprise knowledge graphs, etc. Ontologies support: interoperability, automated reasoning, data sharing and integration, data search and retrieval, and communicating the meaning of data. The Orbital Debris Ontology (ODO), and related ontologies [Rovetto & Kelso 2016] [Rovetto 2016, 2017], were proposed to help achieve this. ODO, for instance, is intended as a domain ontology that can be used across federated databases, offering an explicitly specified set of concepts describing the orbital debris domain. Its meaning-rich taxonomy will provide a sharable semantics for orbital debris data to, in part, consistently communicate the meaning of data to both humans and machines, and tag data elements in space object catalogs to help afford inference tasks, decision support, knowledge discovery, and information integration. ODO and the SSA ontology (SSAO) is part of the overall Orbital Space Domain Ontology concept, which is conceived as a broader domain reference ontology. It aims to provide a knowledge representation structure of the orbital space environment, a common semantic model, and develop a sharable terminology. Collectively this will provide common meaning for datasets, a high-level taxonomy or classification for orbital space objects, and thus means to characterize space objects. Ongoing efforts have included using visualizations, R, JSON-LD, and contemporary semantic technologies. Potential applications and interdisciplinary partnerships include web-based platforms, web apps, visualizations, and academia projects. Community input and participation may yield a more widely understood domain model as well as facilitate terminological standards. For example, the proposed conceptual, terminological and ontological analysis may contribute to such efforts as the Space Debris Mitigation Requirements in the International Standards Organization by developing more precise, consistent and coherent terms and definitions. Projects that seek to develop in-house ontologies can use ODO and related ontologies as domain reference ontologies. This paper was developed independent of author affiliations. Readers are encouraged to contact corresponding author(1) with general interest and potential opportunities to support or realize the described project.

Robert J. Rovetto↗
DOE OSTI2019

Structural Study of Analogues of Titan’s Haze by Trapped Ion Mobility Coupled with a Fourier Transform Ion Cyclotron Mass Spectrometer

The aerosols present in the atmosphere of the Saturn’s moon Titan are of particular planetary science interest and several spacecraft missions are already allowed to gather spectroscopic data. Titan haze’s analogs, so-called tholins, were produced on earth to push forward the comprehension of their formation and properties. In this study, this highly complex mixture was analyzed here for the first time by trapped ion mobility spectrometry coupled to ultra-high resolution mass spectrometry (FTICR MS). Electrospray ionization revealed the characteristic CHN{sub x}-class components, with CHN{sub 5–6} and DBE 6–7 most abundant. Deploying specialized visualization, enabled by accurate mass measurements and elemental composition assignments, the adapted Kendrick mass defect analysis highlights the C{sub 2}H{sub 3}N homolog series, whereas the nitrogen-modified van Krevelen diagram exhibits a clear trend towards H/C 1.5 and N/C 0.5. More interestingly, the representation of m/z versus collision cross section (CCS) allowed hypothesizing a ramified N-PAH structural motif. State-of-the-art IMS is currently not able to resolve the isomeric continuum of ultra-complex mixtures; thus, peak parameters other than the CCS value are explored. As such, analyzing the mobility peak width versus m/z shows a linear increase in isomeric diversity between m/z 170 and 350 and a near plateau in diversity at higher m/z for the N-PAH-like structure. Due to the high complexity of the sample, these structural insights are only to be revealed by TIMS-FTICR MS. .

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2021

Capturing the Physics of MaNGA Galaxies with Self-supervised Machine Learning

As available data sets grow in size and complexity, advanced visualization tools enabling their exploration and analysis become more important. In modern astronomy, integral field spectroscopic galaxy surveys are a clear example of increasing high dimensionality and complex data sets, which challenges the traditional methods used to extract the physical information they contain. Here, we present the use of a novel self-supervised machine-learning method to visualize the multidimensional information on stellar population and kinematics in the MaNGA survey in a 2D plane. Our framework is insensitive to nonphysical properties such as the size of the integral field unit and is therefore able to order galaxies according to their resolved physical properties. Using the extracted representations, we study how galaxies distribute based on their resolved and global physical properties. We show that even when exclusively using information about the internal structure, galaxies naturally cluster into two well-known categories, rotating main-sequence disks and massive slow rotators, from a purely data-driven perspective, hence confirming distinct assembly channels. Low-mass rotation-dominated quenched galaxies appear as a third cluster only if information about the integrated physical properties is preserved, suggesting a mixture of assembly processes for these galaxies without any particular signature in their internal kinematics that distinguishes them from the two main groups. The framework for data exploration is publicly released with this publication, ready to be used with the MaNGA or other integral field data sets.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1982

Aircraft geometry verification with enhanced computer-generated displays

A method for visual verification of aerodynamic geometries using computer-generated, color-shaded images is described. The mathematical models representing aircraft geometries are created for use in theoretical aerodynamic analyses and in computer-aided manufacturing. The aerodynamic shapes are defined using parametric bi-cubic splined patches. This mathematical representation is then used as input to an algorithm that generates a color-shaded image of the geometry. A discussion of the techniques used in the mathematical representation of the geometry and in the rendering of the color-shaded display is presented. The results include examples of color-shaded displays, which are contrasted with wire-frame-type displays. The examples also show the use of mapped surface pressures in terms of color-shaded images of V/STOL fighter/attack aircraft and advanced turboprop aircraft.

Cozzolongo, J. V.↗
NASA NTRS1986

Visual and motion cueing in helicopter simulation

The visual cues presented in the simulator are compared with those of flight in an attempt to identify deficiencies. For the low-amplitude maneuvering tasks normally associated with the hover mode, the unique motion capabilities of the Vertical Motion Simulator (VMS) at Ames Research Center permit nearly a full representation of vehicle motion. Especially appreciated in these tasks are the vertical-acceleration responses to collective control. For larger-amplitude maneuvering, motion fidelity must suffer diminution through direct attenuation through high-pass filtering washout of the computer cockpit accelerations or both. Experiments were conducted in an attempt to determine the effects of these distortions on pilot performance of height-control tasks.

Bray, Richard S.↗
DOE OSTI2026

Viskores: Integrating Parallel Scientific Visualization Research into Applications

Viskores is a scientific visualization library that is the primary deployment of such algorithms to the parallel accelerated processors of modern DOE supercomputers. In this paper, we review the capabilities provided by Viskores and how these capabilities are leveraged by other software in the high-performance computing ecosystem. We discuss the Viskores data representation and pay particular attention to array management. Through this array management we describe how data is adapted between Viskores and other software along with strategies for converting dynamic, polymorphic objects to static representations better suited to GPU processing. We conclude with several examples of Viskores integrating with high-performance software that is used in production today.

Moreland, Ken [ORNL] (ORCID:0000000270513288)↗
NASA NTRS1987

NAECON 87; Proceedings of the IEEE National Aerospace and Electronics Conference, Dayton, OH, May 18-22, 1987. Volumes 1, 2, 3, & 4

The present conference discusses topics in VLSI components and their packaging, signal processing, uses of cartographic data, data transmission, advanced avionics architectures, fiber-optics, information control and display, image processing, airborne radar and fire control, navigation, air data, Kalman filtering, power generation and control, spacecraft power structures, aircraft flying qualities, flight management, fault-tolerant computer architectures, actuation technologies, self-repairing flight control system technology, multivariable control, stability and control methods, and AFTI/F-16 flight test reports. Also discussed are the ADA/JOVIAL language and its applications, software acquisition and testing, advanced software concepts, software management, computer graphics and visual systems softwear, ADA in embedded avionics, 16- and 32-bit architectures, voice interaction applications, human/machine systems analysis, human factors and AI, mental workloads and displays, pilot acceleration protection research, communications system technology, space communications, reliability and maintainability, managerial techniques, engineering management, EM compatibility and nuclear hardening, expert systems, AI language/knowledge representation, expert system implementation, machine vision/optical processing, and advanced AI concepts and architectures.

Avionics↗
NASA NTRS2000

Numerical Zooming Between a NPSS Engine System Simulation and a One-Dimensional High Compressor Analysis Code

Within NASA's High Performance Computing and Communication (HPCC) program, NASA Glenn Research Center is developing an environment for the analysis/design of aircraft engines called the Numerical Propulsion System Simulation (NPSS). NPSS focuses on the integration of multiple disciplines such as aerodynamics, structures, and heat transfer along with the concept of numerical zooming between zero-dimensional to one-, two-, and three-dimensional component engine codes. In addition, the NPSS is refining the computing and communication technologies necessary to capture complex physical processes in a timely and cost-effective manner. The vision for NPSS is to create a "numerical test cell" enabling full engine simulations overnight on cost-effective computing platforms. Of the different technology areas that contribute to the development of the NPSS Environment, the subject of this paper is a discussion on numerical zooming between a NPSS engine simulation and higher fidelity representations of the engine components (fan, compressor, burner, turbines, etc.). What follows is a description of successfully zooming one-dimensional (row-by-row) high-pressure compressor analysis results back to a zero-dimensional NPSS engine simulation and a discussion of the results illustrated using an advanced data visualization tool. This type of high fidelity system-level analysis, made possible by the zooming capability of the NPSS, will greatly improve the capability of the engine system simulation and increase the level of virtual test conducted prior to committing the design to hardware.

Follen, Gregory↗
NASA NTRS2026

Proactive Wildfire Management: A Remote Sensing and Multimodal CNN-MLP Architecture for Ignition Risk Forecasting

As the frequency and intensity of wildfires increase, with fire seasons now starting earlier and ending later than they have over the past decades, current monitoring systems, such as lookout towers and satellites, are hindered by cloud cover, low-resolution imagery, and static data gaps that fail to track vegetation moisture levels fast enough to catch rapid pre-ignition changes. This report proposes a Machine Learning-enabled Wildfire Ignition Prediction framework that combines satellite monitoring with dynamic and high-resolution remote sensing from Unmanned Aerial Vehicle (UAV) swarms. The method would use multispectral and thermal data from the Landsat program to create a baseline for vegetation health, calculating a two-band Enhanced Vegetation Index (EVI2) and the moisture content of the vegetation. These inputs will later be fused with microscale UAV weather data, including thermal hotspots found through thick canopies, hyperspectral chemical signatures of pre-visual combustion, and local weather streams. The multispectral satellite, multispectral Light Detection and Ranging (LiDAR), and thermal data would then be processed through a Convolutional Neural Network (CNN), alongside a Multilayer Perceptron (MLP) for the micro-weather telemetry. The outputs of these networks would be fused into a single feature representation and passed through a final prediction network to generate real-time ignition risk scores and hotspot alerts. Model performance would be assessed using standard classification metrics, including a Receiver Operating Characteristic - Area Under the Curve (ROC AUC) and F1 score. This system would allow first responders to identify high-risk zones and intervene before ignition occurs, improving emergency response time compared to current approaches.

machine learning↗
NASA NTRS2012

QuakeSim 2.0

QuakeSim 2.0 improves understanding of earthquake processes by providing modeling tools and integrating model applications and various heterogeneous data sources within a Web services environment. QuakeSim is a multisource, synergistic, data-intensive environment for modeling the behavior of earthquake faults individually, and as part of complex interacting systems. Remotely sensed geodetic data products may be explored, compared with faults and landscape features, mined by pattern analysis applications, and integrated with models and pattern analysis applications in a rich Web-based and visualization environment. Integration of heterogeneous data products with pattern informatics tools enables efficient development of models. Federated database components and visualization tools allow rapid exploration of large datasets, while pattern informatics enables identification of subtle, but important, features in large data sets. QuakeSim is valuable for earthquake investigations and modeling in its current state, and also serves as a prototype and nucleus for broader systems under development. The framework provides access to physics-based simulation tools that model the earthquake cycle and related crustal deformation. Spaceborne GPS and Inter ferometric Synthetic Aperture (InSAR) data provide information on near-term crustal deformation, while paleoseismic geologic data provide longerterm information on earthquake fault processes. These data sources are integrated into QuakeSim's QuakeTables database system, and are accessible by users or various model applications. UAVSAR repeat pass interferometry data products are added to the QuakeTables database, and are available through a browseable map interface or Representational State Transfer (REST) interfaces. Model applications can retrieve data from Quake Tables, or from third-party GPS velocity data services; alternatively, users can manually input parameters into the models. Pattern analysis of GPS and seismicity data has proved useful for mid-term forecasting of earthquakes, and for detecting subtle changes in crustal deformation. The GPS time series analysis has also proved useful as a data-quality tool, enabling the discovery of station anomalies and data processing and distribution errors. Improved visualization tools enable more efficient data exploration and understanding. Tools provide flexibility to science users for exploring data in new ways through download links, but also facilitate standard, intuitive, and routine uses for science users and end users such as emergency responders.

Donnellan, Andrea↗
NASA NTRS1992

Spatial orientation of the vestibular system

1. A simplified three-dimensional state space model of visual vestibular interaction was formulated. Matrix and dynamical system operators representing coupling from the semicircular canals and the visual system to the velocity storage integrator were incorporated into the model. 2. It was postulated that the system matrix for a tilted position was a composition of two linear transformations of the system matrix for the upright position. One transformation modifies the eigenvalues of the system matrix while another rotates the pitch and roll eigenvectors with the head, while maintaining the yaw axis eigenvector approximately spatially invariant. Using this representation, the response characteristics of the pitch, roll, and yaw eye velocity were obtained in terms of the eigenvalues and associated eigenvectors. 3. Using OKAN data obtained from monkeys and comparing to the model predictions, the eigenvalues and eigenvectors of the system matrix were identified as a function of tilt to the side or of tilt to the prone positions, using a modification of the Marquardt algorithm. The yaw eigenvector for right-side-down tilt and for downward pitch cross-coupling was approximately 30 degrees from the spatial vertical. For the prone position, the eigenvector was computed to be approximately 20 degrees relative to the spatial vertical. For both side-down and prone positions, oblique OKN induced along eigenvector directions generated OKAN which decayed to zero along a straight line with approximately a single time constant. This was verified by a spectral analysis of the residual sequence about the straight line fit to the decaying data. The residual sequence was associated with a narrow autocorrelation function and a wide power spectrum. 4. Parameters found using the Marquardt algorithm were incorporated into the model. Diagonal matrices in a head coordinate frame were introduced to represent the direct pathway and the coupling of the visual system to the integrator. Model simulations predicted the behavior of yaw and pitch OKN and OKAN when the animal was upright, as well as the cross-coupling in the tilted position. The trajectories in velocity space were also accurately simulated. 5. There were similarities between the monkey eigenvectors and human perception of the spatial vertical. For side-down tilts and downward eye velocity cross-coupling, there was only an Aubert (A) effect. For upward eye velocity cross-coupling there were both Muller (E) and Aubert (A) effects. The mean of the eigenvectors for upward and downward eye velocities overlay human 1 x g perceptual data.(ABSTRACT TRUNCATED AT 400 WORDS).

NASA Discipline Neuroscience↗
NASA NTRS1996

Multiresolution and Explicit Methods for Vector Field Analysis and Visualization

We first report on our current progress in the area of explicit methods for tangent curve computation. The basic idea of this method is to decompose the domain into a collection of triangles (or tetrahedra) and assume linear variation of the vector field over each cell. With this assumption, the equations which define a tangent curve become a system of linear, constant coefficient ODE's which can be solved explicitly. There are five different representation of the solution depending on the eigenvalues of the Jacobian. The analysis of these five cases is somewhat similar to the phase plane analysis often associate with critical point classification within the context of topological methods, but it is not exactly the same. There are some critical differences. Moving from one cell to the next as a tangent curve is tracked, requires the computation of the exit point which is an intersection of the solution of the constant coefficient ODE and the edge of a triangle. There are two possible approaches to this root computation problem. We can express the tangent curve into parametric form and substitute into an implicit form for the edge or we can express the edge in parametric form and substitute in an implicit form of the tangent curve. Normally the solution of a system of ODE's is given in parametric form and so the first approach is the most accessible and straightforward. The second approach requires the 'implicitization' of these parametric curves. The implicitization of parametric curves can often be rather difficult, but in this case we have been successful and have been able to develop algorithms and subsequent computer programs for both approaches. We will give these details along with some comparisons in a forthcoming research paper on this topic.

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