THE NOMENCLATURE OF MAP PROJECTIONS

Abstract

Map Projections.—A matter that should have been mentioned in the original article under this title (E.S.R., vii, 51, 190) is the definition of a map projection. In the list of carefully worded “Definitions of Terms used in Surveying and Mapping” prepared by the American Society of Photogrammetry (Photogrammetrie Engineering, vol. 8,1942, pp. 247–283), a map projection is defined as “a systematic drawing of lines on a plane surface to represent the parallels of latitude and the meridians of longitude of the earth or a section of the earth”, and most other published works in which a definition appears employ a somewhat similar wording. This, however, is an unnecessary limitation of the term. Many projections are (and all projections can be) plotted from rectangular grid co-ordinates, and meridians and parallels need not be drawn at all; but a map is still on a projection even when a graticule is not shown. Objection could be raised also to the limitation to “plane surface”, since we may speak of the projection of the spheroid upon a sphere, or of the sphere upon a hemisphere. Hence, it is suggested that “any systematic method of representing the whole or a part of the curved surface of the Earth upon another (usually plane) surface” is an adequate definition of a map projection.     

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 ORTHOMORPHIC PROJECTION OF THE SPHEROID—IV

Abstract

We now turn to a question which has received much attention of recent years; the possibility of transforming angular and linear field measures to an orthomorphic projection so that the results of a survey may be computed directly in plane Co-ordinates without having to go through the spheroid at all. Initially, orthomorphic projections were introduced into surveying practice for this very object. Over short lines they import so little distortion of angles that minor surveys, whose error of angular measurement is comparable with such distortion, may be reduced in the rectangular co-ordinate system of an orthomorphic projection just as if the earth were flat. But the present application goes far beyond that. We no longer ignore distortions of angles and lengths, but systematically introduce them into the field measures so that work of higher precision and of considerable extent may also be computed and adjusted in plane co-ordinates.     

NORMAL EQUATIONS RESOLVED BY APPROXIMATION

Abstract

We are indebted to Professor R. V. Southwell for the approximate method of computation known as the systematic relaxation of constraints. In an article to the Empire Survey Review, 1938, Mr A. N. Black showed how Southwell's ideas could be applied to the adjustment of the co-ordinates of a point.     

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 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.     

CHECKING TRAVERSE COMPUTATIONS

Abstract

Traverse Computations must be Checked.—A traverse is a chain of points connected by angular and linear measurements. The check on observations is provided by the agreement, obtained in computations, between the terminals of the traverse (terminal bearings and terminal co-ordinates) taken as fixed. This check is not sufficient, however, to serve as a check on the computations. As a matter of principle, computations should be free of errors; there are no limits of tolerance in computational work except for discrepancies arising from inaccuracy of last figures. Secondly, errors in computation may occur that are not revealed by the traverse misclosures, not to speak of compensational errors, the field for which is very favourable in traverse work.     

MINOR TRIANGULATION ON THE TRANSVERSE MERCATOR PROJECTION

Abstract

1. Computation of a minor triangulation as if it were executed on a plane surface of course ignores spherical excess, an omission not strictly rigorous so far as azimuths are concerned. Further the reduction of such a triangulation to a system of plane coordinates again assumes that the earth's surface is a plane.     

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.     

MACHINE METHOD OF INTERSECTION

Abstract

The following method will be found better and quicker than the usual logarithmic process in computing the co-ordinates of intersected points in minor triangulation and traverse work. Let A and B be two stations whose co-ordinates (x ₁ y ₁), (x ₂ y ₂) are known. Let P be an intersected point whose co-ordinates (x, y) we wish to determine. Let α and β be the observed angles at A and B respectively.     

THE FIXATION OF MINOR TRIANGULATION ON THE TRANSVERSE MERCATOR PROJECTION

Abstract

The fixation of Minor Triangulation in a Primary system does not, in general, warrant rigorous adjustments of figures; less laborious methods are desirable. For Secondary work a least square adjustment to approximate coordinates is quite sufficient, while, for Tertiary, graphical solutions are amply accurate. Apart from that, cases may arise to which a figure adjustment is not applicable, as in the small net shown in Fig. 2, p. 464. The line BC cannot be equated to the line AB in the ordinary way since it is not the side of a triangle. In this case an adjustment to approxima te coordina tes will overcome the difficulty.     

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 Oblique Zenithal Equidistant Projection

Abstract

The idea of constructing an equidistant map centred on Glasgow originated as a school project. The amount of calculation required meant that it was necessary to use a computer to calculate the co-ordinates for the projection on a 15° graticule. Subsequently it was decided to write a program to instruct another computer, equipped with a graph plotter, to plot the same graticule automatically. The basic mathematical constructions and the procedure for programming the computer for automatic plotting are fully explained.     

A TRIGONOMETRICAL UPSET

Abstract

If the geographical co-ordinates, Φ₀, L 0, and the azimuth A ₀ at a station O of a triangulation undergo corrections, ?Φ₀, ?L ₀ and ?A ₀, the geographical co-ordinates, Φ, L, and the azimuth A have to be re-computed for all the vertices throughout the whole triangulation. This is a tedious operation. It may be vastly simplified, however, by the employment of differential formulae. The derivation of these formulae would consume considerable space, so that the results alone are given here.     

A MACHINE METHOD OF RESECTION

Abstract

With the modern calculating machine in easy reach of every computer, the problem of determining the position of an occupied point from which direction observations have been made to three or more known points has become quite simple. The method outlined below is quite elegant in form and exceedingly simple on the machine. Let A, B, C be the three points whose co-ordinates (X₁Y₁), (X₂Y₂), (X₃Y₃) are known, and let (XY) be the co-ordinates of the point P which we wish to fix.     

A COMBINED PROJECTION

Abstract

The projection in question is a mean between the Cylindrical Equivalent (Equal-Area) Projection and its Transverse Projection. The position of any point on the earth's surface is defined by the mean x and the mean y of the two constituent projections.     

A generalisation of the least-squares method for the adjustment of free networks

Summary The system of normal equations for the adjustment of a free network is a singular one. Therefore, a number of coordinates has to be fixed according to the matrix. The mean square errors and the error ellipses of such an adjustment are dependent on this choice. This paper gives a simple, direct method for the adjustment of free networks, where no coordinates need to be fixed. This is done by minimizing not only the sum of the squares of the weighted errorsV ^T PV=minimun but also the Euclidean norm of the vectorX and of the covariance matrixQ X ^T X=minimum trace (Q)=minimum This last condition is crucial for geodetic problems of this type.     

Geodetic grids in authoritative maps – new findings about the origin of the UTM Grid

ABSTRACT

Recent 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.     

Change of projection following readjustment

Abstract

The readjustment of a major geodetic control network results in a new set of spheroidal coordinates for the network stations. Those new coordinates followed by an appropriate control densification serve as input for computing new plane coordinates. There are many surveying and mapping products which are based on the existing 'old' plane coordinates system. This paper deals with considerations and procedures aiming at the introduction of a new projection defined in such a way as to minimise the detrimental consequences of readjustment through the use of a synthetic point of origin for the new projection.     

LAPLACE AZIMUTHS

Abstract

The adjustment of a chain of triangulation for scale between measured bases is a comparatively simple matter. If it were possible to provide a similar method of adjustment for swing between measured azimuths and if the triangulation were based on a fixed datum, then there would be some hope of holding the chain fixed, without waiting until the completion of further chains rendered a net adjustment possible. This would be well worth doing.