THE TRANSVERSE MERCATOR PROJECTION: A CRITICAL EXAMINATION

Abstract

In 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.     

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 HUNTER SHUTTER EYEPIECE FOR LONGITUDE AND AZIMUTH

Abstract

The shutter eyepiece is a device for eliminating the effects of personality in observations involving the timing of a moving star crossing graticule lines. It is of importance in longitude observations and in any mixed observations involving timing of star passages over cross-lines—vertical, horizontal or oblique.     

Unfolding the Earth: Myriahedral Projections

Abstract

Myriahedral projections are a new class of methods for mapping the earth. The globe is projected on a myriahedron, a polyhedron with a very large number of faces. Next, this polyhedron is cut open and unfolded. The resulting maps have a large number of interrupts, but are (almost) conformal and conserve areas. A general approach is presented to decide where to cut the globe, followed by three different types of solution. These follow from the use of meshes based on the standard graticule, the use of recursively subdivided polyhedra and meshes derived from the geography of the earth. A number of examples are presented, including maps for tutorial purposes, optimal foldouts of Platonic solids, and a map of the coastline of the earth.     

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.     

Mapping Cyberspace: Visualizing,Analysing and Exploring Virtual Worlds

Abstract

With the development of computer technologies such as the Internet and reality, the notion of cyberspace has been emerging and it has been increasingly studied by researchers in various disciplines involving the computer sciences, sociology, geography, and cartography. Cybermaps, as special maps for cyberspace, have been produced and used as a tool for understanding various aspects of cyberspace virtual worlds. Virtual worlds can be distinguished in many ways from the physical world we live in. Because of these distinctions, it is a big challenge for cartographers to offer some clarification. This paper addressed various mapping issues such as visualizing, analysing and exploring cyberspace from different aspects.     

Measuring the Coastline from Maps: A Study of the Scottish Mainland

Abstract

While attempting to devise a methodology for the assessment of the impact of planning decisions on the coastline of Scotland, it became apparent that a means of mapping information concerning the coastline and coastal zone was necessary. This meant that the maps had to be capable of being produced in bulk, they should be easy to alter and produce at different scales and with different information. It was therefore decided to investigate the potential of computational techniques to produce the required maps. This was performed by employing a d-Mac digitising table to generate the required coordinates and by processing the information on the University of Aberdeen's ICL system 4/70 computer, the output being on drum plotter.     

TRAVERSE REDUCTION WITH AN ELECTRONIC DIGITAL COMPUTER

Abstract

The article describes work carried out by the authors on the Manchester Universal Digital Computer at Manchester University. The general characteristics of the computer are discussed. The requirements for traverse reduction for Ordnance Survey work are summarised, with the help of a specific example. The technique required to present the details of a traverse to a computer for reduction is then described, and some details of the operation of the programme are given. The article is concluded with some details of the time taken to carry out the reduction, and remarks about special features of computer design which would have proved advantageous for this work.     

SOME NOTES ON COMPUTING

Abstract

Probably the most important matter connected with computing is checking. Far more time is wasted by an incorrect result which has been passed as correct by the checker than by any other fault in computing. It is not necessary that a computer should always be perfectly accurate, but it is necessary that the method of checking used should always detect his errors. If the computer is sure that his accidental errors will be picked up he is relieved of anxiety and his work will be quicker and eventually more accurate for that reason.     

An Evaluation of the Use of Grid Squares in Computerised Choropleth Maps

Abstract

There are numerous computer programs to produce choropleth maps and some work has also been published on the use of a grid matrix as a way of dividing an area into discrete units. The purpose of this paper is to combine the two approaches and evaluate the suitability of using a network of grid cells, each containing a representative value of the variable being mapped, as a way of producing choropleth maps on a computer.     

FMC cameras,high resolution films and very large scale mapping

Abstract

Very large scale mapping (1/250) was experimented on the basis of FMC camera, high resolution film and total station surveying.

The future attractive combination of precision photogrammetry and personal computer assisted terrestrial surveying was investigated from the point of view of accuracy, time effectiveness and total procedures control.     

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.     

An Agricultural Atlas of Scotland

Abstract

An account is given of the theoretical and practical problems of producing a series of maps from available statistical data. The source material and in particular the disparities in area of the statistical units are treated in detail. A simple program was developed for use with the KDF9 computer with line printer output, and eventually this was used to provide a compilation for drawing rather than a finished product.     

KOMPLOT "Do-it-yourself" Computer Cartography

Abstract

After describing the changes which program packages, such as SYMAP, can make to the production of thematic maps the author briefly considers the aesthetics of computer graphics. The simplicity and versatility of KOMPLOT are then discussed with the aim of bringing computer aided cartography to a wider field of users.     

CONVENTIONS AND GENERALIZED FORMULAE FOR THE ASTRONOMICAL TRIANGLE

Abstract

In the October 1953 issue of this Review (E.S.R. xii, 90, 174), Mr. J. G. Freislich has written of the difficulties of a southern hemisphere computer attempting to use astronomical formulae from a textbook prepared for use in the northern hemisphere. He proposes a solution in which different conventions are adopted in the two hemispheres, leading to different formulae for the two cases, a solution which the present writer does not favour.     

THE COMPUTER—A COG IN THE SURVEY MACHINE

Abstract

1. Lest any reader should be put off by assuming from the title of this article that it would contain a mass of technical matter, it may be stated at the outset that no computations whatever are involved. In fact it is a collection of very ordinary observations pertaining to the efficient functioning of the computing section of a Survey Department. In the opinion of the writer nothing clarifies one's ideas more than sitting down at a table and putting them actually on paper. Things that might otherwise have been overlooked or forgotten are brought to light, and can then be given their proper place in the whole mosaic. There is probably nothing original in this article, yet it is hoped that a comprehensive survey may be of assistance to other computers, as the making of it certainly has been to the writer. It has been written from the point of vievvof a computing section, rather than that of an individual computer. Each individual computer is a cog in the computing machine, whose duties are more those of routine calculation by methods approved by his superior.     

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.     

TRIANGULATION: BEARINGS VERSUS ANGLES

Abstract

There has always been a marked difference of opinion on the relative merits of the methods of bearings and of angles as applied to triangulation, though it is probable that the majority of writers prefer the method of bearings for first-order work. The subject was mentioned in a recent issue of this Review (vii, 47, 19).     

Numerically Controlled Methods of Automatic Plotting and Draughting

Abstract

The equipment available for numerically controlled plotting and draughting is reviewed and certain devices described in some detail. Several systems suited to the digitising of cartographic material exist, while the available plotters may be classified broadly into three categories—those derived from surveying and photogrammetric practice, those associated with computer technology and a group designed substantially for cartographic applications. Finally the various input and output devices associated with cartographic digitising are discussed with special emphasis on their storage capacity, speed of operation and economy.