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Flattening the Earth: Two Thousand Years of Map Projections
... Stereographic projection is a mapping that projects a 2D manifold onto a 2D plane. Such a technique is well developed in the field of Topology and Geography to project surface of the earth to a 2D planar map [30]. Since then, various projection functions have been proposed to improve the quality of mapping. ...
... UV Projection [30]: ...
... Kavrayskiy VII Projection [30]: ...
We propose an efficient Stereographic Projection Neural Network (SPNet) for learning representations of 3D objects. We first transform a 3D input volume into a 2D planar image using stereographic projection. We then present a shallow 2D convolutional neural network (CNN) to estimate the object category followed by view ensemble, which combines the responses from multiple views of the object to further enhance the predictions. Specifically, the proposed approach consists of four stages: (1) Stereographic projection of a 3D object, (2) view-specific feature learning, (3) view selection and (4) view ensemble. The proposed approach performs comparably to the state-of-the-art methods while having substantially lower GPU memory as well as network parameters. Despite its lightness, the experiments on 3D object classification and shape retrievals demonstrate the high performance of the proposed method.
... For animated transitions and scale-space navigation in geographical data, a suitable projection [35,36] from geographical space to two-dimensional Cartesian space is essential. While some approaches utilize a three-dimensional representation of the world, information visualization approaches typically work better in combination with two-dimensional data representation spaces, and require less resources. ...
... In addition, geodetic lines are in general not straight lines when projected into Mercator. Obviously, the choice of projection is also dependent on the application domain and use case [35,36]. In our experiments, we found that animated transitions between two geographical locations should follow a straight line in the image space (i.e., the projected space). ...
... Despite its shortcomings, Mercator projection should still be considered for animated point-to-point transitions because of its familiarity and the efficient pre-generation and retrieval of map material. Many other choices for geographical projections exist [36], all with their individual advantages and drawbacks. For our analysis, we considered two choices (TPEQD and AZEQD) based on two requirements: (1) Distances should be represented without distortion from the start point of the transition, and (2) the straight-line connection between start and end point of the transition should be a geodetic line. ...
The visualization and interactive exploration of geo-referenced networks poses challenges if the network's nodes are not evenly distributed. Our approach proposes new ways of realizing animated transitions for exploring such networks from an ego-perspective. We aim to reduce the required screen estate while maintaining the viewers' mental map of distances and directions. A preliminary study provides first insights of the comprehensiveness of animated geographic transitions regarding directional relationships between start and end point in different projections. Two use cases showcase how ego-perspective graph exploration can be supported using less screen space than previous approaches.
... Cartographers have defined a plethora of projections to send the three dimensional surface of the earth onto a two dimensional map (the word 'map' now being used here in the cartographic sense) [5,6]. Due to inherent complexities in translating a sphere's surface to a flat map, distortions of geometric properties are unavoidable. ...
... A map projection, loosely defined, is any projection on R 3 which is used primarily to map some subset of a sphere to a subset of a plane. Map projections are named for their use in cartography for creating maps of the earth [5]. For myriad topological reasons, distortions of certain geometric properties of a sphere are inevitable under a map projection. ...
We introduce the novel class of reverse map projection embeddings, each one defining a unique new method of encoding classical data into quantum states. Inspired by well-known map projections from the unit sphere onto its tangent planes, used in practice in cartography, these embeddings address the common drawback of the amplitude embedding method, wherein scalar multiples of data points are identified and information about the norm of data is lost. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning. Using these methods, we can leverage symmetries in classical datasets to significantly strengthen performance on quantum machine learning tasks. Finally, we select four values of with which to perform a simple classification task, taking as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.
... After that, 310 the evaluation method is introduced. As an approach to presenting a whole sphere as a 2D image, equirectangular projection 313 maps longitude lines and latitude circles to equally spaced vertical and horizontal lines, 314 respectively (Snyder, 1997). In contrast, perspective projection only maps part of a sphere 315 (within the FOV) from the centre to a tangent plane (Yang et al., 2018). ...
... In the following subsections, each module of the application is described in detail and 487 an experiment is carried out to evaluate the detection accuracy of the application. According to the definition of equirectangular projection (Snyder, 1997), in a panoramic 490 video, objects behind the camera are shown on the left and right sides, while objects in 491 the front are shown in the middle. Therefore, if an object's current position is closer to 492 the centre line compared to its position in the last frame, it is moving forwards relative to 493 the camera. ...
Panoramic cycling videos can record 360{\deg} views around the cyclists. Thus, it is essential to conduct automatic road user analysis on them using computer vision models to provide data for studies on cycling safety. However, the features of panoramic data such as severe distortions, large number of small objects and boundary continuity have brought great challenges to the existing CV models, including poor performance and evaluation methods that are no longer applicable. In addition, due to the lack of data with annotations, it is not easy to re-train the models. In response to these problems, the project proposed and implemented a three-step methodology: (1) improve the prediction performance of the pre-trained object detection models on panoramic data by projecting the original image into 4 perspective sub-images; (2) introduce supports for boundary continuity and category information into DeepSORT, a commonly used multiple object tracking model, and set an improved detection model as its detector; (3) using the tracking results, develop an application for detecting the overtaking behaviour of the surrounding vehicles. Evaluated on the panoramic cycling dataset built by the project, the proposed methodology improves the average precision of YOLO v5m6 and Faster RCNN-FPN under any input resolution setting. In addition, it raises MOTA and IDF1 of DeepSORT by 7.6\% and 9.7\% respectively. When detecting the overtakes in the test videos, it achieves the F-score of 0.88. The code is available on GitHub at github.com/cuppp1998/360_object_tracking to ensure the reproducibility and further improvements of results.
... In order to adapt to these coding schemes as much as possible, the panoramic videos have to be projected onto a 2D plane. Different projection formats have been explored, including equirectangular projection (ERP) [1,2], cube-map projection (CMP) [2,3], rhombic dodecahe-B Ning Li ning_li@mail.nwpu.edu.cn Shuai Wan swan@nwpu.edu.cn 1 School of Electronics and Information, Northwestern Polytechnical University, Xian, China dron projection (RDP) [4], etc. Regardless of the projection format used, it is inevitable to introduce content distortion when projecting panoramic video onto a 2D plane. ...
... Shuai Wan swan@nwpu.edu.cn 1 School of Electronics and Information, Northwestern Polytechnical University, Xian, China dron projection (RDP) [4], etc. Regardless of the projection format used, it is inevitable to introduce content distortion when projecting panoramic video onto a 2D plane. An example using the ERP format is shown in Fig. 1, in which the textures located in the polar regions of the spherical surface are stretched toward both sides, and the direction of texture changes. ...
Geometric partitioning mode (GPM) is introduced into H.266/versatile video coding (VVC) as a new inter-frame prediction technology, in which one block can be divided into two non-rectangular blocks to increase partitioning precision of moving objects. In this paper, a three-dimensional (3D) motion vector (MV) translation based on spherical coordinate transformation is introduced into H.266/VVC and applied to GPM, in which the MV candidate list is constructed separately for each GPM partition. Firstly, the number of MVs in the GPM MV candidate list is expanded to 12, and all MVs in the merge mode candidate list are reserved, which can prevent the MVs with the best encoding performance from being removed. Secondly, the MVs of neighboring blocks of the current block are transformed by 3D MV translation and put into the blank positions of the newly constructed GPM partition MV candidate list. Thirdly, an independent motion compensation prediction based on the MV of each GPM partition is performed to improve the prediction accuracy. The experimental results demonstrated the effectiveness of the proposed method, achieving up to 0.44% reduction in BD rate in rate-distortion performance compared to H.266/VVC.
... Картографічна проекціяце систематичне перетворення широт і довгот місцевостей з поверхні сфери або еліпсоїда в місця на площині [1]. Будь які карти неможливо створити без застосування картографічних проекцій. ...
... Якщо сфера використовується як основа для якогось тіла сонячної системи, то доцільно застосовувати якусь картографічну проекцію [1,5], для того, щоб можна було нанести існуючі картографічні зображення на модель сфери. Найбільш відомою є циліндрична проекція Ламберта, яка є основою сімейства циліндричних рівноплощинних проекцій. ...
У статті розглянуто метод створення безшовної текстурованої моделі сфери та видалення артефактів пов'язаних з особливостями розрахунку текстурних координат при циліндричному проеціюванні. Метою статті є розробка методу побудови та візуалізації сферичних текстурованих об'єктів який би не мав недоліків існуючих методів, які пов'язані з виникненням наочно помітних візуальних артефактів при накладанні текстури на поверхню сфери. Наведено метод який дозволяє коректувати текстурні координати як на геометричному рівні, так і в процесі проведення візуалізації, в якому сфера умовно розділяється на частини відповідно до відстані від полюсів. Для кожної частини використовується найбільш оптимальний алгоритм текстурування, що забезпечує швидкодію методу. Обчислення за допомогою запропонованих алгоритмів можна повністю реалізувати за рахунок ресурсів GPU. За результатами дослідження визначено, що запропонований метод дозволяє видалити артефакти які виникають при розрахунку текстурних координат для циліндричного проеціювання як безпосередньо у процесі візуалізації, так і на етапі синтезу моделі сфери, що дозволяє значно покращити відображення текстур на поверхні сфери.
... The map uses a scale of 1:2,993,200 and slightly shifts to the left of 10°. This lack of north orientation is common in Ptolemaic maps (Snyder, 1997). ...
... The results obtained seem to indicate that it was the conic projection. However, the lack of precision of the map does not allow us to rule out the use of the pseudoconic projection, the one preferred by Ptolemy and, as far as we know, first used by Nicolaus Germanus in his copies of the Geographia made from 1470 onwards (Snyder, 1997). Figure 3 shows the planimetric errors made using pairs of circles linked by a line. ...
In this paper, we study two singular maps, the Modern Map of Spain, attributed to Cardinal Margarit, and Spagna con le distantie de li loci; made in northern Italy in the late 15th and early 16th centuries. Like the British Gough map, both already show communication networks with expressed distances. Using formal analysis, statistical methodology, and computer processing , we present the cartographic characteristics of each one and relate them to their historical context, updating the scarce information available until now. We explain their relationship as two milestones of the same cartographic process. At the same time, we study the routes represented, finding out the units of measurement used and the communication networks that both maps show us in the context of the Revolution of Communications that the Renaissance represented in Europe. The research has allowed us to attribute a new dating to them, question the traditional authorship, and advance a theory on the transport networks' functionality, demonstrating that both maps are part of a cartographic and historical process at the European level. All these updates to the vision on the first maps of communications in Western Europe established new contributions to the relationship between maps and itineraries. It contributes to filling a void occupied in solitary, until now, by the map of Gough of Great Britain.
... On the contrary, having performed the equations for the projection, he said that a map produced in it could be rolled up into a cone. • According to Snyder (1993), developable surfaces are mentioned for the first time in the cartographic literature in 1863 by D'Avezac in Coup d'oeil historique sur la projection des cartes de géographie. D'Avezac deals with the classifications of map projections and proposes the general term constructions for all map projections, dividing them into projections which follow the perspective law, developments under the condition of correlation between the sphere and the developable surface and representation systems based on purely conventional combinations. ...
... For cylindrical and most other projections, he uses the terms normal (regular), transverse, and oblique aspects. In his book on the history of map projections (Snyder 1993) he rarely uses the term aspect. Instead, he prefers the terms transverse and oblique projections. ...
There is no standard or generally accepted terminology of aspect in the theory of map projections. The term is probably derived from the concept that a graticule is produced by perspective projection of the meridians and parallels on a sphere onto a developable surface. Developable surfaces are widely accepted, and it is almost impossible to find a publication that deals with map projections in general and without developable surfaces story. If found, it usually classifies projections as cylindrical, conical and azimuthal, and applies developable surfaces to define the projection aspect. This paper explains why applying developable surfaces in the interpretation of map projections is not recommended, nor defining the aspect of all projections by the position of a midpoint as polar, equatorial, or oblique. In fact, defining a projection aspect this way is invalid in general, and obscures the fact that the aspect depends on the class to which a particular map projection belongs.
... Corresponding considerations were made quite early on, for example, by Robinson (1951) who examined a variety of projections in relation to different aspects of distortion. To select suitable projections, comprehensive lists of projections suitable for world maps were compiled (Canters and Decleir 1989;Snyder 1987Snyder , 1997Snyder and Voxland 1989) and corresponding software was developed (Jenny, Patterson, and Hurni 2008). Projections that were developed exclusively for world maps should be emphasized, such as the representation of the Earth's surface as an essentially complete and 'uninterrupted' disc for thematic maps (DeLucia and Snyder 1986); a balanced equal-area pseudocylindrical projection (Šavrič, Patterson, and Jenny 2019); and a compromise pseudocylindrical projection (Šavrič, Patterson, and Jenny 2015). ...
Projections for world maps can only be a compromise between different requirements as it is impossible to construct a projection that preserves distances, areas, and angles. Accordingly, choosing a suitable projection is difficult – the wish list of requirements for such a projection is sometimes long and in most cases impossible to fulfil. This article presents a mechanism to translate such a wish list into a suitable compromise projection. For this purpose, an already known projection is initially selected and then subsequently optimized step by step, following principles of classical mechanics and using a discrete global grid system. This mechanism is effective and can, in principle, easily be adapted to other possible requirements for a map projection. In particular, this makes it possible to create projections that are optimized for the land areas and therefore exhibit less relevant distortion.
... He thought that a map of a whole oikoumene should be made in a projection on which the convergence of meridians is displayed to reflect the sphericity of the Earth and proposed two map projections whose updated modifications are still in use: the equidistant conic projection (φ 0 = 36°N; the parallel of Rhodes), known as Ptolemy's first projection (Ptolemy I) and the pseudoconic projection (φ 0 = 23.8°N; the parallel of Syene), so-called Ptolemy's second projection (Ptolemy II). In addition, Ptolemy's assumed circumference of the Earth was 180,000 stadia (29,000 km), a value that is 1.4 times smaller in comparison to Eratosthenes's accurate estimation (Keuning 1955;Snyder 1993;Berggren and Jones 2000) and that may have unintentionally affected the cartography of later eras-a hypothetical scenario proposed by this research and discussed in more detail in the following sections. ...
Some of the earliest-made known portolan charts (Pisane, Cortona, Avignon, and Lucca charts) and Qutb al-Din al-Shirazi’s schematic (1282) resembling “a map of the Mediterranean that was drawn by the sages of Greece and the ancient geometers” were cartometrically analysed. The results show that the Mediterranean was drawn nearly identically on the Carte Pisane and Lucca chart and that the preserved fragment of the Avignon chart is somewhat of an intermediate step between the Pisane–Lucca model and Pietro Vesconte’s model. The coasts of al-Shirazi’s schematic fit well with the Pisane and Cortona charts, whereas its 40×30 grid is similar in size to the grid drawn on the Pisane and Avignon charts. The length of its sides along the latitude φ = 36° (125 km) corresponds to 100 portolan miles (miglia) and a 2-degree interval along the parallel φ = 36° according to Ptolemy’s incorrect estimation of the Earth’s size. Hypothetically, such a misunderstanding could have occurred if a late medieval cartographer (unfamiliar with the differences between spherical and Euclidean geometry) erroneously combined different spatial datasets from classical antiquity by unwittingly superimposing Ptolemy’s longitudes onto maps or charts made according to Eratosthenes’s correctly obtained size of the Earth by treating them as distances.
... Cerezo's work pointed to the idea that a center of the projection must be defined and that the azimuths are maintained through the projection: the ions fly within a plane that contains the main specimen axis and its original position at the specimen surface. If we assume azimuths are conserved, then we are dealing with azimuthal equidistant projection, which is a well-known and well-described projection used, e.g. by geographers (Snyder, 2007), and is the projection model of the Earth represented on the United Nations emblem. More recently, Miller and Forbes (2014) also discussed the Hawkes-Kasper approach, which is an equivalent to this projection model. ...
There are two main projections used to transform, and reconstruct, field ion micrographs or atom probe tomography data into atomic coordinates at the specimen surface and, subsequently, in three-dimensions. In this article, we present a perspective on the strength of the azimuthal equidistant projection in comparison to the more widely used and well-established point-projection(or pseudo-stereographic projection), which underpins data reconstruction in most software packages currently in use across the community. After an overview of the reconstruction methodology, we demonstrate that the azimuthal equidistant is more robust with regards to errors on the parameters used to perform the reconstruction and is therefore more likely to yield more accurate tomographic reconstructions.
... Typically 360°video is shot using a collection of cameras that cover a 360°FOV. The captured frames are then stitched together and projected onto a 2D plane using the Equi-Rectangular Projection (ERP) [21]. The overall video has to have a very high resolution such as 8K in order to provide high quality for a given viewport, which is only a portion of the overall view. ...
360{\deg} video is attracting an increasing amount of attention in the context of Virtual Reality (VR). Owing to its very high-resolution requirements, existing professional streaming services for 360{\deg} video suffer from severe drawbacks. This paper introduces a novel end-to-end streaming system from encoding to displaying, to transmit 8K resolution 360{\deg} video and to provide an enhanced VR experience using Head Mounted Displays (HMDs). The main contributions of the proposed system are about tiling, integration of the MPEG-Dynamic Adaptive Streaming over HTTP (DASH) standard, and viewport-aware bitrate level selection. Tiling and adaptive streaming enable the proposed system to deliver very high-resolution 360{\deg} video at good visual quality. Further, the proposed viewport-aware bitrate assignment selects an optimum DASH representation for each tile in a viewport-aware manner. The quality performance of the proposed system is verified in simulations with varying network bandwidth using realistic view trajectories recorded from user experiments. Our results show that the proposed streaming system compares favorably compared to existing methods in terms of PSNR and SSIM inside the viewport.
... In order to process the surface of the cornea further, the texture is unwrapped. For the unwrapping, we use a equirectangular projection [30]. This type of projection has higher distortions in the polar region than around the equator. ...
Around-device interaction techniques aim at extending the input space using various sensing modalities on mobile and wearable devices. In this paper, we present our work towards extending the input area of mobile devices using front-facing device-centered cameras that capture reflections in the human eye. As current generation mobile devices lack high resolution front-facing cameras we study the feasibility of around-device interaction using corneal reflective imaging based on a high resolution camera. We present a workflow, a technical prototype and an evaluation, including a migration path from high resolution to low resolution imagers. Our study indicates, that under optimal conditions a spatial sensing resolution of 5 cm in the vicinity of a mobile phone is possible.
... This has to be done prior the the cartogram generation. An overview of map projection can be found in Snyder (1997). Map projections aim to optimize towards different objectives, e.g., conformal mapping (preservation of local angles) or area mapping (preservation of area). ...
Cartogram drawing is a technique for showing geography-related statistical information, such as demographic and epidemiological data. The idea is to distort a map by resizing its regions according to a statistical parameter by keeping the map recognizable. This article describes an R package implementing an algorithm called RecMap which approximates every map region by a rectangle where the area corresponds to the given statistical value (maintain zero cartographic error). The package implements the computationally intensive tasks in C++. This paper's contribution is that it demonstrates on real and synthetic maps how recmap can be used, how it is implemented and used with other statistical packages.
... Furthermore, to facilitate the computation of these distances and the subsequent travel times for each vehicle, we converted geographic coordinates into Cartesian coordinates using Miller projections [12]: ...
Parking challenges escalate significantly during large events such as concerts and sports games, yet few studies address dynamic parking lot assignments in these occasions. This paper introduces a smart navigation system designed to optimize parking assignments efficiently during major events, employing a mixed search algorithm that considers diverse drivers characteristics. We validated our system through simulations conducted in Berkeley, CA during the "Big Game" showcasing the advantages of our novel parking assignment approach.
... Furthermore, to facilitate the computation of these distances and the subsequent travel times for each vehicle, we converted geographic coordinates into Cartesian coordinates using Miller projections [11]: ′ = ( ) = ′ = ( ) = 1.25 ⋅ ln ( ( 4 + 0.4 ⋅ )) = 6381372 ⋅ ⋅ 2 ...
Parking challenges escalate significantly during large events such as concerts and sports games, yet few studies address dynamic parking lot assignments in these occasions. This paper introduces a smart navigation system designed to optimize parking assignments efficiently during major events, employing a mixed search algorithm that considers diverse drivers characteristics. We validated our system through simulations conducted in Berkeley, CA during the "Big Game" showcasing the advantages of our novel parking assignment approach.
... Since shapely requires projected coordinates in meters, geographic coordinates (latitude/longitude) are transformed before any area calculations. Folium, which specializes in visualizing geographic data, typically assumes coordinates in degrees (ellipsoidal axes, WGS84 format-EPSG code 4326, unless specified otherwise), automatically converting them to meters in the pseudo-Mercator projection (cartesian axes, EPSG code 3857) for map display [23,24]. To perform these coordinate transformations and accurately compute the coverage area in square meters, pyproj is used to establish a transformer that converts coordinates from the WGS84 system to the projection used for the Universal Transverse Mercator (UTM)-specific zone that corresponds to the area under study. ...
This study introduces a novel method that addresses the challenge of visualizing and quantifying detection coverage areas in wireless sensor networks. The method involves projecting a network of range-free sensors and pre-existing transmitters, located within a predefined area of interest, onto a global coordinate system. Detection areas are defined as those covered by the sensing range of at least three sensors. Pre-existing transmitters located within the detection range of the sensors are assumed to degrade the networks’ performance by causing coverage gaps. Interactive satellite maps facilitate the dynamic exploration of coverage via the calculation and visualization of the resulting detection areas. The algorithmic structure of the proposed tool is explained in detail, and four example scenarios demonstrate the tool’s capabilities, as well as its flexibility, adaptability, and effectiveness in identifying the triangulated detection areas. Designed primarily as a geometry calculation and visualization tool that allows for the adjustment of sensor parameters such as locations, ranges, and angular ranges of detection, the proposed tool has the potential to enhance decision-making in sensor network configuration, prior to final sensor placement, across a wide range of applications.
... This has included advances into the various ways of representing spherical images in existing 2D formats. Such representations are not new, for they have been used by painters and in the cartography space for a while [17], where map projections are used to represent the globe on a 2D surface [30]. Different projections will exhibit different distortions [4]. ...
Spherical or omni-directional images offer an immersive visual format appealing to a wide range of computer vision applications. However, geometric properties of spherical images pose a major challenge for models and metrics designed for ordinary 2D images. Here, we show that direct application of Fr\'echet Inception Distance (FID) is insufficient for quantifying geometric fidelity in spherical images. We introduce two quantitative metrics accounting for geometric constraints, namely Omnidirectional FID (OmniFID) and Discontinuity Score (DS). OmniFID is an extension of FID tailored to additionally capture field-of-view requirements of the spherical format by leveraging cubemap projections. DS is a kernel-based seam alignment score of continuity across borders of 2D representations of spherical images. In experiments, OmniFID and DS quantify geometry fidelity issues that are undetected by FID.
... Equirectangular Projection (ERP) [19]: It streams 360 • video in the naive way as same as the ordinary video. The entire 360 • scene is streamed in a constant quality, without prioritizing the quality within the user's viewport. ...
360∘ videos have become increasingly popular recently, but consume much more bandwidth than non-360∘ videos. Usually, 360∘ video streaming partitions the video surface into multiple tiles and encodes the tiles independently to effectively and flexibly use limited link bandwidth. However, current bitrate adaptive algorithms generally aim to maximize the bitrate, rather than perceptual quality, resulting in degradation of user experience. More importantly, we reveal that the distribution of tile size is very skewed, that is, a small number of large tiles consumes more bandwidth than a large number of small tiles, further hurting the overall viewing quality. Therefore, in this paper, we propose a tile-size aware bitrate allocation scheme TSA for adaptive 360∘ video streaming to improve the viewing experience of users. Specifically, TSA cautiously decreases the quality of a few large tiles to allocate more bandwidth to large number of small tiles, thus improving the perceptual quality of overall video, without sacrificing large tiles excessively. Experiments over real-world datasets show that TSA effectively improves V-VMAF by up to 39% compared with several state-of-the-art adaptive bitrate algorithms.
... Hence, trade-offs are necessary. Likewise, the problem of mapping spheres to 2D pictures has been thoroughly studied in cartography, with hundreds of different projections developed (Snyder, 1993). ...
This paper reviews projection models and their perception in realistic pictures, and proposes hypotheses for three-dimensional (3D) shape and space perception in pictures. In these hypotheses, eye fixations, and foveal vision play a central role. Many past theories and experimental studies focus solely on linear perspective. Yet, these theories fail to explain many important perceptual phenomena, including the effectiveness of nonlinear projections. Indeed, few classical paintings strictly obey linear perspective, nor do the best distortion-avoidance techniques for wide-angle computational photography. The hypotheses here employ a two-stage model for 3D human vision. When viewing a picture, the first stage perceives 3D shape for the current gaze. Each fixation has its own perspective projection, but, owing to the nature of foveal and peripheral vision, shape information is obtained primarily for a small region of the picture around the fixation. As a viewer moves their eyes, the second stage continually integrates some of the per-gaze information into an overall interpretation of a picture. The interpretation need not be geometrically stable or consistent over time. It is argued that this framework could explain many disparate pictorial phenomena, including different projection styles throughout art history and computational photography, while being consistent with the constraints of human 3D vision. The paper reviews open questions and suggests new studies to explore these hypotheses.
... The grid resolution is 1°× 5°in latitude and longitude, is explicitly shown in the figure. Since the bins are located on a spherical grid, bin areas become slightly larger at lower latitudes (Snyder, 1993). Therefore, this will bring more importance to lowlatitude areas with intense ground dB/dt variations as will be shown below in Figures 8-10. ...
In this study, we investigate the effects caused by interplanetary (IP) shock impact angles on the subsequent ground dB/dt variations during substorms. IP shock impact angles have been revealed as a major factor controlling the subsequent geomagnetic activity, meaning that shocks with small inclinations with the Sun‐Earth line are more likely to trigger higher geomagnetic activity resulting from nearly symmetric magnetospheric compressions. Such field variations are linked to the generation of geomagnetically induced currents (GICs), which couple to artificial conductors on the ground leading to deleterious consequences. We use a sub‐set of a shock data base with 237 events observed in the solar wind at L1 upstream of the Earth, and large arrays of ground magnetometers at stations located in North America and Greenland. The spherical elementary current system methodology is applied to the geomagnetic field data, and field‐aligned‐like currents in the ionosphere are derived. Then, such currents are inverted back to the ground and dB/dt variations are computed. Geographic maps are built with these field variations as a function of shock impact angles. The main findings of this investigation are: (a) typical dB/dt variations (5–10 nT/s) are caused by shocks with moderate inclinations; (b) the more frontal the shock impact, the more intense and the more spatially defined the ionospheric current amplitudes; and (c) nearly frontal shocks trigger more intense dB/dt variations with larger equatorward latitudinal expansions. Therefore, the findings of this work provide new insights for GIC forecasting focusing on nearly frontal shock impacts on the magnetosphere.
... The usage of developable surfaces in teaching map projections is often justified by the term conceptual [2,12,13], but this makes scientific verification impossible. It is usually said that this makes understanding map projections easier [14]. ...
The literature usually calls downscaled versions of basic conformal map projections “secant”, referring to conceptual developable map surfaces that intersect the reference frame. However, recent studies pointed out on the examples of various mappings of the sphere that this model may lead to incorrect conclusions. In this study, we examine the paradigm of secant surfaces for two popular map projections of the ellipsoid, the UTM (Universal Transverse Mercator) and the UPS (Universal Polar Stereographic) projections. Results will show that ellipsoidal map projections can exhibit further anomalies. To support the shift to a paradigm avoiding developable map surfaces, this study recommends the new term reduced map projection with a proper and simple definition to be used instead of secant map projections.
... The study of map projections is primarily about the characterization of their distortions. There is no limit to the number of possible map projections (36). More generally, projections are considered in several fields of pure mathematics, including differential geometry, projective geometry, and manifolds. ...
Background : The outbreak of the new coronavirus, also known as COVID-19, began in Wuhan, China, in January 2020, becoming a sudden public health crisis and a severe threat to lives in most parts of the world. This study aimed to use a Geographic Information System (GIS) to study the spatial distribution of COVID-19 incidence in Ho Chi Minh city. Methods: the histogram was first used to study the distribution of the COVID-19 cases and COVID-19 incidence. A GIS was then employed to map the spatial distribution of the COVID-19 cases and COVID-19 incidence. Finally, the study results were discussed and concluded. Results: a large proportion of COVID-19 infections mainly appeared in districts in the eastern region, then followed by the western districts of the city, while the average rate of infections was concentrated in districts near the city center. Low rates of COVID-19 infection were detected in the northern and southern districts of the city and some central districts of the city. Conclusions: the study results indicated the effectiveness of a GIS for mapping COVID-19 incidence. Findings in this study provide an insight into the spatial distribution of infectious diseases and make great contributions to the control of the COVID-19 pandemic.
... where R ≈ 6378137 m denotes the radius of earth. The approximation in (8) comes from the equirectangular projection, transforming spherical coordinates into planar coordinates [25], and ψ 0 denotes the standard parallels decided by the scale of the projection. Note that since the size of the area of the experiment site is small, the approximation is sufficiently accurate. ...
Ensuring reliable and seamless wireless connectivity for unmanned aerial vehicles (UAVs) has emerged as a critical requirement for a wide range of applications. The increasing deployment of UAVs has increased the significance of cellular-connected UAVs (C-UAVs) in enabling beyond-visual line of sight (BVLOS) communications. To ensure the successful operation of C-UAVs within existing terrestrial networks, it is vital to understand the distinctive characteristics associated with air-to-ground signal propagation. In this paper, we investigate the impact of 3D antenna patterns on a UAV air-toground path loss model, utilizing datasets obtained from a measurement campaign. We conducted UAV experiments in a rural area at various fixed heights, while also characterizing the 3D antenna radiation pattern by using an anechoic chamber facility. By analyzing reference signal received power (RSRP) using path loss models that account for antenna patterns, we observed that our measurement results, obtained at different UAV heights, aligned well with the two-ray path loss model when incorporating the measured antenna pattern. We propose an RSRP-based localization algorithm at a UAV that takes into account antenna patterns in both offline and online scenarios. Through our experimentation dataset, we show that incorporating measured antenna patterns significantly enhances the source localization accuracy.
... W). This equivalent projection allowed us to preserve the surface areas locally (Snyder 1993) and compare the HR sizes for all individuals, regardless of the latitude of their wintering sites. In view of the high fidelity of curlews to their wintering sites across years (Bainbridge and Minton 1978;Delany et al. 2009;Brown 2015;Schwemmer et al. 2016;Sanders and Rees 2018), we compared the HRs of all individuals regardless of the year. ...
Coastal, and to a lesser extend inland wetlands, are critical habitats for wintering shorebirds. Given the significant population declines of most shorebird species worldwide, the current degradation of coastal habitats through climate change and human activities raises severe conservation concerns. In order to ensure sufficient and adequate habitats and maintain the populations, a thorough understanding of space use by wintering shorebirds is urgently required. However, overwintering strategies have rarely been investigated throughout the entire range of a shorebird species. This study thus aimed to investigate the spatio-temporal use of wintering habitats by Eurasian curlew Numenius arquata on a European scale, using a large international dataset. A total of 204 adult curlews were tagged with GPS devices at different wintering and breeding sites across Europe between 2014 and 2021, and the data were used to analyse home range size, habitat use, and phenology. The birds were faithful to their wintering sites throughout the winter. Their home ranges were small compared with other shorebirds but highly variable between individuals (533 ± 449 ha). Winter home range areas did not differ in relation to sex or body mass, but were weakly related to the wintering latitude, particularly linked to the many birds wintering in the Wadden Sea. Curlews used a high diversity of coastal and inland habitats, with higher occurrences on mudflats and saltmarshes. Despite the inter-individual variability in space use, the high wintering-site fidelity shown by this near-threatened species raises concerns about its capacity to respond to environmental modifications in coastal regions.
... More possible projections exist which are not special cases of the general perspective projection, for example the azimuthal equidistant projection, which renders the distances and angles from the chosen central point correctly [OCH19]. Dozens of spherical projections are used in cartography [Sny97]; many of them have hyperbolic analogs that are available in the HyperRogue engine [KCv17]. ...
In this paper we discuss computer simulations of de Sitter and anti de Sitter spacetimes. We present prototype games played in these spacetimes, discuss the technical challenges in creating such a simulation, and discuss the geometric and relativistic effects that can be witnessed by people playing our games and watching our visualizations.
... where R ≈ 6378137 m denotes the radius of earth. The approximation in (8) comes from the equirectangular projection, transforming spherical coordinates into planar coordinates [18], and ψ 0 denotes the standard parallels decided by the scale of the projection. Note that since the size of the area of the experiment site is small, the approximation is sufficiently accurate. ...
Ensuring reliable and seamless wireless connectivity for unmanned aerial vehicles (UAVs) has emerged as a critical requirement for a wide range of applications. The increasing deployment of UAVs has increased the significance of cellular-connected UAVs (C-UAVs) in enabling beyond-visual line of sight (BVLOS) communications. To ensure the successful operation of C-UAVs within existing terrestrial networks, it is vital to understand the distinctive characteristics associated with air-to-ground signal propagation. In this paper, we investigate the impact of 3D antenna patterns on a UAV air-to-ground path loss model, utilizing datasets obtained from a measurement campaign. We conducted UAV experiments in a rural area at various fixed heights, while also characterizing the 3D antenna radiation pattern by using an anechoic chamber facility. By analyzing reference signal received power (RSRP) using path loss models that account for antenna patterns, we observed that our measurement results, obtained at different UAV heights, aligned well with the two-ray path loss model when incorporating the measured antenna pattern. we propose an RSRP-based localization algorithm at a UAV that takes into account antenna patterns in both offline and online scenarios. Through our experimentation dataset, we show that incorporating measured antenna patterns significantly enhances the source localization accuracy.
... A map projection represents flattening the surface of a globe into a plane (Snyder, 1987;Snyder and Voxland, 1989). Several map projection approaches have been proposed (Snyder, 1997). However, all of them distort the surface, resulting in distortions of regions, shapes, directions and distances. ...
In this paper, we propose a novel approach to visualizing global geographical information: a panoramic sphere in an immersive environment. The whole geographical surface can be observed through the rotating of heads as the viewpoint of the panoramic sphere is inside the sphere. We compared three approaches to visualizing the earth for rendering the geographical information in a virtual reality environment. On the tasks of terrestrial and marine geographical information, we compare the visualization effects on a) a globe, b) a flat map and c) a panoramic sphere. Terrestrial geographical information tasks include the area comparison and direction determination. Marine geographical information tasks contain the visualization of sea surface temperature and sea surface currents. For terrestrial geographical information tasks, the experimental results show that the panoramic sphere outperforms the globe and the flat map, with a higher average accuracy and a shorter time. On marine geographical information task, the panoramic sphere visualization is also superior to the flat map and the globe in an immersive environment for the sea surface temperature data and the sea surface current fields. In all three visualization experiments, the panoramic sphere is most preferred by the participants, particularly for global, transcontinental and transoceanic needs.
In this paper, we present a new spherical image representation, called unisphorm , and show its strong potential in robotic vision. unisphorm provides an accurate and distortion-free representation of a 360-degree image, by relying on multiple subdivisions of an icosahedron and its associated Voronoi diagrams. The geometric mapping procedure is described in detail, and the trade-off between pixel accuracy and computational complexity is investigated. To demonstrate the benefits of unisphorm in real-world problems, we applied it to direct visual attitude estimation and (vpr), by considering dual-fisheye images captured by a camera mounted on multiple robotic platforms. In the experiments, we measured the impact of the number of subdivision levels of the icosahedron on the attitude estimation error, time efficiency, and size of convergence domain of an existing visual gyroscope, using unisphorm and three competing mapping algorithms. A similar evaluation procedure was carried out for vpr . Finally, two new omnidirectional image datasets, one recorded with a hexacopter, called SVMIS+ , the other based on the Mapillary platform, have been created and released for the entire research community.
Extensive cartometric analyses were performed on the portolan atlases of Pietro Vesconte (1313), Andrea Bianco (1436), and Battista Agnese (1538), along with their manually assembled composites and contemporaneously made portolan charts. The results indicate that none of the small-scale atlas sheets were fully aligned with the contemporary magnetic north, that they typically exhibit coastline contours mapped to different scales, and that the spatiums (50 miglia intervals) in Vesconte's and Bianco's atlases differ in length subsequent to the assembly of their composites. Consequently, the atlas sheets could not provide sailors with consistent navigational accuracy across the whole displayed area. Moreover, notable geometric differences can be observed between the composite of Vesconte's atlas and his 1311 portolan chart, as well as between the composite of Bianco's atlas and his smaller-scale chart bound in the same atlas, indicating that their creators had a limited understanding of their geometry. The discovery of cartometrically determined underlying mosaics of subsections encapsulated within their composites with accuracy twice as great—whose spatial extents have remained almost unchanged during more than two centuries of their production and geometrically correspond to portolan charts created before them—suggests that portolan atlases are medieval and early modern copies of earlier-made cartographic sources.
Mollweide projeksiyonu, yeryüzünün tamamının gösterimi amaçlı, alan koruyan gerçek anlamda olmayan silindirik bir projeksiyondur. Yeryüzünü bir elips içerisinde temsil eder. Çoğu kaynakta yalnızca projeksiyon bağıntıları verilmiştir. Bu çalışmada projeksiyon geometrik yapısı ayrıntılı olarak irdelenmiş, projeksiyon fonksiyonlarının nasıl elde edildiği gösterilmiştir. Projeksiyonun yol açtığı deformasyonlar tartışılmıştır. Projeksiyonda kullanılan yardımcı değişkenin bulunması için Python SciPy modülü fonksiyonlarının kullanımı ele alınmış, üç fonksiyonun karşılaştırması yapılmıştır. Son olarak projeksiyon hesaplamalarına yönelik örnek bir sınıf kodu verilmiştir. Burada ele alınan yazılımsal çözümler enleme bağlı yardımcı değişken kullanan diğer projeksiyonlara da uyarlanabilir.
We present a simulation protocol that allows for efficient sampling of the degrees of freedom of a solute in explicit solvent. The protocol involves using a non-equilibrium umbrella sampling method, in this case the recently developed adaptively biased molecular dynamics method, to compute an approximate free energy for the slow modes of the solute in explicit solvent. This approximate free energy is then used to set up a Hamiltonian replica exchange scheme that samples both from biased and unbiased distributions. The final accurate free energy is recovered via the WHAM technique applied to all the replicas, and equilibrium properties of the solute are computed from the unbiased trajectory. We illustrate the approach by applying it to the study of the puckering landscapes of the methyl glycosides of -L-iduronic acid and its C5 epimer -D-glucuronic acid in water. Big savings in computational resources are gained in comparison to the standard parallel tempering method.
This chapter investigates the potential of graph neural networks (GNNs) in multi-image super-resolution (MISR), positioning them against current state-of-the-art techniques. It introduces foundational concepts of super-resolution reconstruction (SRR), with a focus on the distinctions between single-image super-resolution (SISR) and MISR, and reviews key models in the field. Theoretical underpinnings of graph theory relevant to SRR are explored, alongside detailed descriptions of the architecture and data preparation methodologies for two GNN models designed specifically for MISR tasks. An evaluation conducted using benchmark datasets assesses these models through a training procedure and a comprehensive set of performance metrics. Findings reveal both the strengths and limitations of GNNs in enhancing image resolution, showcasing their effectiveness, and identifying areas where improvements are needed compared to leading methods. The chapter concludes with a discussion of the implications of these results, highlighting the promising yet challenging role of GNNs in MISR and suggesting directions for future research to advance this field.
Seven portolan charts and atlases (including the anonymous Rex Tholomeus chart) made between 1311 and 1538 were cartometrically analysed. The findings indicate that their geometry was not improved chronologically and that the accuracy of Pietro Vesconte's charts (1311–13) was never surpassed. Portolan charts and atlases appear to be mosaics composed of subsections, each with variable metrics and accuracy levels that are, on average, twice as great in comparison to their Mediterranean and Black Sea areas (as a whole), and it is highly unlikely that they are genuine late medieval products, which is also supported by historical evidence. Their anticlockwise tilts fit well with the tilt of the Gibraltar-Antioch line on modern maps that closely approximate Ptolemy I and Ptolemy II projections centred near Gibraltar and with their meridian tilts for areas around Genoa. If late medieval Italian cartographers had acquired a collection of maps or charts from before the Middle Ages, it is theoretically feasible that they could have achieved the typical geometry of portolan charts by graphically combining their manuscript copies of large-scale regional maps in Mercator or Mercator-like projection, using a small-scale map in conic or pseudoconic projection (on which the convergence of meridians is displayed) as a template.
This paper elaborates on the results of an exhaustive study regarding the Earth's position in the universe based on the Cosmic Microwave Background (CMB) radiation map, which is the latest discovery in modern astronomy. CMB radiation provides crucial insights into the early distribution of mass and energy in the universe. The aim of this research is to understand the theory and mechanisms behind the formation of polarity structures in the CMB and analyze their correlation with Earth's position in the overall structure of the universe. Intensity measurement data of CMB radiation published by COBE, WMAP, and Planck present temperature distribution data in coordinates in FITS (Flexible Image Transport System) format files. Subsequently, a spherical harmonic transformation is performed to obtain spherical harmonic coefficients a_lm as equations that represent dipole, quadrupole, octupole models, and various other multipole models. The analysis of the correlation in the temperature distribution of CMB radiation involves detailing various patterns found in the dipole, quadrupole, and octopole models, demonstrating quasi-symmetry characteristics with Earth at its center. An analysis of anisotropic CMB data yields an interesting hypothesis that the Earth's position plays a role in shaping the structure of the universe on a certain scale. Even more extremely, it can be said that Earth is at the center of the universe. This finding prompts profound contemplation about Earth's position in the structure of the cosmos, opening the door for further research in this field.
Three-dimensional cryo electron microscopy reconstructions are obtained by extracting information from a large number of projections of the object. These projections correspond to different `views' or `orientations', i.e. directions in which these projections show the reconstructed object. Uneven distribution of these views and the presence of dominating preferred orientations may distort the reconstructed spatial images. This work describes the program VUE (views on uniform grids for cryo electron microscopy), designed to study such distributions. Its algorithms, based on uniform virtual grids on a sphere, allow an easy calculation and accurate quantitative analysis of the frequency distribution of the views. The key computational element is the Lambert azimuthal equal-area projection of a spherical uniform grid onto a disc. This projection keeps the surface area constant and represents the frequency distribution with no visual bias. Since it has multiple tunable parameters, the program is easily adaptable to individual needs, and to the features of a particular project or of the figure to be produced. It can help identify problems related to an uneven distribution of views. Optionally, it can modify the list of projections, distributing the views more uniformly. The program can also be used as a teaching tool.
360-degree videos, as a type of media that offers highly immersive experiences, often result in significant bandwidth waste due to incomplete views by users. This places a heavy demand on streaming systems to support high-bandwidth requirements. Recently, tile-based streaming systems combined with viewport prediction have become popular to improve bandwidth efficiency. However, since the viewport prediction is only reliable in the short term, maintaining a long buffer to avoid rebuffering is challenging. We propose
JUST360
, a joint utility based two-tier 360-degree video streaming system in this paper. To better improve the accuracy of utility evaluation, a utility model incorporates image quality and prediction accuracy is proposed to evaluate the contribution of each tile so that longer buffer and bandwidth efficiency can coexist. The optimal bitrate allocation strategy is determined by using
model predictive control
(MPC) to dynamically select the tiles according to their characteristics. Experiments show that our method successfully achieves higher PSNR and less rebuffering. Compared with other state-of-the-art methods, our proposed method can outperform the other methods by 3%-20% in terms of QoE.
This paper focuses on digitally-supported research methods for an important group of cultural heritage objects, the Greek pottery, especially with figured decoration. The design, development and application of new digital methods for searching, comparing, and visually exploring these vases need an interdisciplinary approach to effectively analyse the various features of the vases, like shape, decoration, and manufacturing techniques, and relationships between the vases. We motivate the need and opportunities by a multimodal representation of the objects, including 3D shape, material, and painting. We then illustrate a range of innovative methods for these representations, including quantified surface and capacity comparison, material analysis, image flattening from 3D objects, retrieval and comparison of shapes and paintings, and multidimensional data visualization. We also discuss challenges and future work in this area.
Ovaj rad ima za cilj istražiti bi li Apian mogao biti autor kartografskih projekcija koje se po njemu zovu. Dokazi sugeriraju da ta preslikavanja nisu trebala biti nazvana po Apianu. Najvjerojatniji izumitelj kartografske projekcije Apian I je Amerigo Vespucci, a projekciju Apian II vjerojatno je tek 300 godina kasnije razvio François Arago. S obzirom na to da slučaj Apianovih kartografskih projekcija nije jedinstveni primjer kada kartografska projekcija nije nazvana po svom pravom izumitelju i da bi autorstvo više kartografskih projekcija moglo biti dovedeno u pitanje u budućnosti, kartografska bi zajednica trebala preispitati je li to dobra konvencija nazvati kartografske projekcije po njihovom navodnom izumitelju.
Anonymous Carte Pisane (c. 1275) and Cortona chart (c. 1300), as well as Pietro Vesconte's charts, made in 1311 and 1313, which represent some of the earliest known portolan charts, were cartometrically analysed to examine the geometric similarities between their coastal renderings. The research results show that not only the majority of their territorial coverage was drawn very similarly to each other but also that certain parts of the coastline are nearly identical between all the examined charts. Also, the magnitude and orientation of displacement vectors of residuals revealed that the Mediterranean and Black Sea areas on charts contain seven subsections which are on average, twice as accurate in comparison to the same area treated as a whole. The fundamental conclusion is that there is a high probability that the coastline renderings on the earliest known portolan charts are, actually, more or less skilfully made copies of the same source material used as a graphic template. The hypothetical source might have been an atlas whose origins date back to before the Middle Ages, containing charts whose extents were, perhaps, similar to the extents of detected subsections, which medieval cartographers were not able to assemble correctly due to their ignorance of map projections.
Satellite Coverage Analysis is a fundamental performance assessment element in remote sensing and communications services projects. Coverage is a key parameter in the constellation operation and design for missions relying on several satellites. Since coverage areas over the surface of the Earth change with time, intersecting and drifting apart, the dynamics of every satellite influence the constellation's behavior as a whole. For this reason, every configuration change, be it in the number of satellites or their relative positions, heavily impacts the cost/performance of the mission. This paper presents a constellation-to-ground coverage analysis model that enables the rapid evaluation of areas on the surface of the Earth. The method leverages geodetic projections and an oblate Earth model and uses dynamic transformation and anti-transformation techniques combined with polygon Boolean operations. Timestamped datasets are obtained to account for the dynamics of the scenario, which can be exploited in statistical coverage analysis of the constellation. Our empirical evaluations show that this approach is superior in accuracy and computation effort compared to traditional net-point techniques. While net-point approaches are at the core of state-of-the-art commercial software, they are approximate. We show that, for finer grid granularity, the net-point schemes converge to our polygon-based results.
This paper documents a famous cartographic controversy played out in the cartographic literature for more than 15 yr. In 1974, a German historian, Arno Peters, introduced what he called The Peters Projection to the cartographic community. It was immediately and roundly condemned as being a copy of one by James Gall, a nineteenth-century evangelist. Both men are discussed in their own contexts. The controversy that then arose is used to illustrate several matters: first, that intellectual traditions proceed through debate and contest, rather than by increasingly accurately describing the world; second, that ideology is an instrumental aspect of cartography. Both these points inform my sense of defining moment. It is argued that post-Peters cartography should engage ideology for a fully informed theory of cartography. -Author
A comprehensive overview of technological advances in cartography, examining the origins of systematic surveys and national survey organisations, present-day map making and publishing, the transition from paper map to geographic data base, and potential future uses of maps. An introductory chapter addresses the past century of technological progress and institutional developments. Chapter 2, location and navigation, investigates navigational techniques ranging from aboriginal way-finding, maritime charts and map projection, through to satellite navigation and warfare systems. Chapter 3, on boundaries and surveys, covers surveying techniques, lasers and electronic distance measuring, aerotriangulation, stereo plotting, geodesy, and surveying networks. Chapter 4 examines aerial surveying, photogrammetry, remote sensing, image analysis, and policy implications. Chapter 5 investigates decision support systems such as computer assisted cartography, geographic information systems, and national digital data banks. Chapter 6 covers map publishing and digital maps, while a final section discusses public policy aspects, data security implications, and humanistic challenges.-after Author
Classification is a fundamental method of science, and seeks to introduce simplicity and order into the bewildering multiplicity of things that confront the human mind by formulating a scheme of mutually exclusive and collectively exhaustive categories based on the important characteristics of the things concerned and on the actual relations between them. The first attempt at any systematic classification of map projections appears to have been that of Tissot, in his “Mémoire sur la Représentation des Surfaces et les Projections des Cartes Géographiques”, 1881, but this side of the subject has been generally neglected by English writers. Many have, in fact, commented upon the impossibility of classifying the known projections into mutually exclusive groups, and Close (“Text-Book of Topographical and Geographical Surveying”, 1905), in listing seven descriptive terms, states “they are not all mutually exclusive, … and to treat them as classes would be to fall into his error who divided the human race into Frenchmen, red-haired people and cannibals”.
320 pages. $21.95. Contents: Preface. 1. Required reading. 2. Excel as workhorse. 3. Tales of the toolbar and other aides-de-camp
- Lee By Woody Leonhard
- T J Hudspeth
- Lee
- O'reilly
- C A Sebastopol
Excel 97 Annoyances. By Woody Leonhard, Lee Hudspeth and T. J. Lee. O'Reilly, Sebastopol, CA. (1997).
320 pages. $21.95.
Contents:
Preface. 1. Required reading. 2. Excel as workhorse. 3. Tales of the toolbar and other aides-de-camp. 4. How
Excel works. 5. Using VBA to customize Excel. 6. Excel strategies. 7. Excel in the office. 8. Where and how to
get help. Index.