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1.
《测量评论》2013,45(87):12-17
AbstractThe excuse for yet another paper on the Transverse Mercator projection, which has already received what should be more than its fair share of space in this Review, can only be that there is a fresh viewpoint to offer. It is the purpose of this paper to show that there are, in fact, two “Transverse Mercator” projections of the spheroid, of which one has hitherto almost escaped notice. 相似文献
2.
AbstractI. Introduction.—Map projection is a branch of applied mathematics which owes much to J. H. Lambert (v. this Review, i, 2, 91). In his “Beyträge zum Gebrauche der Mathematik und deren Anwendung” (Berlin, 1772) he arrived at a form of projection whereof the Transverse Mercator is a special case, and pointed out that this special case is adapted to a country of great extent in latitude but of small longitudinal width. Germain (“Traité des Projections”, Paris, 1865) described it as the Projection cylindrique orthomorphe de Lambert, but he also introduced the name Projection de Mercator transverse or renversée; he shows that Lambert's treatment of the projection was remarkably simple. 相似文献
3.
《测量评论》2013,45(61):267-271
AbstractSome publications that have dealt with the question of convergence of meridians seem, to the present writer, to be clouded with misconception, and these notes are intended to clarify some points of apparent obscurity. For instance, A. E. Young, in “Some Investigations in the Theory of Map Projections”, I920, devoted a short chapter to the subject, and appeared surprised to find that the convergence on the Transverse Mercator projection differs from the spheroidal convergence; the explanation which he advanced can be shown to be faulty. Captain G. T. McCaw, in E.S.R., v, 35, 285, derived an expression for the Transverse Mercator convergence which is equal to the spheroidal convergence, and described this as “a result which might be expected in an orthomorphic system”. Perhaps McCaw did not intend his remark to be so interpreted, but it seems to imply that the convergence on any orthomorphic projection should be equal to the spheroidal convergence, and it is easily demonstrated that this is not so. Also, in the second edition of “Survey Computations” there is given a formula for the convergence on the Cassini projection which is identical, as far as it goes, with that given for the Transverse Mercator, while the Cassini convergence as given by Young is actually the spheroidal convergence. Obviously, there is some confusion somewhere, and it is small wonder that Young prefaced his remarks with the admission that the subject had always presented some difficulty to him. 相似文献
4.
《测量评论》2013,45(58):142-152
AbstractIn January 1940, in a paper entitled “The Transverse Mercator Projection: A Critical Examination” (E.S.R., v, 35, 285), the late Captain G. T. McCaw obtained expressions for the co-ordinates of a point on the Transverse Mercator projection of the spheroid which appeared to cast suspicion on the results originally derived by Gauss. McCaw considered, in fact, that his expressions gave the true measures of the co-ordinates, and that the Gauss method contained some invalidity. He requested readers to report any flaw that might be discovered in his work, but apparently no such flaw had been detected at the time of his death. It can be shown, however, that the invalidities are in McCaw's methods, and there seems no reason for doubting the results derived by the Gauss method. 相似文献
5.
《测量评论》2013,45(32):85-89
AbstractThe necessity of transforming rectangular co-ordinates from one system of projection to another may arise from, various causes, One case, for example, with which the present writer is concerned involves the transformation, to the standard belt now in use, of the co-ordinates of some hundreds of points of a long existing triangulation projected a quarter of a, century ago on a, belt of Transverse Mercator projection, In this case conversion is complicated by the fact that the spheroid used in the original computation differs from that now adopted, and, also, the geodetic datums are not the same, The case in fact approaches the most general that can occur in practice, One step in one solution of this problem, however, is of perhaps wider Interest: that is, the transformation from one belt of Transverse Mercator projection to another when the spheroids and datums are identical. It is this special case which will be discussed here. 相似文献
6.
《测量评论》2013,45(60):220-221
AbstractThe problem of computing marginal scales of latitude and longitude on a rectangular map on the Transverse Mercator projection, where the sheet boundaries are projection co-ordinate lines, may be solved in various ways. A simple method is to compute the latitudes and longitudes of the four corners of the sheet, and then, assuming a constant scale, to interpolate the parallels and meridians between these corner values. Although it is probably sufficiently accurate for practical purposes, this method is not precise. It is not difficult to adapt the fundamental formulce of the projection to give a direct solution of the problem. 相似文献
7.
ABSTRACTRecent discoveries of Wehrmacht Maps in the Military Archive of the Federal Archive of Germany in Freiburg im Breisgau raised the motivation for further investigations into the history of the internationally employed Universal Transverse Mercator (UTM) projection which actually represents a prerequisite for the global use of Global Positioning System (GPS) – and thus of any type of navigation – instruments. In contrast to the frequently stated opinion that this map projection was first operationally used by U.S. Americans it turned out that presumably the first operational maps with indication of the orthogonal UTM grid were produced by German Wehrmacht officers prior to the post World War (WW) II triumph of this projection. Based on the authors´ recent discoveries this article reveals some hitherto hardly known facts concerning the history of cartography of the 1940s. 相似文献
8.
《测量评论》2013,45(70):357-363
AbstractThere are no proper projections for use in geodetic work in a country which has great extensions both in latitude and longitude. For, if a single projection of any kind be applied in such a case, the linear and angular distortions would be so great at the boundary that it is very difficult or even impossible to apply the corrections to them. In order to render it possible for any projection to be applied, the area in question should be divided either into strips bounded by meridians or into zones bounded by parallels. In the former case the Transverse Mercator or Gauss’ projection may be used, while in the latter, the Lambert conformal projection is the most suitable. China is such a country as that mentioned above. It covers an area extending from 16°N. to 53°N. in latitude and of no less than sixty-five degrees in longitude. The problem of choosing a projection for geodetic work depends only on how the area is to be divided. It has been decided by the Central Land Survey of China to adopt the Lambert conformal projection as the basis for the co-ordinate system, and, in order to meet the requirements of geodetic work, the whole country is subdivided into eleven zones bounded by parallels including a spacing of 3½ degrees in latitude-difference. To each of these zones is applied a Lambert projection, properly chosen so as to fit it best. The two standard parallels of the projection are situated at one-seventh of the latitude-difference of the zone from the top and bottom. Thus, the spacing between the standard parallels is 2½ degrees. This gives a maximum value of the scale factor of less than one part in four thousand, thus reducing the distortions of any kind to a reasonable amount. The area between these parallels belongs to the zone proper, while those outside are the overlapping regions with the adjacent ones. All the zones can be extended indefinitely both eastwards and westwards to include the boundaries of the country. 相似文献
9.
AbstractIn January 1938 the writer decided against holding up for more years some work on the Transverse Mercator Projection (E.S.R., 27, 275). The extension to the spheroid was not then complete, nor is the present paper to be regarded as a logical continuance. It is first proposed to show the results of “transplanting” orthomorphically upon the spheroid a spherical configuration forming a graticule. 相似文献
10.
《测量评论》2013,45(60):221-227
AbstractIn a previous article in this Review, the writer endeavoured to show that chains of minor triangulation could be adjusted by plane rectangular co-ordinates ignoring the spherical form of the earth with little loss of accuracy, provided that the two ends were held fixed in position. It was demonstrated that the plane co-ordinates produced by the rigorous adjustment between the fixed starting and closing sides, differ by only a comparatively small amount from the projection co-ordinates produced by a rigorous adjustment on the Transverse Mercator projection. The saving in time when computing by plane co-ordinates as opposed to rigorous computation on the projection by any method will be apparent to any computer with experience of both methods. 相似文献
11.
《测量评论》2013,45(69):318-322
AbstractThe Transverse Mercator Projection, now in use for the new O.S. triangulation and mapping of Great Britain, has been the subject of several recent articles in the “Empire Surpey Review. The formulae of the projection itself have been given by various writers, from Gauss, Schreiber and Jordan to Hristow, Tardi, Lee, Hotine and others—not, it is to be regretted, with complete agreement, in all cases. For the purpose for which these formulae have hitherto been employed, in zones of restricted width and in relatively low latitudes, the completeness with which they were given was adequate, and the omission of certain smaller terms, in the fourth and higher powers of the eccentricity, was of no practical importance. In the case of the British grid, however, we have to cover a zone which must be considered as having a total width of some ten to twelve degrees of longitude at least, and extending to latitude 61 °north. This means, firstly, that terms which have as their initial co-efficients the fourth and sixth powers of the longitude ω (or of y) will be of greater magnitude than usual, and secondly that tan2 ? and tan4 ? are likewise greatly increased. Lastly, an inspection of the formulae (as hitherto available) shows a definite tendency for the numerical co-efficients of terms to increase as the terms themselves decrease—e.g. terms in η4, η6, etc. 相似文献
12.
As a conformal mapping of the sphere S 2
R
or of the ellipsoid of revolution E 2
A
,
B
the Mercator projection maps the equator equidistantly while the transverse Mercator projection maps the transverse metaequator,
the meridian of reference, with equidistance. Accordingly, the Mercator projection is very well suited to geographic regions
which extend east-west along the equator; in contrast, the transverse Mercator projection is appropriate for those regions
which have a south-north extension. Like the optimal transverse Mercator projection known as the Universal Transverse Mercator Projection (UTM), which maps the meridian of reference Λ0 with an optimal dilatation factor &ρcirc;=0.999 578 with respect to the World Geodetic Reference System WGS 84 and a strip [Λ0−Λ
W
,Λ0 + Λ
E
]×[Φ
S
,Φ
N
]= [−3.5∘,+3.5∘]×[−80∘,+84∘], we construct an optimal dilatation factor ρ for the optimal Mercator projection, summarized as the Universal Mercator Projection (UM), and an optimal dilatation factor ρ0 for the optimal polycylindric projection for various strip widths which maps parallel circles Φ0 equidistantly except for a dilatation factor ρ0, summarized as the Universal Polycylindric Projection (UPC). It turns out that the optimal dilatation factors are independent of the longitudinal extension of the strip and depend
only on the latitude Φ0 of the parallel circle of reference and the southern and northern extension, namely the latitudes Φ
S
and Φ
N
, of the strip. For instance, for a strip [Φ
S
,Φ
N
]= [−1.5∘,+1.5∘] along the equator Φ0=0, the optimal Mercator projection with respect to WGS 84 is characterized by an optimal dilatation factor &ρcirc;=0.999 887 (strip width 3∘). For other strip widths and different choices of the parallel circle of reference Φ0, precise optimal dilatation factors are given. Finally the UPC for the geographic region of Indonesia is presented as an
example.
Received: 17 December 1997 / Accepted: 15 August 1997 相似文献
13.
《测量评论》2013,45(21):417-422
AbstractThe Transverse Mercator Projection is also called the Conformal of Gauss since it was devised by him in the early part of the nineteenth century in connexion with the Triangulation of Hanover. It belongs to the class of cylindrical orthomorphic projections. That is to say, the Earth's surface, or part thereof, is developed on the surface of a cylinder, and there is practically no angular distortion, an angle on the surface of the Earth being represented on the map by almost precisely the same angle. The representation of meridians and parallels, for instance, shows them intersecting at right angles as they actually do on the Earth's surface; but this orthotomic condition, though essential, is not in itself sufficient for orthomorphism. 相似文献
14.
《The Cartographic journal》2013,50(2):106-108
AbstractIn 1967 Dr Arno Peters made his first public claim to have designed a world map superior to Mercator and all others. Although professional cartographers have pointed out that the projection is not original or unique, the map has gained a not inconsiderable measure of political acceptance. 相似文献
15.
Simple and highly accurate formulas for the computation of Transverse Mercator coordinates from longitude and isometric latitude 总被引：2，自引：0，他引：2
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.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献
16.
《测量评论》2013,45(26):230-234
AbstractTHE resolutions and pious hopes (væux) passed by the International Union of Geodesy and Geophysics at Edinburgh in September 1936 have just been circulated in a formidable document of 8 pages and XXIX commandments. Of these, two affect the Cape-to-Cairo line particularly and they seem to deserve special study. The first of these, Number III—on systems of Projections—applies the meridional strips of the Transverse Mercator Projection apparently to all maps, topographical as well as cadastral. 相似文献
17.
Shapes on a plane: evaluating the impact of projection distortion on spatial binning 总被引：1，自引：0，他引：1
ABSTRACTOne method for working with large, dense sets of spatial point data is to aggregate the measure of the data into polygonal containers, such as political boundaries, or into regular spatial bins such as triangles, squares, or hexagons. When mapping these aggregations, the map projection must inevitably distort relationships. This distortion can impact the reader’s ability to compare count and density measures across the map. Spatial binning, particularly via hexagons, is becoming a popular technique for displaying aggregate measures of point data sets. Increasingly, we see questionable use of the technique without attendant discussion of its hazards. In this work, we discuss when and why spatial binning works and how mapmakers can better understand the limitations caused by distortion from projecting to the plane. We introduce equations for evaluating distortion’s impact on one common projection (Web Mercator) and discuss how the methods used generalize to other projections. While we focus on hexagonal binning, these same considerations affect spatial bins of any shape, and more generally, any analysis of geographic data performed in planar space. 相似文献
18.
Abstract Introductory Remarks.—A line of constant bearing was known as a Rhumb line. Later Snel invented the name Loxodrome for the same line. The drawing of this line on a curvilinear graticule was naturally difficult and attempts at graphical working in the chart-house were not very successfuL Consequently, according to Germain, in 1318 Petrus Vesconte de Janua devised the Plate Carree projection (“Plane” Chart). This had a rectilinear graticule and parallel meridians, and distances on the meridians were made true. The projection gave a rectilinear rhumb line; but the bearing of this rhumb line was in general far from true and the representation of the earth's surface was greatly distorted in high latitudes. For the former reason it offered no real solution of the problem of the navigator, who required a chart on which any straight line would be a line not alone of constant bearing but also of true bearing; the first condition necessarily postulated a chart with rectilinear meridians, since a meridian is itself a rhumb line, and for the same reason it postulated rectilinear parallels. It follows, therefore, that the meridians also must be parallel inter se, like the parallels of latitude. The remaining desideratum—that for a true bearing—was attained in I569 by Gerhard Kramer, usually known by his Latin name of Mercator, in early life a pupil of Gemma Frisius of Louvain, who was the first to teach triangulation as a means for surveying a country. Let us consider, then, that a chart is required to show a straight line as a rhumb line of true bearing and let us consider the Mercator projection from this point of view. 相似文献
19.
A new companion for Mercator 总被引：1，自引：1，他引：0
Waldo Tobler 《制图学和地理信息科学》2018,45(3):284-285
The inappropriate use of the Mercator projection has declined but still occasionally occurs. One method of contrasting the Mercator projection is to present an alternative in the form of an equal area projection. The map projection derived here is thus not simply a pretty Christmas tree ornament: it is instead a complement to Mercator’s conformal navigation anamorphose and can be displayed as an alternative. The equations for the new map projection preserve the latitudinal stretching of the Mercator while adjusting the longitudinal spacing. This allows placement of the new map adjacent to that of Mercator. The surface area, while drastically warped, maintains the correct magnitude. 相似文献
20.
针对球体横墨卡托投影与基于地球椭球体的导航设备结合使用存在误差以及传统椭球横墨卡托投影依据经差分带不适用于极区的问题,在分析双重投影可用于极区存在计算奇异和计算溢出问题的基础上,研究了一种基于双重投影的横墨卡托投影极区应用改进方法。首先利用函数等效变换和经线长度比计算公式推导出椭球投影到球体上的坐标变换、球体半径和长度比计算公式,然后利用分段函数的方法研究了球体横墨卡托投影计算公式,综合两个阶段给出了完整的坐标变换公式和长度比计算公式,最后推导了子午线收敛角计算公式。理论分析和算例仿真表明,该改进方法能够解决极区投影计算奇异和计算溢出问题,近极点地区长度变形较小,且与导航设备采用的地球模型一致,可消除由于地球模型不同引起的误差,提高航海绘算精度。 相似文献