No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
LIII. Explanation of a projection by balance of errors for maps applying to a very large extent of the earth's surface; and comparison of this projection with other projections
... The algorithm proposed in Reference 2 introduces new distortion-minimizing parameterizations of attitude inspired by Airy map projections [11][12][13] that are generalized to higher dimensions. The resulting parameterizations reside in the three-dimensional projected space in which Euclidean distances between points closely resemble rotational displacements between the attitudes that these points represent. ...
... if any point on path outside admissible region (11) Here i is the count, starting at 0, of the latest grid point r(i) added to the evaluated path, and i n , i n are the integer Cartesian indexes assigned to the current and target grid points, respectively, within the regular three-dimensional grid. ...
... (10),(11) become f (r (i)) = i + 2 n=1 |i n − i n | if path inside admissible region ∞if any point on path outside admissible region(27) ...
A new algorithm suitable for fast on-board implementation is presented for designing attitude maneuver paths that achieve desired boresight alignment in the presence of multiple three-axis attitude constraints. It is based on a recently proposed algorithm in which possible attitudes are discretized in a three-dimensional distortion-minimizing projected space. In that space grid points are enumerated and graph search pathfinding algorithms are employed to find the shortest constrained path between selected initial and target grid points. The new algorithm extends this approach from a single target point to a continuum of target points representing all possible attitudes that maintain the desired boresight alignment. It augments the original algorithm with the transformation that makes it possible to assign a single value to estimated distances for the whole continuum of target points. This, in turn, allows the pathfinding algorithms to efficiently determine the shortest overall path to the whole continuum, which corresponds to the shortest attitude maneuver path that achieves the desired boresight alignment.
... Distortions of projections are traditionally measured by Tissot's indicatrix (Tissot, 1878). Airy (1861) was the first to apply distortion criteria based on the semi-axes of Tissot's indicatrix. The theory of such infinitesimal distortions was later developed by Kavrayskiy (1934), Frančula (1971) and Meshcheryakov (1968). ...
... This method was first used by Airy (1861). Bayeva (1987) showed that this method does not treat areal scales 2 and 1/2 the same, although they equally distort the map. ...
The traditional way to calculate the global distortion of a given area in a map projection is to create what we call a local distortion criterion that is a function of the infinitesimal semi-axes of the Tissot's indicatrix. Some contemporary scholars criticize this method, saying that the map readers face distortion of the finite type. These researchers suggest taking plenty of simple random spherical elements (line sections, triangles) and average the distortion on them. Although the aforementioned researchers all state that their approach is something fundamentally different from the traditional method, the major disadvantage is that this method is irreproducible. Therefore, it has to be investigated whether the difference is really significant between these methods and if it is, what its nature is. At first, different distortion values are evaluated on a huge number of various projections showing the whole Earth. Correlation analysis shows that there exists a strong linear dependence between the corresponding infinitesimal and finite measures. A considerable difference can be observed if the examined area is not the whole globe rather a part of it. After optimizing a projection for different distortion measures, the isolines of equal distortion follow the boundary lines significantly closer using the traditional approach.
... Cartographic projections can be defined as a mathematical transformation of a surface of an ellipsoid (defined by its semimajor axis a and eccentricity e) or a sphere (defined by its radius R) onto a plane [4]. A point on the surface of an ellipsoid is referenced by its latitude φ and longitude λ or by isometric coordinates q, λ: (1) The mathematical transformation between its ellipsoidical coordinates φ, λ and planar coordinates x, y is given by map equations: ...
... Cartographic projections can be evaluated (with respect to extremal and minimax criteria) by the maximal value of the scale distortion |m -1|max or using the RMS -13 -value of scale distortion throughout the territory according to Airy's, Jordan's and Kavraiskii's variational criteria [1], [5], [9]. The most popularized variational criterion for the valuation of map projections is Airy-Kavraiskii's criterion, where the characteristic value of the cartographic projection of the domain Δ with area pΔ on reference surface is: ...
The choice of the optimal cartographic projection, especially for large-scale maps, is an actual problem affected by the precision of positioning geodetic points using the new GNSS technologies in the coordinate systems. In this contribution we describe the map projections designed by minimax type criteria, Airy-Kavraiskii's variational criterion and map projections with a minimal RMS distortion in the category of conic, azimuthal and cylindrical projections. The aim of this paper is to compare the mentioned criteria based on the achieved values of scale distortion in the selected European countries. © 2017, Budapest Tech Polytechnical Institution. All rights reserved.
... L'histoire des représentations de déformations minimales (Lee [5], Hendrickz [3]) apparaît revenir au Sir George Airy [1] dont le concept pour minimiser la distosion totale du module linéaire d'une carte, quoique à l'origine appliquée pour trouver un compromis entre les représentations conformes et équivalentes, peut être appliquée pour une seule représentation conforme. ...
... Par analogie avec la représentaion de la "balance of errors" d'Airy [1], nous permettons de définir la distorsion totale T 2 de la représentation d'une surface arbitraire S de l'ellipsoïde par : ...
It is a translation in French of the paper " A Conformal Mapping Projection With Minimum Scale Error", written by
W.I. Reilly and published in Survey Review, Volume XXII, n°� 168 - April 1973, with more details of some formulas and giving some critics .
... Airyjeva i Jordanova ocjena KiG 2009, 11 The criterion used was the Airy/Jordan criterion for conformal projections. The Airy and Jordan distortion mesure is often used for evaluating distortions over the area (Canters 2002, Frankiae 1982, Reilly 1973, Franèula 1971, Airy 1861). ...
... A choice of map projection for an area does not substantially depend on the high accuracy of boundary coordinates of the area, but primarily on the shape and size of the area. Data for defining the administrative area were taken from the Euro Global Map 1:1 000 000 for Croatia published by the State Geodetic Administration deformacija èesto se upotrebljava prilikom ocjene deformacija na nekon podruèju (Canters 2002, Frankiae 1982, Reilly 1973, Franèula 1971, Airy 1861. ...
The paper describes optimal conformal polynomial projections for Croatia according to the Airy/Jordan criterion. A brief introduction of history and theory of conformal mapping is followed by descriptions of conformal polynomial projections and their current application. The paper considers polynomials of degrees 1 to 10. Since there are conditions in which the 1st degree polynomial becomes the famous Mercator projection, it was not considered specifically for Croatian territory. The area of Croatia was defined as a union of national territory and the continental shelf. Area definition data were taken from the Euro Global Map 1:1 000 000 for Croatia, as well as from two maritime delimitation treaties. Such an irregular area was approximated with a regular grid consisting of 11 934 ellipsoidal trapezoids 2'large. The Airy/Jordan criterion for the optimal projection is defined as minimum of weighted mean of Airy/Jordan measure of distortion in points. The value of the Airy/Jordan criterion is calculated from all 11 934 centres of ellipsoidal trapezoids, while the weights are equal to areas of corresponding ellipsoidal trapezoids. The minimum is obtained by Nelder and Mead's method, as implemented in the fminsearch function of the MAT-LAB package. Maps of Croatia representing the distribution of distortions are given for polynomial degrees 2 to 6 and 10. Increasing the polynomial degree results in better projections considering the criterion, and the 6th degree polynomial provides a good ratio of formula complexity and criterion value.
... Airyjeva i Jordanova ocjena KiG 2009, 11 The criterion used was the Airy/Jordan criterion for conformal projections. The Airy and Jordan distortion mesure is often used for evaluating distortions over the area (Canters 2002, Frankiae 1982, Reilly 1973, Franèula 1971, Airy 1861). ...
... A choice of map projection for an area does not substantially depend on the high accuracy of boundary coordinates of the area, but primarily on the shape and size of the area. Data for defining the administrative area were taken from the Euro Global Map 1:1 000 000 for Croatia published by the State Geodetic Administration deformacija èesto se upotrebljava prilikom ocjene deformacija na nekon podruèju (Canters 2002, Frankiae 1982, Reilly 1973, Franèula 1971, Airy 1861. ...
... Međutim, ako prijeđemo na numeričko integriranje onda oba ova problema nestaju. Zapravo, u prvim radovima optimizacije (Airy, 1861) i (Young, 1920) autori su uzimali analitički definirane granice domena, ali u današnje vrijeme modernih i brzih elektroničkih kompjutora integriranje je vrlo jednostavan, pouzdan i stabilan postupak i analitička integracija može se potpuno zamijeniti numeričkom. Granice kartografskih domena sastoje se većinom od zatvorenog poligona točaka, a originalna površina sfere zamijeni se velikim brojem malih površinskih elemenata, sfernih trapeza, koje ograničava mreža meridijana i paralela i u gravitacijskim točkama tih površinskih elemenata računaju se lokalni elementi linearne kvalitete projekcija. ...
Cartographic projections are basis for the graphical representation of various territories in small scale mapping. Proper selection of projection reduces the deformation of the presented territory, which is bounded by a boundary line. In most cases, this border line is not a mathematically defined curve, which is most easily displayed in the form of a closed polygon. The optimal cartographic projections based on a selected criterion of quality are those whose constants lead to the smallest value of the criterion. In the presented work it is recommended to use Airy-Kavrajski criterion whose minimization is actually minimization of the second Euclidean norm. The solution of optimal projections of various classes is reduced to the method of least squares. Fast modern computers enable the optimization of an arbitrary territory by evaluating the selected criterion in a finite number of points.
... It is also possible, in the cases where the desired property cannot be preserved everywhere, to optimize the map projection to have the minimum possible distortion. In highly constrained cases, this can be done analytically using the calculus of variations (Airy, 1861). However, in general, such a task requires numerical approaches (Tobler, 1977). ...
In order to transform the curved surface of the earth onto the planar surface of a map, some distortion must be introduced. Conventional map projections are defined by mathematical formulas, and are thereby limited in their ability to control that distortion by the complexity of those formulas. Piecewise map projections based on interpolation onto unstructured meshes have no such constraints, and can therefore create maps that are better suited to many use cases. A method using multi-dimensional optimization to optimize such map projections is presented here, and demonstrated by the generation of several new map projections. These map projections are presented as the Danseiji projections, along with their potential applications.
Using the Euler‒Urmaev system of equations, it is shown that only that surface allows one to obtain projections onto a plane without distortion, the coefficient of the metric form of which is a harmonic function. Since this quantity is not harmonious for the surface of an ellipsoid, the problem arises of finding such a projection in which the length distortions are minimal. To solve the problem of finding ideal projections unambiguously, it is necessary: first, to select a measure of linear distortion at a given point in a given direction; second, to select a general measure of linear distortion at a point; and third, to select a length distortion criterion for the entire mapping area. Definitions for the ideal and best projections are formulated.
The problem of finding the best projection from a set of projections in which the sum of the extremal scales is equal to two is an optimal control problem with an equality type constraint, which is solved by the Lagrange multiplier method with subsequent integration of the corresponding Euler‒Ostrogradsky system. The resulting best projection is related to the best conformal projection and to the ideal projection according to the Airy criterion. In particular, the mapping functions of an Airy ideal projection are equal to half the sum of the mapping functions of the best conformal projection and the best of the set of close-to-equal-area projections. The three projections mentioned above provide a triad of related objects.
The properties of the Airy projection as an ideal projection according to the Airy criterion for a region bounded by an almucantar are examined. It is shown that the Airy projection constructed for the entire globe is not one-to-one; two points on the ellipsoid’s surface correspond to a single point in the projection. Two versions of the projection with the one-to-one property have been developed: the first version covers 98.5% of the globe, and the second version covers 93%. The first version has the drawback that the minimum length scale near the boundary of the region approaches zero. Therefore, the second version is preferable. The first ideal projection known to mankind according to the Airy criterion is the azimuthal projection proposed by the Royal Astronomer G.B. Airy (Airy (1861) Explanation of a projection by balance of errors for maps applying to a very large extent of the Earth’s surface and comparison of this projection with other projections. London, Edinburgh, and Dublin Philosophical Magazine 4, 22(149):409–421. 10.1080/14786446108643179). This projection is ideal for areas limited by almucantar. The polar distance of this projection is determined by the minimum condition of the Airy criterion (Lapaine (2016) George Biddell Airy and his Contribution to Map Projections Theory. Paper presented at the 6th International Conference on Cartography and GIS, 13–17 June 2016, Albena, Bulgaria ISSN: 1314-0604).
It is demonstrated that the ideal projection according to the Airy criterion for an arbitrary mapping area is one in which two functions, one associated with the sum of the extremal scales and the other with the difference between the azimuth of the main direction on the surface and the grid azimuth of the main direction on the plane, are conjugate harmonic functions. At the boundary of the region, one of the extremal scale factors of the ideal projection equals unity, and the Tissot’s indicatrixes are oriented either along the normals or tangents to the boundary. The fundamental properties of the resulting ideal projection are examined.
The Korn-Lichtenstein partial differential equations subject to an integrability condition of Laplace-Beltrami type which govern conformal mapping are reviewed. They are completed by an extensive review of deformation measures (Cauchy-Green deformation tensor, Euler-Lagrange deformation tensor, simultaneous diagonalization of a pair of symmetric matrices) extending the Tissot deformation portrait. W.r.t. one system of isometric parameters which cover a surface (oriented two-dimensional Riemann manifold) the d’Alemberi-Euler equations (Cauchy-Riemann equations) subject to an integrability condition of Laplace-Beltrami type are solved in real analysis by various systems of functions (fundamental solution: 2d-polynomial, separation of variables) plus a properly chosen boundary value problem, namely the equidistant mapping of one parameter line. Finally the optimal universal transverse Mercator projection is outlined by solving a boundary value problem of the d’Alembert-Euler equations (Cauchy-Riemann equations) of a biaxial ellipsoid (ellipsoid of revolution) where a dilatation factor of a central meridian is to be determined. It is proven that for a non-symmetric and a symmetric UTM strip the total areal distortion approaches zero once the total departure from an isometry is minimized. According to the “Geodetic Reference System 1980” for a strip [−lE, +lE] × [BS, BN] = [−3.5, +3.5°] × [80°S, 84°N - the standard UTM strip - an optimal dilatation factor is ϱ = 0.999, 578, while for a strip [−2°, +2°] × [80°S, 84°N] - the standard Gauβ-Krüger strip - an optimal dilatation factor is ϱ = 0.999, 864.
Bu çalışmada, projeksiyonlardan kaynaklanan deformasyonların değerlendirilmesinde kullanılan enlem-boylam gridi yerine geometrik ve izotropik özellikleri açısından daha uygun olan Fibonacci kafesinin kullanımı incelenmiştir. Fibonacci kafesi, küresel yüzeylerde daha homojen bir nokta dağılımı sağlayarak, ölçek deformasyon katsayılarının değerlendirilmesinde enlem-boylam gridine kıyasla daha yüksek performans sunmaktadır. Çalışma kapsamında her iki yöntemle oluşturulan modellemelerde, Lambert konform konik projeksiyonu ölçek deformasyon katsayıları değerlendirilmiştir. Yöntemler arasında karşılaştırma yapmak amacıyla farklı standart paralel seçimleriyle Türkiye için deformasyon ölçek katsayıları değerlendirilmiştir. Analizlerde, projeksiyon karakteristik deformasyon ölçütü ∆k_1 ile E_P ve E_G klasik deformasyon kestiricileri, 40°40'–43°20', 35°–41° ve 36°24'–40°36' standart paralelleri için hesaplanmıştır. Karakteristik deformasyon ölçüt değerleri bu standart paraleller kullanılarak Fibonacci kafesi ile sırasıyla 2401.66, 1008.32, ve 544.14ppm olarak hesaplanmıştır. Benzer sonuçlar elde etmek için, coğrafi grid ile yaklaşık 10 kat fazla noktaya ihtiyaç duyulduğu belirlenmiştir. Fibonacci kafesinin daha düzenli nokta dağılımı sağladığı ve bu sayede deformasyon analizlerinde daha yüksek performansa sahip olduğu gözlemlenmiştir. Özellikle, daha az sayıda nokta ile daha yüksek doğrulukta modelleme yapılabilmesi, Fibonacci kafesinin öne çıkan avantajlarından biridir.
Optimal mapping is one of the longest-standing problems in computational mathematics. It is natural to measure the relative curve length error under map to assess its quality. The maximum of such error is called the quasi-isometry constant, and its minimization is a nontrivial max-norm optimization problem. We present a physics-based quasi-isometric stiffening (QIS) algorithm for the max-norm minimization of hyperelastic distortion.
QIS perfectly equidistributes distortion over the entire domain for the ground truth test (unit hemisphere flattening) and, when it is not possible, tends to create zones where all cells have the same distortion. Such zones correspond to fragments of elastic material that became rigid under stiffening, reaching the deformation limit. As such, maps built by QIS are related to the de Boor equidistribution principle, which asks for an integral of a certain error indicator function to be the same over each mesh cell.
Under certain assumptions on the minimization toolbox, we prove that our method can build, in a finite number of steps, a deformation whose maximum distortion is arbitrarily close to the (unknown) minimum. We performed extensive testing: on more than 10,000 domains QIS was reliably better than the competing methods. In summary, we reliably build 2D and 3D mesh deformations with the smallest known distortion estimates for very stiff problems.
Over the last nine years, the ongoing armed conflict in Syria has had a devastating impact on properties and infrastructure, making it necessary to rebuild everything from scratch. As spatial data are a pillar to enhance sustainable socio-economic development, it should be georeferenced and reformed to keep up with the tremendous growth in information and communication technology. On the other hand, the low-distortion conformal mapping contributes significantly to changing the position of any feature on the earth's surface onto a plane to create accurate large-scale spatial data. The current research conducts an analytical study using spatial statistics tools to investigate an optimal conformal projection for representing the Syrian area in a single zone that preserves angles locally, reduces linear distortions, and fulfills modern geospatial technologies' requirements. The findings have shown the applicability of the proposed model as an alternative solution to minimize the scale error over the interest area. In addition, the isograms pattern is close to the shape of the area, and better fits the quality criteria of the chosen projection.
In map projection theory, it is usual to utilize numerical quadrature rules to estimate the overall map distortion. However, it is not known which method is the most efficient to approximate this integral. In this paper, overall map distortion is calculated analytically by a computer algebra system. Various integration methods are compared to the exact results. Some calculations are also performed on irregular spherical polygons. Considering the experiments, the author suggests utilizing the first-order Gaussian quadrature as it always gave reasonable results, although it is not the best for all cases.
In map projections theory, various criteria have been proposed to evaluate the mean distortion of a map projection over a given area. Reports of studies are not comparable because researchers use different methods for estimating the deviation from the undistorted state. In this paper, statistical methods are extended to be used for averaging map projection distortions over an area. It turns out that the measure known as the Airy–Kavrayskiy criterion stands out as a simple statistical quantity making it a good candidate for standardization. The theoretical arguments are strengthened by a practical map projection optimization exercise.
Although the theory of the minimum-distortion projections is well known, there were only a few attempts to develop such maps. This is mostly due to the fact that a solution of this problem usually connects to differential equations, which are difficult to solve. In this paper, the author shows how to approximate the best projection even for irregular areas using numeric methods by the example of equal-area and pseudo-conic projections for smallscale maps. Distortion values of optimal solutions are displayed in a table for several regions. Furthermore, the paper suggests various applications of their usage.
In the study of map projections, it is relatively simple to obtain meaningful estimators of distortion for a small area. The definition and especially the evaluation of global distortion measures (i.e., estimators representing the distortion worldwide or in a continent-like area) are undoubtedly more troublesome. Therefore, it is relatively common to find that recommendations for the parameters to use in a particular map projection, be it devised for a continent or a country, are based on simple rules (like the one-sixth rule of thumb for conic projections), with no possibility of further improvement in terms of resulting distortions and sometimes even with no knowledge at all of the sizes of these distortions. Although the choice of map defining parameters is normally made for reasons other than distortion minimization, such as ease of use (e.g., integer or half-integer numbers may be preferable), preservation of conventional or traditional definitions, and uniformity of parameters between neighboring regions, it is always worthwhile to know the optimal set of parameters in terms of minimal distortion. Then, the cartographer may mindfully deviate from this optimal set, documenting the differences in defining parameters and in the resulting distortions. The present research provides a means to do this by extending a related work presented in a previous contribution, where the evaluation and optimization of distortions were studied for a single map projection and only two areas of interest. To this end, a new tool has been developed and presented in this paper. This tool allows users to evaluate several measures of distortion for the most common conformal and equal-area projections within user-defined geographic boundaries of interest. Also embedded in the tool and transparent to users are global optimization techniques operating on Fibonacci grids, which permit the optimization of parameters for the particular map projection and area of interest under two possible criteria: minimization of typical distortion or minimization of extreme distortions. This tool and the associated techniques are applied to several official projections to analyze their original performance and to propose new parameters that significantly improve the resulting distortions while leaving room for users to easily evaluate and optimize the tool for the lowest distortions of these projections within their regions of interest.
A recent study on map projections expanded two new measures of distortion, namely flexion and skewness. However, it introduced them only for the unit sphere. The present paper derives formulas for the rotational ellipsoid and demonstrates that these kinds of distortion always have a unit of measurement. A new method of illustration is described, by which these quantities can be visualized in an expressive way.
Harmonic maps are a certain kind of an optimal map projection which has been developed for map projections of the sphere. Here we generalize it to the “ellipsoid of revolution”. The subject of an optimization of a map projection is not new for a cartographer. For instance, in Sect. 5-25, we compute the minimum distortion energy for mapping the “sphere-to-plane”.
Conventionally, conformal coordinates, also called conformal charts, representing the surface of the Earth or any other Planet as an ellipsoid-of-revolution, also called the Geodetic Reference Figure, are generated by a two-step procedure. First, conformal coordinates (isometric coordinates, isothermal coordinates) of type UMP (Universal Mercator Projection, compare with Example 15.1) or of type UPS (Universal Polar Stereographic Projection, compare with Example 15.2) are derived from geodetic coordinates such as surface normal ellipsoidal longitude/ellipsoidal latitude. UMP is classified as a conformal mapping on a circular cylinder, while UPS refers to a conformal mapping onto a polar tangential plane with respect to an ellipsoid-of-revolution, an azimuthal mapping.
In this chapter, we present a collection of most widely used map projections in the polar aspect in which meridians are shown as a set of equidistant parallel straight lines and parallel circles (parallels) by a system of parallel straight lines orthogonally crossing the images of the meridians. As a specialty, the poles are not displayed as points but straight lines as long as the equator. First, we derive the general mapping equations for both cases of (i) a tangent cylinder and (ii) a secant cylinder and describe the construction principle.
At the beginning of this chapter, let us briefly refer to Chap. 8, where the data of the best fitting “ellipsoid-of-revolution to Earth” are derived in form of a table. Here, we specialize on the mapping equations and the distortion measures for mapping an ellipsoid-of-revolution to a cylinder, equidistant on the equator. Section 14-1 concentrates on the structure of the mapping equations, while Sect. 14-2 gives special cylindric mappings of the ellipsoid-of-revolution, equidistant on the equator. At the end, we shortly review in Sect. 14-3 the general mapping equations of a rotationally symmetric figure different from an ellipsoid-of-revolution, namely the torus.
A special mapping, which was invented by Gauss (1822, 1844), is the double projection of the ellipsoid-of-revolution to the sphere and from the sphere to the plane. These are conformal mappings. A very efficient compiler version of the Gauss double projection was presented by Rosenmund (1903) (ROM mapping equations) and applied for mapping Switzerland and the Netherlands, for example. An alternative mapping, called “authalic”, is equal area, first ellipsoid-of-revolution to sphere, and second sphere to plane.
In the world of conformal mappings of the Earth or other celestial bodies, the Mercator projection plays a central role. The Mercator projection of the sphere or of the ellipsoid-of-revolution beside conformality is characterized by the equidistant mapping of the equator. In contrast, the transverse Mercator projection is conformal and maps the transverse meta-equator, the meridian of reference, equidistantly. Accordingly, the Mercator projection is very well suited for regions which extend East–West around the equator, while the transverse Mercator projection fits well to those regions which have a South–North extension. Obviously, several geographical regions are centered along lines which are neither equatorial, parallel circles, or meridians, but may be taken as central intersection of a plane and the reference figure of the Earth or other celestial bodies, the ellipsoid-of-revolution (spheroid).
Up to now, we treated various mappings of the ellipsoid and the sphere, for instance of type conformal, equidistant, or equal areal or perspective and geodetic.
Among cylindrical projections, mappings in the transverse aspect play the most important role. Although many worldwide adopted legal map projections use the ellipsoid-of-revolution as the reference figure for the Earth, the spherical variant forms the basis for the Universal Transverse Mercator (UTM) grid and projection. In the subsequent chapter, we first introduce the general concept of a cylindrical projection in the transverse aspect. Following this, three special map projections are presented: (i) the equidistant mapping (transverse Plate Carrée projection), (ii) the conformal mapping (transverse Mercator projection), and (iii) the equal area mapping (transverse Lambert projection). The transverse Mercator projection is especially appropriate for regions with a predominant North-South extent. As in previous chapters, the two possible cases of a tangent and a secant cylinder are treated simultaneously by introducing the meta-latitude B = ±B1 of a meta-parallel circle which is mapped equidistantly. For a first impression, have a look at Fig. 11.1.
In Chap. 21, we already transformed from a global three- dimensional geodetic network into a regional or local geodetic network. We aimed at the analysis of datum parameters, namely seven parameters of type translation, rotation and scale, as elements of the global conformal group C7(3).
Pseudo-cylindrical projections have, in the normal aspect, straight parallel lines for parallels. The meridians are most often equally spaced along parallels, as they are on a cylindrical projection, but on which the meridians are curved. Meridians may be mapped as straight lines or general curves.
Cylindrical projections in the oblique aspect are mainly used to display regions which have a predominant extent in the oblique direction, neither East-West nor North-South. In addition, they form the most general cylindrical projections because mapping equations for projections in the polar and the transverse aspect can easily be derived from it. This is done by setting the corresponding latitude of the meta-North Pole Φ
0 to a specific value: Φ
0 = 90∘ generates cylindrical projections in the polar aspect, Φ
0 = 0∘ result in cylindrical projections in the transverse aspect. As an introductory part, we present the equations for general cylindrical mappings together with the equations for the principal stretches, before derivations for specific cylindrical map projections of the sphere (oblique equidistant projection, oblique conformal projection and oblique equal area projection) are given. For a first impression, have a look at Fig. 12.1.
Mapping the ellipsoid-of-revolution to a tangential plane. Azimuthal projections in the normal aspect (polar aspect): equidistant, conformal, equiareal, and perspective mapping.
Mapping the sphere to a tangential plane: meta-azimuthal projections in the oblique aspect. Equidistant, conformal (oblique UPS), and equal area (oblique Lambert) mappings.
Geodesics, geodetic mapping. Riemann, Soldner, and Fermi coordinates on the ellipsoid-of-revolution, initial values, boundary values. Initial value problems versus boundary value problems.
A new algorithm suitable for fast onboard implementation is presented for designing attitude maneuver paths that achieve desired boresight alignment in the presence of multiple three-axis attitude constraints. It is based on a recently proposed algorithm in which possible attitudes are discretized in a three-dimensional distortion-minimizing projected space. In that space, grid points are enumerated and graph search pathfinding algorithms are employed to find the shortest constrained path between selected initial and target grid points. The new algorithm extends this approach from a single target point to a continuum of target points, representing all possible attitudes that maintain the desired boresight alignment. It augments the original algorithm with the transformation that makes it possible to assign a single value to estimated distances for the whole continuum of target points. This, in turn, allows the pathfinding algorithms to efficiently determine the shortest overall path to the whole continuum, which corresponds to the shortest attitude maneuver path that achieves the desired boresight alignment.
A conformal (orthomorphic) mapping projection of the spheroid can be constructed to give minimum scale error over a given arbitrary area, and in this respect has an advantage over more regular projections such as the transverse Mercator or the Lambert conformal conic. Geodetic coordinates on the spheroid are first transformed into isometric coordinates, and the latter are then transformed into the rectangular cartesian coordinates of the desired projection by means of a polynomial expression in complex variables. The total distortion of the projection is expressed as the integral of the squared scale error over the given area. After fixing the values of the rectangular coordinates and of the meridian convergence at the origin of the projection, the remaining coefficients of the complex polynomial are adjusted to minimise the total distortion. This set of coefficients can be used directly in formulae to carry out the direct and inverse transformations between geodetic and rectangular coordinates, and to calculate the scale factor, the meridian convergence, and the geodesic curvatures of projected curves (including meridians and parallels) at any point. In the reduction of the observations of local surveys in rectangular coordinates, the minimum scale error property means that corrections to bearings and distances are often negligible, or if required they can be interpolated from small-scale contour maps. As an example, coefficients have been calculated for a projection designed to give minimum distortion over the land area of New Zealand, using a complex polynomial to order six. The range of scale error for this projection is about 4 x 10-4, less than can be obtained with any conventional projection.
The Theory of Map Projections is based here on the transformation of Riemann manifolds to Riemann manifolds. Section 2 offers some orientation based on simultaneous diagonalization of two symmetric matrices. We separate simply connected regions. In detail, we review the pullback versus pushforward operations. Section 3 introduces the first multiplicative measure of deformation: the Cauchy-Green deformation tensor, its polar decomposition, as well as singular value decomposition. An example is the Hammer retroazimuthal projection. A second multiplicative measure of deformation is presented in Sect. 4: stretch, or length distortion, and Tissot portrait as well. The Euler-Lagrange deformation tensor presented in Sect. 5 is the first additive measure of deformation based on the difference of the metrics (ds2; dS2). Section 6 introduces a reviewof 25 differentmeasures of deformation. First, angular shear is the second additive measure, also called angular distorsion, left and right. Section 8 introduces a third multiplicative measure of deformation, called relative angular shear. In contrast, the equivalence theorem of conformal mapping in Sect. 9 is based on Korn-Lichtenstein equations. Areal distortion in Sect. 10 offers a popular alternative based on the fourth multiplicative and additive measure of deformation, namely, dual deformation called areomorphism. Section 11 offers an equivalence theorem of equiareal mapping. The highlight is our review of canonical criteria in Sect. 12: (i) isometry; (ii) equidistant mapping of submanifolds; (iii) in particular, canonical conformism, areomorphism, isometry, and equidistance; and finally, (iv) optimal map projections. Please study Sect. 13, the exercise: the Armadillo double projection. © Springer-Verlag Berlin Heidelberg 2010, 2015. All rights are reserved.
The chorographic world maps cannot go missing from the atlases for the general public and for schools. Among other requirements, their map distortions are usually expected to be minimal. In the aphylactic projections both the angular and area distortions can be reduced by the principle of “balance of errors”. Equations for the mapping functions help us to search pseudocylindrical projections showing minimum distortions according to the Kavrayskiy criterion, while the outline shape of the mapped Earth is monitored. Some of the best solutions are demonstrated in this paper.
The paper gives an answer to the question: if the maritime area of Croatia (territorial sea and continental shelf) is involved, which is the best Lambert conic conformal map projection for the territory ofCroatia? Numerical realization demanded the northernmost and the southernmost coordinates on the land and sea of Croatia. Since the coordinates ofthe southernmost point on the sea, i.e. those on the line of the continental shelf, had never been calculated before, it had to be done in this work. Thereafter eight variants of conic conformal map projections with respect to the distribution of linear distortions were analyzed. Each variant was clearly defined, and they differ in the constants of projection which were obtained by setting initial conditions. On the basis of performed research the best Lambert conic conformal map projection for the territory of Croatia including its territorial sea and continental shelf was proposed.
The main objective of this paper is to review previous and current trends in small-scale map projection research, and to discuss the work that has been done by this author in relation to similar work by others. Attention is focused on three issues: (a) evaluation of map projection distortion, (b) development of low-error map projections, (c) automated (and semi-automated) map projection selection. It is hoped that this paper will give the reader a good impression of the marked evolution of map projection science over the last forty years, of changes in the way map projections are used (or will be used in the near future), and of the important role the rapidly increasing, widespread use of computers has played in this process.
Disadvantages of the currently used Křovák’s map projection in the Slovak Republic, such as large scale distortion, became evident after the division of Czechoslovakia. The aim of this paper is to show the results of the optimization of cartographic projections using Chebyshev’s theorem for conformal projections and its application to the territory of the Slovak Republic. The calculus used, the scale distortions achieved and their comparison with the scale distortions of currently used map projections will be demonstrated.
Disadvantages of currently used Křovák`s projection in Slovak Republic such as large scale distortion became evident after dividing of Czechoslovakia. The new proposal of the cartographic projection for the Slovak Republic (Lambert`s conformal conic projection in normal position) was created in 2010. The aim of this paper is to define an optimal oblique position of map projections for the Slovak Republic territory and to design two cartographic projections in different positions (normal, oblique) using variational criteria modified for conformal projections for the Slovak Republic. Used calculus, the achieved scale distortions and their comparison with scale distortions of currently used map projections will be demonstrated.
The Theory of Map Projections is based here on the transformation of Riemann manifolds to Riemann manifolds. Section 2 offers some orientation based on simultaneous diagonalization of two symmetric matrices. We separate simply connected regions. In detail, we review the pullback versus pushforward operations. Section 3 introduces the first multiplicative measure of deformation: the Cauchy–Green deformation tensor, its polar decomposition, as well as singular value decomposition. An example is the Hammer retroazimuthal projection. A second multiplicative measure of deformation is presented in Sect. 4: stretch, or length distortion, and Tissot portrait as well. The Euler–Lagrange deformation tensor presented in Sect. 5 is the first additive measure of deformation based on the difference of the metrics {ds2, dS2}. Section 6 introduces a review of 25 different measures of deformation. First, angular shear is the second additive measure, also called angular distorsion, left and right. Section 8 introduces a third multiplicative measure of deformation, called relative angular shear. In contrast, the equivalence theorem of conformal mapping in Sect. 9 is based on Korn–Lichtenstein equations. Areal distortion in Sect. 10 offers a popular alternative based on the fourth multiplicative and additive measure of deformation, namely, dual deformation called areomorphism. Section 11 offers an equivalence theorem of equiareal mapping. The highlight is our review of canonical criteria in Sect. 12: (i) isometry; (ii) equidistant mapping of submanifolds; (iii) in particular, canonical conformism, areomorphism, isometry, and equidistance; and finally, (iv) optimal map projections. Please study Sect. 13, the exercise: the Armadillo double projection.
Obrađene su koordinate, oblik i veličina Zemlje, opća teorija i podjela kartografskih projekcija.Za konusne, azimutalne, cilindrične, pseudokonusne, pseudocilindrične, polikonusne, kružne i mješovite projekcije dane su osnovne kartografske jednadžbe. Detaljnije od drugih obrađena je Gauss-Krugerova projekcija. Udžbenik završava pregledom matematičke osnove topografskih karata Hrvatske, izborom projekcije i primjenom računala.
Mappings from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold, simultaneous diagonalization of two matrices, mappings (isoparametric, conformal, equiareal, isometric, equidistant), measures of deformation (Cauchy–Green deformation tensor, Euler–Lagrange deformation tensor, stretch, angular shear, areal distortion), decompositions (polar, singular value), equivalence theorems of conformal and equiareal mappings (conformeomorphism, areomorphism), Korn–Lichtenstein equations, optimal map projections.
The Theory of Map Projections is based here on the transformation of Riemann manifolds to Riemann manifolds. Section 2 offers some orientation based on simultaneous diagonalization of two symmetric matrices. We separate simply connected regions. In detail, we review the pullback versus pushforward operations.Section 3 introduces the first multiplicative measure of deformation: the Cauchy–Green deformation tensor, its polar decomposition, as well as singular value decomposition. An example is the Hammer retroazimuthal projection.A second multiplicative measure of deformation is presented in Sect. 4: stretch, or length distortion, and Tissot portrait as well. The Euler–Lagrange deformation tensor presented in Sect. 5 is the first additive measure of deformation based on the difference of the metrics {ds2, dS2}. Section 6 introduces a review of 25 different measures of deformation. First, angular shear is the second additive measure, also called angular distorsion, left and right.Section 8 introduces a third multiplicative measure of deformation, called relative angular shear. In contrast, the equivalence theorem of conformal mapping in Sect. 9 is based on Korn–Lichtenstein equations. Areal distortion in Sect. 10 offers a popular alternative based on the fourth multiplicative and additive measure of deformation, namely, dual deformation called areomorphism. Section 11 offers an equivalence theorem of equiareal mapping. The highlight is our review of canonical criteria in Sect. 12: (i) isometry; (ii) equidistant mapping of submanifolds; (iii) in particular, canonical conformism, areomorphism, isometry, and equidistance; and finally, (iv) optimal map projections. Please study Sect. 13, the exercise: the Armadillo double projection.
Another interesting distortion measurement is the distortion energy that it can be seen intuitively as the necessary energy to transform the real surface (portion of sphere) into the map plane (portion of plane). The physical modification of the surface.
ResearchGate has not been able to resolve any references for this publication.