MEAN SQUARE ERRORS OF POINT DETERMINATIONS

Abstract

This, the third and last, instalment on this subject has been curtailed considerably to save printing expenses on graphs and on mathematical symbols. Only the barest outline of the more interesting results are given.     

THE ADJUSTMENT OFA BLOCK OF AERIAL TRIANGULATION EVALUATED WITH THE WILD A5

Abstract

The procedure for aerotriangulation on the Wild A5 and similar plotting instruments is well known. The first overlap is set up in absolute orientation on well spaced plan and height control and successive overlaps are set up relatively, each to the previous overlap, by eliminating want of correspondence and preserving the height agreement of points falling in the common portion of successive overlaps. When each overlap is correctly set, the co-ordinates of selected points are measured on the instrument (machine co-ordinates). These co-ordinates differ from true ground co-ordinates only in origin, azimuth and scale, provided the settings and measurements are precisely done on error-free models, precisely connected together. However, such ideal conditions are never obtained, and the errors in azimp.th, scale and height datum increase with the number of overlaps added along a strip.     

LONG LINES ON THE EARTH: VARIOUS FORMULAE

Abstract

Extensions were given for all these formulae, so that precise results may now be obtained even for lines of 500 miles in latitudes above 45°. The present instalment gives the extension of the Clarke approximate (sic) formulae to lines of 500miles, with a practical example and general conclusions: the great advantage of the method is that 8-figuretables sufficeto give rigorous results.     

COMMONWEALTH SURVEY OFFICERS CONFERENCE, 1951

Abstract

As already noted in the last instalment of this report, the whole of Monday, 16th July, was devoted to all-day visits to (i) a surveying ship of the Royal Navy, or (ii) the Royal Observatory at Hurstmonceaux Castle, or (iii) the School of Military Survey at Newbury, Berks, delegates being divided into three groups with one group going on each visit.     

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.     

THE CLARKE FORMULÆ FOR LATITUDE,LONGITUDE AND REVERSE AZIMUTH,WITH CORRECTIVE TERMS FOR USE ON VERY LONG LINES

Abstract

This extremely simple and elegant method of computing geographical co-ordinates, given the initial azimuth and length of line from the standpoint, was published by Col. A. R. Clarke in 1880. There is no other known method giving the same degree of accuracy with the use of only three tabulated spheroidal factors. Clarke himself regarded this as an approximate formula (vide his remark in section 5, p. 109, “Geodesy”); but as this article demonstrates, it is capable of a high degree of precision in all occupied lati tudes when certain corrections are applied to the various terms. These corrections are comparatively easy to compute, require no further spheroidal factors, and some of them may be tabulated directly once and for all.     

THE ORTHOMORPHIC PROJECTION OF THE SPHEROID–III

Abstract

In the last instalment we were able to obtain most of the surveyor's projections in common use by applying simple scale conditions to the meridians and parallels. This method of approach naturally suggests that results of some value might be obtained by applying similar conditions to the plane co-ordinate lines. If we do so, we are immediately led to consider curves on the surface known as geodesics, which are the nearest approach to straight lines it is possible to draw on a curved surface. Accordingly, we give some account of these curves for the benefit of surveyors who have not hitherto made their acquaintance.     

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 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—II

Abstract

The reader will play this round better if he remembers to bring the bag of clubs provided in the last instalment. On the assumption that he will do so, we proceed straight to the first tee without further explanation or recapitulation.     

COMPUTATION OF LONG ARCS FOR RADIO PURPOSES

Abstract

In the last instalment of this article I showed how, by computing the difference in height between the spheroid and the sphere at the mid-point of the line, the third order term could be obtained and a more accurate correction to the spherical length applied. This allows the formula to be used for the determination of distances and azimuths for lines far exceeding 1,000 kilometres in length.     

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.     

The Ordnance Survey of Northern Ireland: An Outline of its History and Present Mapping Tasks

Abstract

This explains the formation of the Ordnance Survey of Northern Ireland as a separate institution in 1921, when it took over the map series which until that time had been part of the Ordnance Survey (U. K.) responsibility. As happened with many mapping organisations, replanning and reorganisation took place after 1945; this included a triangulation for Northern Ireland to provide a basis for a new Irt"sh Grid, and the introduction of the 1:1250 scale for major urban areas.

The technical problems of reproduction and revision are then considered, and details given for each of the main series.     

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.     

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.     

Systematic Geometric Image Measurement Errors of Circular Object Targets: Mathematical Formulation and Correction

The use of circular object targets is very common in spatial photogrammetric object reconstruction. An object circle is projected on to the image plane as an ellipse if the object plane and the image plane are not parallel to each other. The image co-ordinates of the centre of the ellipse are usually determined automatically by means of digital image processing. These co-ordinates are then used as observations for a subsequent reconstruction of the three dimensional object point. The image co-ordinates of the centre of the ellipse and the true co-ordinates of the projected centre of the circular object target are not identical; thus eccentricity is caused, resulting in systematic geometric image measurement errors. This paper describes the functional context of this eccentricity for a typical target/camera set-up. The possible solutions for a correction of this systematic eccentricity error are derived. Guidelines for the correct combination of object target sizes and exposure distances, guaranteeing reliable image point measurements and accurate object point determination, are proposed.     

SOME SURVEYS OF RIVERS IN CEYLON

Abstract

The city of Colombo, capital of Ceylon, is situated at the mouth of the Kelani Ganga and about 25 miles north of another considerable river, the Kalu Ganga, which enters the sea at Kalutara. Though its proximity to two of the largest of the perennial rivers of Ceylon provides some advantages of inland water transport by river craft, it also places it in the zone of periodical floods which cause damage to buildings, communications and agriculture within the city limits and over extensive areas of low-lying paddy fields and gardens beyond. Except for a spur of high land running roughly parallel to the eastern shore of the harbour through Hultsdorp, Blomendahl and Kotahena to the loop of the Kelani Ganga near its mouth, the greater part of the city is very flat and only a matter of 20 feet elevation above sea-level. On its eastern flank lies a series of connected shallow lakes, marshes and paddy fields which extend in a loop from the banks of the Kelani Ganga at Ambatalenpahala to Dehiwela on the city's southern limit; there are, in addition, canals through this area with two outlets into the sea at Wellawatta and Dehiwela. Navigable canals connect the Kelani Ganga with the Beira Lake and Colombo Harbour and, with the canals already mentioned, form a continuous waterway which, in fact, extends northwards as far as Puttalam, 80 miles from Colombo, and southwards to the Kalu Ganga and Kalutara. About half a mile east of Mount Lavinia another series of shallow lakes, marshes and paddy fields spreads its irregular shape roughly parallel to the coast through the Bolgoda Lakes to a low ridge north of the Kalu Ganga and close to its right bank. The whole of these lowlands is referred to as the Bolgoda Basin. A canal called the Kepu Ela, constructed by the Dutch during their occupation of Ceylon, passes through it to join the Kalu Ganga at Diyagama about 5 miles from the river mouth.     

SOME NOTES ON THE ADJUSTMENT OF TRAVERSES

Abstract

It has now become a truism that angles in traverses are observed with a far greater accuracy than the sides, and this applies to all categories of traverse work: precise, transit-and-tape, tacheometric traverses, etc. The result of this is that the effect of angular errors on misclosures in co-ordinates is sometimes so small when compared with that of linear errors that it may be considered as negligible.     

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.     

AIR SURVEY: THE RAND METHOD OF HEIGHTING FROM NEAR VERTICAL PHOTOGRAPHS

Abstract

The simplest approach is that of spherical trigonometry. In the following proof I have departed from Hotine's sign conventions in an attempt to obtain consistency between the spherical quantities and the photo rectangular co-ordinates.