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NASA NTRS1977

Reflective liquid crystal light valve with hybrid field effect mode

There is disclosed a high performance reflective mode liquid crystal light valve suitable for general image processing and projection and particularly suited for application to real-time coherent optical data processing. A preferred example of the device uses a CdS photoconductor, a CdTe light absorbing layer, a dielectric mirror, and a liquid crystal layer sandwiched between indium-tin-oxide transparent electrodes deposited on optical quality glass flats. The non-coherent light image is directed onto the photoconductor; this reduces the impedance of the photoconductor, thereby switching the AC voltage that is impressed across the electrodes onto the liquid crystal to activate the device. The liquid crystal is operated in a hybrid field effect mode. It utilizes the twisted nematic effect to create a dark off-state (voltage off the liquid crystal) and the optical birefringence effect to create the bright on-state. The liquid crystal thus modulates the polarization of the coherent read-out or projection light responsively to the non-coherent image. An analyzer is used to create an intensity modulated output beam.

Boswell, Donald D.↗

NASA NTRS2007

Comparison of Different Global Information Sources Used in Surface Radiative Flux Calculation: Radiative Properties of the Surface

Direct estimates of surface radiative fluxes that resolve regional and weather-scale variabilty over the whole globe with reasonable accuracy have only become possible with the advent of extensive global, mostly satellite, datasets within the past couple of decades. The accuracy of these fluxes, estimated to be about 10-15 W per square meter is largely limited by the accuracy of the input datasets. The leading uncertainties in the surface fluxes are no longer predominantly induced by clouds but are now as much associated with uncertainties in the surface and near-surface atmospheric properties. This study presents a fuller, more quantitative evaluation of the uncertainties for the surface albedo and emissivity and surface skin temperatures by comparing the main available global datasets from the Moderate-Resolution Imaging Spectroradiometer product, the NASA Global Energy and Water Cycle Experiment Surface Radiation Budget project, the European Centre for Medium-Range Weather Forecasts, the National Aeronautics and Space Administration, the National Centers for Environmental Prediction, the International Satellite Cloud Climatology Project (ISCCP), the Laboratoire de Meteorologie Dynamique, NOAA/NASA Pathfinder Advanced Very High Resolution Radiometer project, NOAA Optimum Interpolation Sea Surface Temperature Analysis and the Tropical Rainfall Measuring Mission (TRMM) Microwave Image project. The datasets are, in practice, treated as an ensemble of realizations of the actual climate such that their differences represent an estimate of the uncertainty in their measurements because we do not possess global truth datasets for these quantities. The results are globally representative and may be taken as a generalization of our previous ISCCP-based uncertainty estimates for the input datasets. Surface properties have the primary role in determining the surface upward shortwave (SW) and longwave (LW) flux. From this study, the following conclusions are obtained. Although land surface albedos in the near near-infrared remain poorly constrained (highly uncertain), they do not cause too much error in total surface SW fluxes; the more subtle regional and seasonal variations associated with vegetation and snow are still on doubt. The uncertainty of the broadband black-sky SW albedo for land surface from this study is about 7%, which can easily induce 5-10 W per square meter uncertainty in (upwelling) surface SW flux estimates. Even though available surface (broadband) LW emissivity datasets differ significantly (3%-5% uncertainty), this disagreement is confined to wavelengths greater than 20 micrometers so that there is little practical effect (1-3 W per square meters) on the surface upwelling LW fluxes. The surface skin temperature is one of two leading factors that cause problems with surface LW fluxes. Even though the differences among the various datasets are generally only 2-4 K, this can easily cause 10-15 W per square meter uncertainty in calculated surface (upwelling) LW fluxes. Significant improvements could be obtained for surface LW flux calculations by improving the retrievals of (in order of decreasing importance): (1) surface skin temperature, (2) surface air and near-surface-layer temperature, (3) column precipitable water amount and (4) broadband emissivity. And for surface SW fluxes, improvements could be obtained (excluding improved cloud treatment) by improving the retrievals of (1) aerosols (from our sensitivity studies but not discussed in this work), and (2) surface (black-sky) albedo, of which, NIR part of the spectrum has much larger uncertainty.

Zhang, Yuanchong↗

NASA NTRS1998

Submillimeter Imaging of Dust Around Main Sequence Stars

This grant was to image circumstellar dust disks surrounding main-sequence stars. The delivery of the SCUBA detector we had planned to use for this work was delayed repeatedly, leading us to undertake a majority of the observations with the UKT14 submillimeter detector at the JCMT (James Clerk Maxwell Telescope) and optical imagers and a coronagraph at the University of Hawaii 2.2-m telescope. Major findings under this grant include: (1) We discovered 5 asymmetries in the beta Pictoris regenerated dust disk. The discovery of these asymmetries was a surprise, since smearing due to Keplerian shear should eliminate most such features on timescales of a few thousand years. One exception is the "wing tilt" asymmetry, which we interpret as due to the scattering phase function of dust disk particles. From the wing tilt and a model of the phase function, we find a disk plane inclination to the line of sight of < 5 degrees. Other asymmetries (e.g. the butterfly asymmetry) suggest a disk that has been recently disturbed. We searched for possible nearby perturbers but found no clear candidates. Low mass stars (M dwarfs) and brown dwarfs would have fallen beneath the sensitivity threshhold of our survey, however. (2) We calculated a set of disk models to assess the detectability of dust disks around stars as a function of (a) distance, (b) disk, inclination (c) dust optical depth/mass, and (d) imaging resolution. These models guided our observational strategy on Mauna Kea. (3) We performed a coronagraphic survey of approx. 100 main-sequence stars in search of additional examples of circumstellar disks. The best new candidate disk, around the 5 M(sun) star BD+31deg.643, is distinguished by its large extent (few x 10( exp 3) AU). This disk, if real, cannot be rotationally supported. We suggest that the dust particles are ejected from a smaller, unseen disk (Kuiper Belt?) by strong radiation pressure forces due to the high luminosity central star. (4) SCUBA images of circumstellar dust disks were obtained at 850 gm in 1997/8. These images show extended, asymmetric emission, but have a signal-to-noise ratio too low to permit disk mapping to large projected distances. Our images of beta Pic, in particular, are in agreement with those obtained by Holland et al., and appear to confirm the blob-structure reported first by these authors. We have not yet been able to confirm that the structure is intrinsic to the disk, since beta Pic is at -50 degree declination, and suitable observing opportunities from northern latitudes are comparatively rare (even at the +20 degree latitude of JCMT). It is possible, for instance, that the main 850 micro-m blob is merely a galaxy or other high-z source projected onto the beta Pic mid-plane.

Jewitt, David↗

NASA NTRS1987

Welding monitoring system

This invention relates to systems for remotely monitoring automatic welding operations, and more particularly to a system wherein the welder is readily positionable, while components of the optical system remain fixed. A welder having an electrode is mounted in an enclosure containing a pair of mirrors. The electrode passes through an opening in the first mirror and a gas cup. The mirror reflects an image of a welding operation taken through the opening of the gas cup to the second mirror. The second mirror then reflects the image through a rotary coupling to a third mirror which, in turn, reflects the image to a receiving lense mounted to a second rotatable coupling. The image is then projected via a fiber optic bundle to a filter unit where selected wavelengths of light are filtered from the welding image. The filter unit is coupled to an enlarger which enlarges the image and passes it to a camera. The camera is connected to an electronic eclipser which selectively darkens the brightest portions of the image. Finally, the image is recorded by a video tape recorder and displayed by a monitor.

Babcock, Stephen G.↗

NASA NTRS2002

RADARSAT: The Antarctic Mapping Project

The first Antarctic Imaging Campaign (AIC) occurred during the period September 9, 1997 through October 20, 1997. The AIC utilized the unique attributes of the Canadian RADARSAT-1 to acquire the first, high-resolution, synthetic aperture imagery covering the entire Antarctic Continent. Although the primary goal of the mission was the acquisition of image data, the nearly flawless execution of the mission enabled additional collections of exact repeat orbit data. These data, covering an extensive portion of the interior Antarctic, potentially are suitable for interferometric analysis of topography and surface velocity. This document summarizes the Project through completion with delivery of products to the NASA DAACs.

Jezek, Kenneth C.↗

DOE OSTI2021

Deep Learning for Fish Identification from Sonar Data: CRADA 481 [Abstract only]

To help solve the challenges of hydropower energy production related to the potential for eel injury and mortality from passage through hydropower turbines, we will develop a deep learning method for identifying migrating eels from imaging sonar. This project continues with a prior project conducted by the Pacific Northwest National Laboratory (PNNL) and the Electric Power Research Institute (EPRI) in FY2018-2019. The proposed method employs Convolution Neural Network (CNN), a powerful deep learning method for image classification, to distinguish between images of eels and non-eel moving objects. We propose to collect more laboratory data and add more existing field data to train a powerful deep learning model. In addition to eels and sticks as classified in previous studies, we will add images containing several non-eel fish species and macrophyte mats to the training data. A multi-class classification model will be developed to distinguish these objects. Object detection algorithm will be explored and developed to locate and identify multiple objects in each sonar frame. Motion analysis will be performed to track the movement of objects in sonar video clips. We will also improve the data conversion algorithm so that it can read in both DIDSON and ARIS (both are imaging sonars developed by Sound Metrics Corp) data files and convert them to images with comparably high resolution, regardless of the varying detection ranges in different environments. The developed algorithms will be packaged as a software with a graphic user interface. The software will be evaluated by external collaborators in the field. The developed framework can be generalized for automatic monitoring of fish passage and migration using other imaging sonars like ARIS and will benefit the design and operation of ecologically friendly hydroelectric projects. The developed wavelet and CNN model configuration parameters can potentially be transferred to lamprey detection in similar riverine environments.

13 HYDRO ENERGY↗

DOE OSTI2023

Low-energy Electron-track Imaging for a Liquid Argon Time-projection-chamber Telescope Concept Using Probabilistic Deep Learning

The GammaTPC is an MeV-scale single-phase liquid argon time-projection-chamber gamma-ray telescope concept with a novel dual-scale pixel-based charge-readout system. It promises to enable a significant improvement in sensitivity to MeV-scale gamma rays over previous telescopes. The novel pixel-based charge readout allows for imaging of the tracks of electrons scattered by Compton interactions of incident gamma rays. The two primary contributors to the accuracy of a Compton telescope in reconstructing an incident gamma-ray’s original direction are its energy and position resolution. In this work, we focus on using deep learning to optimize the reconstruction of the initial position and direction of electrons scattered in Compton interactions, including using probabilistic models to estimate predictive uncertainty. We show that the deep-learning models are able to predict locations of Compton scatters of MeV-scale gamma rays from simulated 500 μm pixel-based data to better than 1 mm rms error and are sensitive to the initial direction of the scattered electron. We compare and contrast different deep-learning uncertainty estimation algorithms for reconstruction applications. Additionally, we show that event-by-event estimates of the uncertainty of the locations of the Compton scatters can be used to select those events that were reconstructed most accurately, leading to improvement in locating the origin of gamma-ray sources on the sky.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS2007

Software for Automated Image-to-Image Co-registration

The project objectives are: a) Develop software to fine-tune image-to-image co-registration, presuming images are orthorectified prior to input; b) Create a reusable software development kit (SDK) to enable incorporation of these tools into other software; d) provide automated testing for quantitative analysis; and e) Develop software that applies multiple techniques to achieve subpixel precision in the co-registration of image pairs.

Benkelman, Cody A.↗

NASA NTRS1995

CNR LARA project, Italy: Airborne laboratory for environmental research

The increasing interest for the environmental problems and the study of the impact on the environment due to antropic activity produced an enhancement of remote sensing applications. The Italian National Research Council (CNR) established a new laboratory for airborne hyperspectral imaging, the LARA Project (Laboratorio Aero per Ricerche Ambientali - Airborne Laboratory for Environmental Research), equipping its airborne laboratory, a CASA-212, mainly with the Daedalus AA5000 MIVIS (Multispectral Infrared and Visible Imaging Spectrometer) instrument. MIVIS's channels, spectral bandwidths, and locations are chosen to meet the needs of scientific research for advanced applications of remote sensing data. MIVIS can make significant contributions to solving problems in many diverse areas such as geologic exploration, land use studies, mineralogy, agricultural crop studies, energy loss analysis, pollution assessment, volcanology, forest fire management and others. The broad spectral range and the many discrete narrow channels of MIVIS provide a fine quantization of spectral information that permits accurate definition of absorption features from a variety of materials, allowing the extraction of chemical and physical information of our environment. The availability of such a hyperspectral imager, that will operate mainly in the Mediterranean area, at the present represents a unique opportunity for those who are involved in environmental studies and land-management to collect systematically large-scale and high spectral-spatial resolution data of this part of the world. Nevertheless, MIVIS deployments will touch other parts of the world, where a major interest from the international scientific community is present.

Bianchi, R.↗

NASA NTRS2012

High-Speed Edge-Detecting Line Scan Smart Camera

A high-speed edge-detecting line scan smart camera was developed. The camera is designed to operate as a component in a NASA Glenn Research Center developed inlet shock detection system. The inlet shock is detected by projecting a laser sheet through the airflow. The shock within the airflow is the densest part and refracts the laser sheet the most in its vicinity, leaving a dark spot or shadowgraph. These spots show up as a dip or negative peak within the pixel intensity profile of an image of the projected laser sheet. The smart camera acquires and processes in real-time the linear image containing the shock shadowgraph and outputting the shock location. Previously a high-speed camera and personal computer would perform the image capture and processing to determine the shock location. This innovation consists of a linear image sensor, analog signal processing circuit, and a digital circuit that provides a numerical digital output of the shock or negative edge location. The smart camera is capable of capturing and processing linear images at over 1,000 frames per second. The edges are identified as numeric pixel values within the linear array of pixels, and the edge location information can be sent out from the circuit in a variety of ways, such as by using a microcontroller and onboard or external digital interface to include serial data such as RS-232/485, USB, Ethernet, or CAN BUS; parallel digital data; or an analog signal. The smart camera system can be integrated into a small package with a relatively small number of parts, reducing size and increasing reliability over the previous imaging system..

Prokop, Norman F.↗

NASA NTRS1997

Registration assisted mosaic generation

This paper presents a general strategy for assembling mosaics from numerous individual images where uncertainty exists in the position and orientation of those images. Both of the presented applications relate to remotely operated camera platforms, the first being the Galileo solid state imaging (SSI) camera presently in orbit around Jupiter, and the second being the Imager for Mars Pathfinder (IMP) stereo camera on Mars. A basic strategy in both applications is to determine the correct relative camera pointing followed by direct map projection of the images. It is assumed that approximate camera pointing exists sufficient to locate adjacent images and to place initial tiepoints within reach of the correlator. Spatial correlation is used to fix tiepoints whose initial locations are predicted by the camera pointing. We use either an fast fourier transform (fft) algorithm or a variant of Gruen's scheme permitting limited image rotation and skew. The Gruen correlator has three hierarchical modes: 1) A classical spatial least squares correlation on integral pixel boundaries used when rotation is small. 2) An annealing non-deterministic search used when rotations are unknown. A simplex deterministic search used for the end game. The correlation operation can be performed either interactively or autonomously. The final camera pointing solution relies upon a simplex downhill search in 2n or 3n dimensions where n is the number of images comprising the mosaic and the objective function to be minimized is the disagreement between tiepoint locations predicted from the camera pointing with those observed by the correlator. For Galileo the 3n unknowns are euler angles defining camera pointing in planet coordinates, and for Mars Pathfinder they are 2n unknowns representing commanded azimuth and elevation in the Lander coordinate system.

Lorre, Jean J.↗

DOE OSTI2025

Optical galaxy cluster mock catalogs with realistic projection effects: Validations with the SDSS clusters

Galaxy clusters identified in optical imaging surveys suffer from projection effects: Physically unassociated galaxies along a cluster’s line of sight can be counted as its members and boost the observed richness (the number of cluster members). To model the impact of projection on cluster cosmology analyses, we apply a halo occupation distribution model to 𝑁-body simulations to simulate the red galaxies contributing to cluster members, and we use the number of galaxies in a cylinder along the line of sight (counts in cylinders) to model the impact of projection on cluster richness. We compare three projection models: uniform, quadratic, and Gaussian, and we convert between them by matching their effective cylinder volumes. We validate our mock catalogs using SDSS redMaPPer clusters’ data vectors, including counts vs richness, stacked lensing signal, spectroscopic redshift distribution of member galaxies, and richness remeasured on a redshift grid. We find the former two are insensitive to the projection model, while the latter two favor a quadratic projection model with a width of ≈180 ℎ −1 Mpc (equivalent to the volume of a uniform model with a width of 100 ℎ −1 Mpc and a Gaussian model with a width of 110 ℎ −1 Mpc, or a Gaussian redshift error of 0.04). Furthermore, our framework provides an efficient and flexible way to model optical cluster data vectors, paving the way for a simulation-based joint analysis for clusters, galaxies, and shear.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS2016

Thomas Leps Internship Abstract

An optical navigation system is being flown as the backup system to the primary Deep Space Network telemetry for navigation and guidance purposes on Orion. This is required to ensure Orion can recover from a loss of communication, which would simultaneously cause a loss of DSN telemetry. Images taken of the Moon and Earth are used to give range and position information to the navigation computer for trajectory calculations and maneuver execution. To get telemetry data from these images, the size and location of the moon need to be calculated with high accuracy and precision. The reentry envelope for the Orion EM-1 mission requires the centroid and radius of the moon images to be determined within 1/3 of a pixel 3 sigma. In order to ensure this accuracy and precision can be attained, I was tasked with building precise dot grid images for camera calibration as well as building a hardware in the loop test stand for flight software and hardware proofing. To calibrate the Op-Nav camera a dot grid is imaged with the camera, the error between the image dot location and the actual dot location can be used to build a distortion map of the camera and lens system so that images can be fixed to display truth locations. To build the dot grid images I used the Electro Optics Lab optical bench Bright Object Simulator System, and gimbal. The gimbal was slewed to a series of elevations and azimuths. An image of the collimated single point light source was then taken at each position. After a series of 99 images were taken at different locations the single light spots were extracted from each image and added to a composite image containing all 99 points. During the development of these grids it was noticed that an intermittent error in the artificial "star" locations occurred. Prior to the summer this error was attributed to the gimbal having glitches in it's pointing direction and was going to be replaced, however after further examining the issue I determined it to be a software issue. I have since narrowed the likely source of the error down to a Software Development Kit released by the camera supplier PixeLink. I have since developed a workaround in order to build star grids for calibration until the software bug can be isolated and fixed. I was also tasked with building a Hardware in the Loop test stand in order to test the full Op-Nav system. A 4k screen displays simulated Lunar and Terrestrial images from a possible Orion trajectory. These images are then projected through a collimator and then captured with an Op-Nav camera controlled by an Intel NUC computer running flight software. The flight software then analyzes the images to determine attitude and position, this data is then reconstructed into a trajectory and matched to the simulated trajectory in order to determine the accuracy of the attitude and position estimates. In order for the system to work it needs to be precisely and accurately aligned. I developed an alignment procedure that allows the screen, collimator and camera to be squared, centered and collinear with each other within a micron spatially and 5 arcseconds in rotation. I also designed a rigid mount for the screen that was machined on site in Building 10 by another intern. While I was working in the EOL we received a $500k Orion startracker for alignment procedure testing. Due to my prior experience in electronics development, as an ancillary duty, I was tasked with building the cables required to operate and power the startracker. If any errors are made building these cables the startracker would be destroyed, I was honored that the director of the lab entrusted such a critical component with me. This internship has cemented my view on public space exploration. I always preferred public sector to privatization because, as a scientist, the most interesting aspects of space for me are not necessarily the most profitable. I was concerned that the public sector was faltering however, and that in order to improve human space exploration I would be forced into private sector. I now know that, at least at JSC, human spaceflight is still progressing, and exciting work is still being done. I am now actively seeking employment at JSC after I complete my Ph.D and have met with my branch chiefs and mentor to discuss transitioning to a grad Co-op position.

Leps, Thomas↗

DOE OSTI2024

The Active Asteroids Citizen Science Program: Overview and First Results

Abstract We present the Citizen Science program Active Asteroids and describe discoveries stemming from our ongoing project. Our NASA Partner program is hosted on the Zooniverse online platform and launched on 2021 August 31, with the goal of engaging the community in the search for active asteroids—asteroids with comet-like tails or comae. We also set out to identify other unusual active solar system objects, such as active Centaurs, active quasi-Hilda asteroids (QHAs), and Jupiter-family comets (JFCs). Active objects are rare in large part because they are difficult to identify, so we ask volunteers to assist us in searching for active bodies in our collection of millions of images of known minor planets. We produced these cutout images with our project pipeline that makes use of publicly available Dark Energy Camera data. Since the project launch, roughly 8300 volunteers have scrutinized some 430,000 images to great effect, which we describe in this work. In total, we have identified previously unknown activity on 15 asteroids, plus one Centaur, that were thought to be asteroidal (i.e., inactive). Of the asteroids, we classify four as active QHAs, seven as JFCs, and four as active asteroids, consisting of one main-belt comet (MBC) and three MBC candidates. We also include our findings concerning known active objects that our program facilitated, an unanticipated avenue of scientific discovery. These include discovering activity occurring during an orbital epoch for which objects were not known to be active, and the reclassification of objects based on our dynamical analyses.

Astronomy & Astrophysics↗

NASA NTRSDate not supplied

Using Digital Image Correlation for Material Testing at Cryogenic Temperatures

Recent efforts were made to parametrize an aluminum alloy at cryogenic temperatures to calibrate finite element models simulating material deformation. Digital image correlation (DIC) was used to measure material properties using front/back strains of specimens during tensile tests. Data acquired from DIC was compared to physical extensometer measurements. Two camera pairs (front and back) were mounted via aluminum extrusion to the exterior of an environmental chamber of a hydraulic test stand. Images were taken through single-pane optical glass windows of the environment chamber. Positive pressure bags encased the cameras, attaching to the perimeter of the glass to keep out moisture and prevent windows from frosting. Figure 1 shows the environmental setup and the camera setup during cryogenic testing. Some of the challenges during testing at cryogenic temperatures include the gaseous flow of liquid nitrogen in the optical path of the mounted cameras, frost buildup on the windows as seen in Figure 1b, and frost on the test coupon. The temperature gradient present during the introduction of liquid nitrogen into the chamber can increase the projection error of images captured of the specimen. Figure 2 shows the estimated projection error from DIC as a function of temperature during the cool down process. This presentation addresses some of the concerns and observations when using DIC for testing at cryogenic temperatures using methods of isolated camera system and techniques for maintaining the glass panel clear from frost.

Joseph Cochran↗

NASA NTRS1992

Instrument requirements for imaging the magnetosphere in extreme-ultraviolet and energetic neutral atoms derived from computer-simulated images

Scientific requirements for He(+)(304 A) and energetic neutral atom (ENA) magnetospheric imaging, as well as the derived instrumental requirements are presented. Both ENA imaging of the hot plasma and EUV imaging of the cold plasma are highlighted. The question of the accuracy with which physically significant parameters can be extracted from actual images using computerized algorithms is addressed. An example of an ENA image analyzed by means of the Powell minimization algorithm is given. Automated unfolding of global magnetospheric images is also discussed. A Mercator projection of a model INA image is shown.

Roelof, Edmond C.↗

NASA NTRS1987

High-resolution CCD imagers using area-array CCD's for sensing spectral components of an optical line image

CCD imagers with a novel replicated-line-imager architecture are abutted to form an extended line sensor. The sensor is preceded by optics having a slit aperture and having an optical beam splitter or astigmatic lens for projecting multiple line images through an optical color-discriminating stripe filter to the CCD imagers. A very high resolution camera suitable for use in a satellite, for example, is thus provided. The replicated-line architecture of the imager comprises an area-array CCD, successive rows of which are illuminated by replications of the same line segment, as transmitted by respective color filter stripes. The charge packets formed by accumulation of photoresponsive charge in the area-array CCD are read out row by row. Each successive row of charge packets is then converted from parallel to serial format in a CCD line register and its amplitude sensed to generate a line of output signal.

Elabd, Hammam↗

NASA NTRS2017

Map Projection Induced Variations in Locations of Polygon Geofence Edges

This Paper under-estimates answers to the following question under various constraints: If a geofencing algorithm uses a map projection to determine whether a position is inside/outside a polygon region, how far outside/inside the polygon can the point be and the algorithm determine that it is inside/outside (the opposite and therefore incorrect answer)? Geofencing systems for unmanned aircraft systems (UAS) often model stay-in and stay-out regions using 2D polygons with minimum and maximum altitudes. The vertices of the polygons are typically input as latitude-longitude pairs, and the edges as paths between adjacent vertices. There are numerous ways to generate these paths, resulting in numerous potential locations for the edges of stay-in and stay-out regions. These paths may be geodesics on a spherical model of the earth or geodesics on the WGS84 reference ellipsoid. In geofencing applications that use map projections, these paths are inverse images of straight lines in the projected plane. This projected plane may be a projection of a spherical earth model onto a tangent plane, called an orthographic projection. Alternatively, it may be a projection where the straight lines in the projected plane correspond to straight lines in the latitudelongitude coordinate system, also called a Plate Carr´ee projection. This paper estimates distances between different edge paths and an oracle path, which is a geodesic on either the spherical earth or the WGS84 ellipsoidal earth. This paper therefore estimates how far apart different edge paths can be rather than comparing their path lengths, which are not considered. Rather, the comparision is between the actual locations of the edges between vertices. For edges drawn using orthographic projections, this maximum distance increases as the distance from the polygon vertices to the projection point increases. For edges drawn using Plate Carr´ee projections, this maximum distance increases as the vertices become further from the equator. Distances between geodesics on a spherical earth and a WGS84 ellipsoidal earth are also analyzed, using the WGS84 ellipsoid as the oracle. Bounds on the 2D distance between a straight line and a great circle path, in an orthographically projected plane rather than on the surface of the earth, have been formally verified in the PVS theorem prover, meaning that they are mathematically correct in the absence of floating point errors.

Neeley, Paula↗