THE CONVERSION OF CO-ORDINATES FROM THE CASSINI TO THE TRANSVERSE MERCATOR PROJECTION

Abstract

1. The Secondary and Tertiary Triangulations of the six counties of Northern Ireland which were observed about 1900 were computed county by county each on its own meridian on a Cassini projection using Airy's figure of the earth. Although a number of points common to two or more counties were fixed no attempt was made to bring the separate counties into sympathy either with each other or even with the old Primary triangulation as adjusted by Clarke in 1856.     

A DIRECT METHOD OF TRANSFORMING RECTANGULAR COORDINATES FROM THE CASSINI PROJECTION TO THE GAUSS CONFORMAL PROJECTION

Abstract

6. Further Expansions.—Equations (4.3) and (5.5) enable a computer to transform coordinates from the Cassini projection to the Gauss projection without recourse to geographical coordinates. If applied to one or two points, no doubt these equations would be quite satisfactory; but if applied to 100,000 points their use would be laborious and it would be difficult to adapt them to machine computing.     

THE MERCATOR PROJECTION AND THE RHUMB LINE

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.     

THE ORTHOMORPHISM OF THE STEREOGRAPHIC PROJECTION

Abstract

When developing the argument leading to the stereographic solution of the spherical triangle and its application to field astronomy (Empire Survey Review, Vol. 2, No. 10, October, 1933, p. 226) A. J. Potter rendered a very useful service in demonstrating how proofs of the two practically useful properties of the stereographic projection can be provided along lines that demand no more than simple geometry in their development. The proof advanced for the unique property that any circle on the. sphere remains a circle in projection is at once simple and complete; but in the attempt to prove that the projection is orthomorphic in the sense that angles everywhere remain true there is the difficulty that the argument was developed for what must be regarded as a special case in that the point was located on the great circle through the origin of the projection normal to the plane of the projection. Treatment of the problem along similar lines for other points away from the central meridian does not seem to admit of such ready solution and the alternative approach suggested here, while still not demanding. anything beyond simple geometry for its understanding, affords a proof for a general case.     

ON THE LAMBERT CONFORMAL PROJECTION AS APPLIED IN CHINA

Abstract

There 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.     

THE TRANSVERSE MERCATOR PROJECTION OF THE SPHEROID

Abstract

In 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.     

THE MODIFIED MERIDIAN DISTANCE IN THE CONICAL ORTHOMORPHIC PROJECTION

Abstract

Recently the writer of this article became interested in the conical orthomorphic projection and wanted to see a simple proof of the formula for the modified meridian distance for the projection on the sphere. Owing to the exigencies of the war, however, he has been separated from the bulk of his books, and, consequently, has had to evolve a proof for himself. Later, this proof was shown to a friend who told him that he had some memory of a mistake in the sign of the spheroidal term in m⁴given in “Survey Computations”, perhaps the first edition. Curiosity therefore suggested an attempt to verify this sign, which meant extending his work to the spheroid. This has now been done, with the result that the formula given in “Survey Computations”, up to the terms of the fourth order at any rate, is found correct after all.     

GRID BEARINGS AND DISTANCES ON THE CONICAL ORTHOMORPHIC PROJECTION

Abstract

When computing and adjusting traverses or secondary and tertiary triangulation in countries to which the Transverse Mercator projection has been applied, it is often more convenient to work directly in terms of rectangular co-ordinates on the projection system than it is to work in terms of geographical coordinates and then convert these later on into rectangulars. The Transverse Mercator projection is designed in the first place to cover a country whose principal extent is in latitude and hence work on it is generally confined to a belt, or helts, in which the extent of longitude on either side of the central meridian is so limited as seldom to exceed a width of much more than about 200 miles.     

THE TRANSVERSE MERCATOR PROJECTION AND ITS APPLICATION TO TANGANYIKA TERRITORY

Abstract

The 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.     

THE CONVERGENCE OF MERIDIANS

Abstract

Some 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.     

THE TRANSVERSE MERCATOR PROJECTION

Abstract

I. 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.     

TRANSFORMATION OF RECTANGULAR CO-ORDINATES ON THE TRANSVERSE MERCATOR PROJECTION

Abstract

The 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.     

THE DERIVATION OF CERTAIN PROJECTION FORMULAE FOR THE SPHERE

Abstract

The purpose of this article is to make available to readers of the Review a simple method of derivation from first principles of the projection farmulae for same of the more impartant normal conical projections.     

MINOR TRIANGULATION ON THE TRANSVERSE MERCATOR PROJECTION

Abstract

In 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.     

A GEOMETRIC ALGORITHM FOR TILTED-CAMERA PERSPECTIVE PROJECTION AND THE COMPARISON WITH EXTERNAL PERSPECTIVE PROJECTION

ABSTRACT

A geometric algorithm for Tilted-Camera Perspective (TCP) projection is proposed in this paper based on the principle of perspective projection. According to that, the difference between TCP projection and External Perspective (EXP) projecton is analyzed. It is put forward prerequisites making these two projections were compatible, and some examples are given.     

极区高斯投影与日晷投影的比较

为解决传统球面高斯投影公式在极点处的奇异问题,通过引入余纬度对原有投影公式进行改进,推导了极区高斯投影非奇异公式;基于该公式推导了极区经纬线投影方程,并结合日晷投影进行长度变形及子午线偏移角分析。结果表明,在余纬度很小时,高斯投影与日晷投影非常接近,即其经纬网与日晷投影近似;在极圈内高斯投影长度变形小于日晷投影,其经线与日晷投影经线的最大偏移角为2.4688°,而在纬度80°以上,最大偏移角为0.4386°。极区非奇异高斯投影公式满足了极区内连续投影的需求,可为极区海图绘制提供理论依据。     

AN OCTAHEDRAL GNOMONIC PROJECTION

Abstract

The problem of projecting the whole sphere on to a polyhedral gnomonic projection has been studied since very early times. Amongst more recent works may be mentioned those of Fisher, Campbell and Taylor, and the present writer.     

ON THE LAMBERT CONFORMAL PROJECTION AS APPLIED IN CHINA

Abstract

In the last instalment of this paper it was explained that, owing to the immense size of the country, the co-ordinate system adopted by the Central Land Survey for the mapping of China consists of a number of zones bounded by parallels of latitude, the survey in each zone being based on the Lambert conical orthomorphic projection. The great extent of each zone in longitude, some sixty-five degrees, necessitated the development of series which would converge reasonably quickly, and, for this purpose, series were obtained in which a vertical distance between the parallel passing through the given point and the central parallel was used instead of the co-ordinates themselves. The series already given provided for the conversion of geographical co-ordinates into rectangulars and the inverse problem, while the present instalment deals with the scale factor and the transformation of co-ordinates from one zone to another, concluding with some numerical examples.     

等距离球面高斯投影

针对传统高斯投影直接基于等角横切圆柱投影而带来的不能直接用球面坐标换算等问题,研究了一种运用等距离球面进行投影的高斯投影,即等距离球面高斯投影。借助等距离纬度正反解公式,推导了等距离球面高斯投影的正反解公式,分析了其经纬线变形情况;基于投影公式,计算和分析了等距离球面高斯投影的长度变形、角度变形、面积变形及子午线收敛角等参数;最后与传统高斯投影进行比较,说明了该投影的可用性。