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Abstract
Chesterton did not, of course, intend this gibe to be taken literally. But the more we consider what he would doubtless have called the “Higher Geodetics”, the more we must conclude that there is some literal justification for it. Not only are straight lines straight. A sufficiently short part of a curved line may also be considered straight, provided that it is continuous (i.e. does not contain a sudden break or sharp corner), and provided we are not concerned with a measure of its curvature. Similarly a square mile or so on the curved surface of the conventionally spheroidal earth is to all intents and purposes flat. We shall achieve a considerable simplification, without any approximation, in the treatment of the present subject by getting back to these fundamental glimpses of the obvious, whether the formalists and conformalists accept them or not.
... Les représentations conformes sont largement utilisées, particulièrement celles de Mercator Transverse (Gauss-Krüger) et les représentations coniques de Lambert. Une étude des représentations conformes régulières d'un ellipsoïde a été faite par Hotine [4]. ...
... Courbes de niveau de l'altération linéaire pour la représentation plane exprimée par un polynôme de degré 6 de la Nouvelle Zélande, en unités 100 mm/km une parabole cubique (x 3 ), Hotine[4] a donné une valeur de δ comme : ...
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 .
... The datum format for West Malaysia and Singapore until 1968 was the Kertau 1948, whereas the datum for East Malaysia, including Sabah and Sarawak was Timbalai 1948 (BT1948). Both mapping of MRT and BT are done using the Rectified Skew Orthomorphic System (RSO), which is an oblique mercator projection technique defined by Hotine in 1947 [1]. ...
... The Rectified Skew Orthomorphic (RSO) projection is an Oblique Mercator projection developed by Hotine in 1947 [1,13]. RSO is used in surveying of Malaysia since it is a country with large extend of areas in one single direction. ...
This paper discusses the design and development of a Malaysian geographical information system (GIS) database server for location data sources available in Malaysia, such as the Global Positioning System (GPS), WGS84, geocentric datum for Malaysia (GDM2000), geodetic reference system 1980 (GRS80), local geodetic measurements such as Malayan revised triangulation 1948 (MRT48 or KERTAU 48), Borneo triangulation 1948 (TIMBALAI 1948), Malayan Revised Triangulation 1968(MRT68), Borneo triangulation 1968 (BT68), local cadastral information such as the ECadastre coordinated cadastral system (CCS) developed by JUPEM and cellular coordinate information based on cell-based ID, E-OTD etc It describes the pros and cons of the above mentioned systems, analyze the accuracies and tolerance of the data source. It also highlights the disparities between different systems, and proposes fast methods to correlate and transform these data into a unified, reliable data source. Lastly it will cover the storage and retrieval process for the textual and graphical GIS data.
... It was very important in the second half of the twentieth century in military maps, and is now probably the most important conformal mapping in geodesy. Its mathematical basis and the formulas for the projection can be found in classical works such as those by Hotine (1946Hotine ( , 1947a, Levallois (1969) and Tardi and Laclavère (1954). For more recent and theoretical work, see Grafarend (1995). ...
... Consider a point with geodetic coordinates (Q, l), where the longitude l is computed from the central meridian. The isometric latitude, q, is defined as follows (Hotine 1946): ...
Problems of precision and accuracy in geographical information systems have often been undervalued. However, these are important issues when trying to avoid inaccurate computations that could distort the final result. A method is described for the Transverse Mercator Projection of the ellipsoid to calculate the necessary number of terms of their series development so that desired accuracy can be achieved. The first part describes one algorithm used to obtain the full development of the formulas by means of a symbolic calculus program. The second part studies the influence of each term in the final result in order to know which terms can be neglected while achieving the desired precision. Finally, the results are applied to a rectangle of 30° width and latitude varying from 30° to 50°.
... Moreover the constants n and k may be fixed choosing a particular parallel or parallels along which the scale is true (i.e., m = 1). This is the traditional approach which include the two following different cases (Hotine, 1947): ...
Two optimization methods are studied for choosing the standard parallels for a conformal conical projection. One is based on a criterion called of minimum error and the other is based on a criterion of minimum mean square dilatation. In both cases, the idea is to minimize, with respect to the two constants that describe the family of conformal conical projections, a function which measures departure from equivalence. The basic parameters are the linear scale error and the area scale error respectively. The second method is shown to be more suitable from a numerical point of view although the first one presents some analytical advantages when the region to be mapped contains the pole. Some numerical examples based on the second criterion are given. -Authors
... Over the years, the Rectified Skew Orthomorphic (RSO) and Cassini Soldner (Cassini) map projection systems have been used for national mapping and cadastral purposes, respectively, based on these two local datums. For example, based on the theory developed byHotine (1947), the Directorate of Colonial Surveys in Teddington, England, prepared the RSO Projection Tables for Malaya. This Table has ever since been used by the DSMM for the RSO topographical mapping projection purposes in the Peninsular. ...
The Department of Survey and Mapping Malaysia (DSMM), in collaboration with the Universiti Teknologi Malaysia(UTM), has carried out a study towards the establishment of a new geodetic framework for the country. The new Geocentric Datum for Malaysia (GDM2000) is being built by GPS space geodetic technology based on International Terrestrial Reference Frame (ITRF2000) and Geodetic Reference System 1980 (GRS80) reference ellipsoid. GDM2000 is connected to ITRF2000 by the inclusion of 17 International GPS Service (IGS) sites from the nearby regions in the precise baselines processing and adjustment of a network of existing Malaysian Active GPS System (MASS) stations. The process of implementing GDM2000 in particular matters related to datum transformation and map projection is being addressed. New sets of transformation parameters concerning conversion from the existing local datum to the new GDM2000 have been developed. Its implication on the existing cadastral and mapping practices, various GPS non-mapping applications, and GIS/LIS related applications have been considered. In addition to that, its socio-economic implications on the implementation of GDM2000 is also presented.
... The theory of surfaces (Hotine, 1947) explains that the distortion in small parts of the surface tends to assume the character of affinity. On a plane surface the unit circle by affine representation is distorted into an ellipse and the (orthogonal) major and minor axes provides the maximum and minimum scale of distortion. ...
... Laborde's development is based on the Gauss-Schreiber Transverse Mercator projection. Hotine (1946Hotine ( , 1947 first introduced the development of the oblique Mercator on the ellipsoid through the "aposphere," a surface of constant curvature and thence to the plane. Hotine's original imple- mentation was for the peninsula of Malaya and the island of Borneo, but this projection also has been used as a grid in numerous areas elsewhere, including Alaska Zone 1 of the State Plane Coordinate Systems on both NAD 27 and NAD 83. ...
... The simple geometrical treatment contained in this paper is adapted from Hotine (1946Hotine ( , 1947a. ...
This paper describes a new algorithm for calculating all equations (direct and inverse transformation, convergence of meridians, linear distortion, calculus of surfaces, and arc‐to‐chord correction) for the Gauss–Krüger projection. Instead of using different equations for each problem, all the calculi are based on the equations used to obtain the direct transformation. These equations are also more accurate than previous ones and can be extended to an arbitrary width. This paper also explains how the Gauss–Krüger projection may satisfy the needs and requirements of civil engineers as well as Geographical Information Systems (GIS) users, particularly if extended beyond its usual 4° longitudinal range, and therefore can be used as a global reference system for GIS.
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.
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.
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 purpose of surveying and mapping is to determine relative positions of points at or near the surface of the earth, which requires the establishment of a system of reference in three-dimensional space. A natural choice of elevation, or height Z, as one spatial coordinate, leaves a set of two horizontal coordinates X and Y subject to definition. In urban surveying and mapping, the horizontal coordinates are most conveniently referred to a rectangular plane coordinate system, or grid, which is oriented so that the positive directions of the coordinate axes point toward the north and the east. Unfortunately, there is no general agreement as to whether X or Y should be taken in the north direction; Figure 2-1 shows the convention accepted throughout in this text.
Top 1: Gravitostatics Geodetic heights, better height differences are not integrable. For instance, every geodesist knows “dH”, the infinitesimal change of geodetic heights.
Geodetic datum (including coordinate datum, height datum, depth datum, gravimetry datum) and geodetic systems (including geodetic coordinate system, plane coordinate system, height system, gravimetry system) are the common foundations for every aspect of geomatics. This course book focuses on geodetic datum and geodetic systems, and describes the basic theories, techniques, methods of geodesy. The main themes include: the various techniques of geodetic data acquisition, geodetic datum and geodetic control networks, geoid and height systems, reference ellipsoid and geodetic coordinate systems, Gaussian projection and Gaussian plan coordinates and the establishment of geodetic coordinate systems. The framework of this book is based on several decades of lecture noted and the contents are developed systematically for a complete introduction to the geodetic foundations of geomatics. © Springer-Verlag Berlin Heidelberg 2014. All rights are reserved.
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While the standardMercator projection / transverse Mercator projecton maps the equator / the transverse metaequator equivalent to the meridian of referenceequidistantly, theoblique Mercator projection aims at aconformal mapping of the ellipsoid of revolution constraint to anequidistant mapping of an oblique metaequator. Obliqueness is determined by the extension of the area to be mapped, e.g. determined by the inclination of satellite orbits: Satellite cameras map the area just under the orbit geometry. Here we derive themapping equations of theoblique Mercator projection being characterized to beconformal andequidistant on the oblique metaequator extending results ofM. Hotine (1946, 1947).
Application of standard map projections to the ellipsoidal Earth is often considered excessively difficult. Using a few symbols for frequently-used combinations, exact equations may be shown in compact form for ellipsoidal versions of conformal, equal-area, and equidistant projections developed onto the cone, cylinder (in conventional position), and plane, as well as for the polyconic projection. Series are needed only for true distances along meridians. The formulas are quite interrelated. The ellipsoidal transverse and oblique Mercator projections remain more involved. An adaptation of the Space Oblique Mercator projection provides a new ellipsoidal oblique Mercator which, unlike Hotine's, retains true scale throughout the length of the central line.
A conformal approximation to the Transverse Mercator (TM) map projection, global in longitude λ and isometric latitude q, is constructed. New formulas for the point scale factor and grid convergence are also shown. Assuming that the true values
of the TM coordinates are given by conveniently truncated Gauss–Krüger series expansions, we use the maximum norm of the absolute
error to measure globally the accuracy of the approximation. For a Universal Transverse Mercator (UTM) zone the accuracy equals
0.21 mm, whereas for the region of the ellipsoid bounded by the meridians ±20° the accuracy is equal to 0.3 mm. Our approach
is based on a four-term perturbation series approximation to the radius r(q) of the parallel q, with a maximum absolute deviation of 0.43 mm. The small parameter of the power series expansion is the square of the eccentricity
of the ellipsoid. This closed approximation to r(q) is obtained by solving a regularly perturbed Cauchy problem with the Poincaré method of the small parameter.
Abstract: The Transverse Mercator projection of the ellipsoid was derived by Gauss as a special case of his general theory of conformal representation, and was introduced by him into the Survey of Hanover between 1820 and 1830. A method for the full development of the formulae, using a symbolic calculus program is developed, then the influence of each term in the final result is studied to know which terms can be neglected while achieving a desired precision and, finally, these results are applied to a particular case. A new set of formulas is described for calculating all the direct and inverse transformation, the convergence of meridians, the linear distortion, calculus of surfaces, and arc-to-chord correction, for the Gauss–Krüger projection. Instead of using different formulas for each problem, all the calculi are based on the formulas used to obtain the direct transformation. These formulas are also more accurate than previous ones and can be extended to an arbitrary width. As an example, an oblique conformal projection is compared with a broaden version of the Gauss Kruger, to see the difference between them and how it can be possible to extend the latter to an arbitrary width. Resumen La proyección Transversa de Mercator en el elipsoide fue desarrollada por Gauss como un caso especial dentro de su teoría general de representaciones conformes, y la introdujo en el Catastro de Hannover entre 1820 y 1830. Utilizamos un programa de cálculo simbólico, para obtener el desarrollo completo de todas las fórmulas y estudiar la influencia de cada término en el resultado final. De esta manera podemos saber qué términos pueden ser o no despreciados cuando queremos alcanzar una precisión determinada. Finalmente estos resultados se aplican a un caso particular. Se describe una colección de algoritmos para el cálculo de todas las fórmulas (transformación directa e inversa, deformación lineal, cálculo de superficies, convergencia de meridianos y reducción a la cuerda) de la Proyección Transversa de Mercator para el elipsoide o proyección Gauss–Krüger. En vez de utilizar fórmulas específicas y directas para cada problema, todos los cálculos se basan en las fórmulas utilizadas en la transformación directa. Estas fórmulas no solo son más precisas que las clásicas, sino que además su rango de validez puede ampliarse de forma arbitraria Como ejemplo se compara la proyección oblicua conforme de Mercator con esta versión ampliada de la proyección Gauss−Krüger para ver la diferencia entre ambas y como ésta última puede extenderse hasta una anchura arbitraria.
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