Systems for automatic computation and plotting of position fixing patterns

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RIJ KSW A TERST AAT COMMUNICATIONS

SYSTEMS

FOR

AUTOMATIC

COMPUTATIONS

AND

PLOTTING

OF

POSITION

FIXING

PATTERNS

by

Ir. H. PH. VAN DER SCHAAF

t

Chief Engineer, Rijkswaterstaat

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Any correspondence should beaddressed10

RIJKSW ATERST AAT

DIRECTIE WATERHUiSHOUDING EN WATERBEWEGING

THE HAGUE - NETHERLANDS

lt isregretted that the author died before this artiele could be published.

Many thanks are due to Ir. C. W.Corbet of the Survey Department ofthe Rijkswaterstaat, Mr. H.E. Oliver ofthe DeccaNavigator Company Limited, Mr. N, van derSchraatofthe Netherlands Geodetic Commission and all ot hers who have collaborated in the realization of th is publication.

The work begun byIr. H. Ph. van der Schaaf will becontinued by Ir.C.W. Corbet. The viewsin this report are theauthor's own.

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Decca pattern for the Netherlands Delta Project as one ofthe first examples ofthe use of electronic computers by the Survey Department of the 'Rijkswaterstaat'.* The introduetion of automatic platting equipment has now made it possible to reduce manual computing and platting work to an absolute minimum. Under the super-vision of Prof. Dr. D.Eckhart, acting as our consultant on computerization problems,

a complete software package has been developed for the computation and plotting of position fixing patterns. The systems on which this software is based are described in this issue of'Communications'.

In the simplest systems, the depietion of the earth's surface by a system of map pro-jection is assumed to provide an undistorted representation, thus enabling plane hyperbolic, circular and are of circle patterns to be used. This mathematical model must, however, be refined if the inaccuracies inherent in the type of projection used are to be taken into account. The plane pattern (see bibl, 1), or alternatively the accompanying grid or graticule, may be modified for this purpose. Systems of this kind are described, not only with regard to the chart patterns but also with regard to the relationship between the chart and the transmitted patterns together with the calibration necessary to determine the pattern constants.

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lotroductioo

Pattern charts are used to fix the position of ships graphicaJly. Position lines are traeed on thern, i.e.loci corresponding to the readings which would be obtained with position fixing instruments at the same location. A position pattern consists theoret-icaJly of an infinite number of such position lines; in practice the presentation is con-fined to a specific pattern of selected readings or points marked at predeterrnined intervals.

Since the position is determined by the intersection of two position lines a pattern chart must contain at least two specific patterns. And since a position is normally expressed in rectangular or geographical coordinates, it is necessary to show either a 'grid' of coordinate intersections based on round values for X and Ycoordinates in the local system of rectangular coordinates or on a 'graticule' of intersections between parallels and meridians based on round values for <pand À.

Position patterns are therefore plotted against a predetermined background consisting either of a grid or of a graticule. Position fixing as such is effected by graphic inter-polation between the plotted position lines, which are based on round values. The rectangular or geographical coordinates of the position can then be read off. Without pattern charts these position fixing systems would entail numerical derivation of the position readings in terms ofX and Yor <pandÀ.

Specific patterns can be plotted for the different position fixing systems on the basis of a simple geometrical relationship bet ween the position lines. The foJlowing pos-sibilities may be used :

A. Hyperbolic patterns, based on confocal hyperbola branches in the plane. These patterns are used,for example, in conjunction with the Decca or Ri-Fix position fixing systems, the readings being taken either on 'decometers' in the case of Decca or on counters in the case of Ri-Fix.

B. Circular patterns, based on concentric circles, used for decometer or counter readings in the two-range Decca or Ri-Fix system. This pattern is a1soused when-ever distances are measured from fixed points; this is done electronically when radio-log, tellurometer, geodimeter or the Cubic and Hydrodist systems are used. C. Arc of circlepatterns, made up of a series of arcs of a circle for sextant readings at

fixed sighting points.

These three types of position fixing patterns are particularly suitable for automatic plotting because of the mathematical relationship between the position lines in a single pattern. Automatic plotting will be considered in detail in this issue of Rijks-waterstaat Communications. It is, therefore, appropriate that a brief review of the

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development of automatic processes for this part ieuIar application should be given in this introduction.

Before the appearance of automatic computing and plotting equipment, the drawing of position patterns on charts was an extremely laborious task. Straight or curved position lines had to be drawn by hand on the paper; the actual plotting work was preceded by equally elaborate manual computing. To draw a straight line the coor-dinates of at least two points had to be calculated. Compasses could be used to draw arcs of circles if the central point was in a convenient position, but the coordinates of this central point and the radius of the circle had to be calculated. In all other cases the coordinates of large numbers of points had to be calculated and the points plotted on the chart. These points then had to be joined manually using a ruler or spline. The first step towards automation was the use of a computer, which led to an enormous reduction in the time needed for calculations. Because of the mathematical relation-ship between the coordinates of the points on each individual position line and also between the position lines themselves, this is an application which produces a maximum saving of labour. Once programmes have been written for different types of pattern, each specific pattern requires very little input data.

The next stage in the development of automation was the appearance of the electron-Ically controlled coordinatograph. Once calculated on the computer and reproduced on punched tape or cards, the coordinates of the points referred to above could be used as direct input for the automatic coordinatograph. The needIe unit controlled by the input then marked out all these points in succession, but manual tracing was still necessary.

In the automatic plotting unit built in the next stage of development, the needIe was replaced by a plotting stylus which automatically connected the points on the chart. The automatic plotting unit could, if desired, be directly controlled by the computer. Rijkswaterstaat Communications No. 2 (see bibl. 1),published in 1960, deals with the method used for preparing the Decca patterns for the Delta Project. Itdescribes one of the first applications of the computer and electronic coordinatograph by the Survey Department of 'Rijkswaterstaat'. Electronic plotting machines were not available commercially at that time. Subsequent developments, however, have made the method described in the above publication obsolete,since, with the equipment available today, the charts could have been plotted even more efficiently.

Allowance was, nevertheless, made for the fact that the plane hyperbolic pattern (type A) does not give a completely accurate representation of the geometrie position of points on the earth's surface when the area considered is large. In principle the geographical position lines forming a spherical, or more precisely a spheroidal, hyperbolic pattern on a sphere or ellipsoid have to be represented on the plane surface by means of a particular map projection. The mathematical model of plane, confocal hyperbolic branches then need refining, i.e. the plane hyperbolic branches must be modified.This correction is necessary if,for example, patterns are being plotted for the North Sea Continental Shelf. A general method of doing this is described later.

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In Chapter 6 of Communications No. 2 (see bib!. 1), reference is also made to the difference between transmitted patterns and chart patterns. If the speed of prop a gation of radio waves were constant, the chart pattern obtained in the manner described above would be a correct representation of the transmitted pattern. In practice, how-ever, the behaviour of radio waves when passing over areas ofvarying electromagnetic conductivity leads to discrepancies between the two patterns.

Practical interpretation of these discrepancies has to be based on calibration measure-ments. For the .measurements the Decometer readings resulting from the transmitted pattern and specific ship positions are compared with synchronous position fixes. The latter have either been made on land by means of direction or distance measure-ments from the shore marks known or are based on readings from a different pattern system the accuracy of which is suflicient to be used as a means of calibration. The above refinements have been developed from experience with hyperbolic position patterns, but they also appJy in principle mutatis mutandis, to circular and are of circle patterns. Up to now, however, there has been no need to appJy the refinements to the Jatter types of pattern because of the much smaller areas of operation in which they are used.

The following are exam pies of the different types of pattern charts used in the Nether-lands. Hyperbolic patterns (type A) we re used in the Delta Project (the RWS Delta Chain OB), the enlargement of the outer harbour at Ymuiden (the Ymuiden Hi-Fix Chain), studies of the Waddenzee along the coast of Groningen (Groningen Hi-Fix Chain), the enlargement of the outer harbour at the Hook of Holland (Europoort Hi-Fix Chain) and ofthe outer harbour at Scheveningen (Scheveningen Hi-Fix Chain), for control of the northern part of the Continental Shelf (Decca Navigator Chain 9B, Frisian Islands Chain), for navigation through the approach and access channel to Europoort and also for control of the southern part of the Continental Shelf (Decca Navigator Chain 2E Europoort) and recently in the Waddenzee studies carried out between the islands and the Frisian coast and between the Ysselmeer dam and the coast of Noord-Holland (Wadden Sea-Fix Chain). Circle pattern charts (type B) have been used in laying a sewage outlet pipe in the sea at Scheveningen building harbour moles at the Hook of Holland and building the dam in the Brouwershavense Gat.

Are of circle patterns (type C) for sextant readings have been used in laying under-water mains in the Western Scheldt, building the Benelux Tunnel; the construction ofthe Hellegat dam, laying mains under the Hollandsch Diep, building the Heinenoord Tunnel and for the dumping of gravel in the Nieuwe Waterweg.

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A. hyperbolic pattern

c.arc of circle pattern

Figure 1. Types ofposition patterns

B. circular pattern

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1. Definition of the

specific position fixing pattern and chart sheet

1.1. Theplanehyperbolicpattern

On a plane surface a hyperbola is an infinite number of points having an identical differencein distance from twogiven points, thefocal points. By varying the differences in distance it ispossible toarrange the points on the plane surfaceinaccordance with aset ofan infinite number of hyperbolae. This set istherefore defined bythe position ofthe focal points.

Aspecific hyperbola intheset can be indicated bythe differencein distance mentioned above or alternatively bythe value assumed by agiven function of this difference in distance. In either case it is essential to distinguish between positive and negative differenees in distance so that thevalues ean be considered ashyperbolic coordinates relating to the correct branches of the hyperbolic system.

In the hyperbolic position fixing system,the focal points are formed by the master (M) and slave(SI) stations; the hyperbolic coordinate isthe lane number L, which can be read off on a decometer. The relationship between such a coordinate L,of the point S, (seefig.1) and the differencein distance (SjM-SISj) isexpressed bythe formula

L, = SjM+MSI-SISj

+

k

), (I)

1

Ifwe defineSjM =aj, MS/=b (base),SISj=cj and-,- = I(whereÀisthe wavelength), A

then Lj=/(aj+b-cj)+k (I)in whichIis a multiplicative constant andk an additive constant (index correction).

When the earth is considered as a flat surface, the general definition of this plane hyperbolic pattern isgiven by the rectangular coordinates XM, YM and XS1, YS1 in a Cartesian system of terrestrial coordinates and by the constants land k, while the specific definition is obtained by choosing a series of discrete values for L,e.g. by taking equalvalues for an interval LIL.

1.2. Thecircular pattern

A circle isthe loci of points situated at an equal distance from a given point. The set of circlesforming a circular pattern is defined bythe common central point M with the coordinates XM, YM(see figure I).

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Discrete values for the radii rof these concentric circles again prod uce aspecific pattern.

The parameter with which a particular circle in the pattern is identified is generally a function of the radius. Since the distance from the fixed point M is often determined electronically, the concept of lane number Lis again used, in linear relationship to the radius r; thus:

1

where1= - is a multiplicative constant and k an additive constant (index correc

-Je

tion).Ifthe reading forL corresponds to the radius r in metres,then 1= 1/2 and k=O.

1.3. The are of circle pattern

The are of a circle is a set of points characterized by an identical difference in

direction from two givenpoints.

When the are of circle pattern (see figure 1) isused, the given points Land R form the directional points for sextant measurements. The measured angle LSR=y has a specific are of circle as position line. In geometrical terms a particular angle"Iwould

give two arcs of circle but by introducing the convention that the angle"Iisidentical to the azimuth from S toR,less the azimuth from S to L, one of these arcs of circle is eliminated. For a point Si it follows that LLSiR=Yi (see fig. I). This sequence of points runs anticlockwise. Jfthelocus excluded in this way is referredto, the order in which Land Rare mentioned is reversed.

Since the actual differences in direction, and not a function of them, are read off on the circle graduation of the sextant, the general definition of the pattern is givenby theterrestrialcoordinates ofthe sightingpoints XL' YL;XR, YR• The specific definition

is obtained by choosing discrete values for"I,e.g. by taking equal values for an interval L1y.

1.4. Definition of the chartsheet

The limits of a section of the specific fixingpattern to be represented on a chart sheet are shown in figure I.The problem is simplyto plot this section to a given scale on

the chart. lts position must therefore also be defined.

It should be possible to give the X and Ycoordinates of the angular points of this area,but if it forms a rectangle it willbe sufficient to work with thelengthpand width

qof the rectangle, together with theX and Ycoordinates of one ofthe angularpoints and the azimuth of one of the sides.

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In this waytwo limiting factors are arrived at, the first relating to the values of the

parameters used todefine theposition linesthat aretobedrawn onthe chart and the

second to the sectionor sectionsofeach position line tobe represented.

1.5. Conclusion

The position pattern and chart sheet have now been defined, while the geometrical

solution to the problem has been given. However, for the elaboration of the com

-putational and plotting techniques an analytic solution of the problems involved is

needed.

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2. Tbe consequences of using an automatic plotting unit

The design ofthe automatic plotting unit influences the computing and plotting system to be used.The explanation given below refers to an automatic unit in which a plotting stylus is moved by step-type motors over a drawing tabie.This movement takes place in steps in eight principal directions which can be distinguished with respect to the azimuth they make with the y-axis of the coordinate system of the instrument (auto-matic plotting units are also available with 16 and 24 operating directions). The y-axis motor generates steps in the

+

y or - y direction, the x-motor steps perpendie-ular to it, i.e. in the

+

x or - x direction. Simultaneous actuation of both motors causes the plotting styles to travel diagonally. Figure 2 iIIustrates these eight principal directions. The step size is fixed; a particular automatic plotting unit may, for example be based on a step of 0.1 mmo

Figure 2. Principal directions inautomatic plotting

The drawing stylus is therefore driven by motors which in their turn are actuated by impulses generated in the computer. If the automatic plotting unit is coup led directly to the computer (on-line system), the resulting chart is to be considered as the computer output. If there is no direct conneetion (off-line system) a system must be designed to use the output results obtained from the computer as the input for the automatic plotting unit. In both cases the stylus must receive successive orders to move aspecific number of steps in one of the main directions. The computer must select both the

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y

,--1../

L...-._..::._""'_'---__ ....X

®

Figure 3. Stepwise plotting of a straight line

direction and the number of steps. The computer also gives the order to lower the stylus onto the drawing table or raise it from the tabie.

Figure 3 is an enlarged illustration of a network of squares having a side length of 0.1 mmo Assuming that the stylus is located at position I, 14 steps in the +x direction bring it from 1 to 2 and 14 steps in the +y axis from I to 3,while 14 combined +x

and

+y

steps bring it from 1 to 4. The lines thus described by the stylus would be 1-2,1-3 and 1-4.

An example of a movement which doesnot coincide with one of the main directions would be from 5 to 6, entailing a total x-movement of 14 steps and any-movement of 2 steps. The line from 5 to 6 can, however,only be drawn by the stylus in basic steps in one of the main directions, in other words the stylus must travel across a polygon. The computer must be programmed to produce the series of elementary steps.Such a program me can be based on the following system.

At the beginning of each step the direction which gives the shortest distance from the finishing point ofthe step to the line connecting the starting and finishing points of the polygon is selected from the main directions which would bring the finishing point of the step closer to the finishing point of the polygon to be described.

The result of this selection is demonstrated for different directions in figure 3, which shows clearly that a choice always has to be made between not more than two

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sibilities. The result is th at zig-zag conneetion lines are obtained whose zig-zag pattern varies with the azimuth of the line to be represented. If the steps are sufficiently small

this raises no practical problems.

Similarly it would be possible to develop systems which approximate curves of the second or higher power and to work out appropriate programmes for them. In the present publication we shall only be concerned with the method discussed above, viz.

approximating the straight line connecting two points that are given in x- and y -coordinates.

The system of operation described has the following irnplication for the design of computing and plotting systems for position patterns:

1. A position line must be built up from chords, although the are is the geometrical form with which we are actually concerned. The distance between the chord and are must therefore not exceed the margin of plotting accuracy.

2. The points on the are serving as the ends of these chords must be calculated in the system of plotting coordinates, viz.x-y system. As the first stage of the calculation it is therefore logical to convert the coordinates of points indicated in the terrestrial

x-

Ysystem, into coordinates of points in thex-y system.

®

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Taking into account the location of the chart concerned, e.g.by theX, Ycoordinates of one angular point and the azimuth of one of the sides, for which in figure 4 point 1 and the side 1-4 have been chosen, the conversion formulae are as folJows:

x=(X -XI)cos104-(Y - YI)sin IA]

y=(Y- YI)cos 1A+(X-XI) sin IA

(2)

Using this coordinate conversion, the methods of calculation to be developed can also be based on thex,y coordinate system of the automatic plotting unit, the origin of which is considered to coincide with the angular point I of the chart.

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3. Elements of system analysis for the specific

plane h

y

perbolic chart

pattern

3.1. Introduetionofparameters for points onaposition line

A position line in a specific pattern is defined by the value allocated to the parameter L. For successive points on line Lh x,y coordinates must be calculated to enable the plotting stylus to draw the connecting lines.

We therefore need a parameter by means of which successive points Sj,j can be deter-mined unambiguously on line Lj.It is not suflicient to use the x orycoordinate because the possibility exists that line x=constant and liney=constant have two points of intersection with line L;. As an appropriate parameter we therefore introduce one of the coordinates based on the oblique angled system ofaxes

ç

-I'], formed by the asymptotes of the hyperbola. The origin of this system is centre 0 of base M SI,

and the equation for the hyperbola is:

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For positive values of

ç

and I']the hyperbola section is indicated to which the lane b2

number Lis added. The epxression -- is a pattern constant illustrating that the 16

shape ofthe pattern is solely dependent onb.All position lines have the same equation in

ç

and '1.Either

ç

or '1 can be used as a parameter to indicate a point on the position line.Each position line has a separate

ç,

'1system.These systems vary with the anglea between the asymptotes and the base MSI. The relationship between the system Çi' '1idefined by angleai and lane number L,is expressed by the formula:

cos «,

=

b-(Lj-k)À

b (4)

whereai < 200 g.The conversion of a point Sj,j given inaj, Çi,j' '1i,j into Xi,j' Yi,j is effected by means of the expressions:

I

(5a)

(5b) Itis necessary to introduce into these conversion formulae (5) either increasing values

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for 11,in which case the Çi,i values corresponding to l1i,i are obtained from (3), or increasing values for

ç,

thel1i,ivalues corresponding toÇi,iagain being obtained from

(3).

4

Figure 5. The parameters tand "Ifor points on a lane

Figure 5 shows that for line L, with 11 as the parameter, the plotting stylus must be brought onto the edge ofthe sheet at point A.The line fromAtoBmust then be drawn and the stylus mustbe raised again at point B. If

ç

is used as the parameter, lineLI would run in the opposite direction, i.e.from B toA.

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3.2. Calculating theparameters for tbe points of intersectien of a positionline witb tbe cbart edge

The first problem is to determine the values of

ç

and '1which apply to the points of

intersection between a position lineL; and the chart edge. Figure 6 shows that there

may be 0, 2, 4 or 6 of these points of intersection, depending on the lane number L,

and on the position of the chart with respect to the pattern. Points of tangency are

irrelevant for our purposes.

If, in accordance with figure 5,the terrestrial distance between angle points 1 and 2 or 3 and 4 isp,and the distance between angle points 1 and 4 or 2 and 3 isq,this implies analytical testing in 2 phases:

1. Does the hyperbola branche L,have real points of intersection with the 4 lines:

x=O, x=p, y=O and y=q?

2. If so, for the first 2lines with x = constant the condition must be met that

°

:s;;y :s;;q

and for the last 2lines withy=constant that

°

:s;;x ~ p.

In the situation illustrated in figure 5, lineL,accordingly has no real points of intersec-tion with the line x=O,but two such intersections with each ofthe other lines; the first

examination therefore gives 6 points of intersection, of which, however, only two

comply with the second criterion ofthe test.

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The following system of formulae gives us the result in terms of

ç

and 'I.If in (3) we substitute from (5a):

we obtain the quadratic equation:

- } 2 b2_

{sin (OM -a.i) 'Ii-(X-Xo}'1i+-- sin(OM +a.;)=O

16 (6a)

and from (5b) in (3):

(y-Yo)-

n,

cos(OM-a.;)

ç

.

=

, cos (OM+a.;)

we obtain the quadratic equation:

- 2 b2_

{cos (OM -a.;)hi -(Y- Yo}'1i+- cos (OM +a.i)=O

16 (6b)

The first test is an examination of the discriminant of equation (6a):

2 b2 _ _

Dx = (x-xo) - -sin(OM+a.;)sin(OM-a.;) >0

4 (7a)

withx=O or x=p,

and for equation (6b):

2 b2 _ _

Dy =(y- Yo) - - cos (OM +C(;) cos (OM-cti) >0

4 (7b)

withy=O or y=q.

The second test is appliedintwophases totheequations remaining from the first test. The first criterion isthat the'Iroots must bepositive, i.e.

(X-Xo)

+

y'Dx 'I= > 0 for x =D, or x=p 2sin (OM-cti) (x - xo) - y'Dx 'I= > 0 for x=O, orx=p 2 sin (OM-cti) 21

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(y - Yo)

+

v

D;

0"

°

'1= > lor y= ,or y=q

2 cos (OM-lXi)

(y - Yo) -

v

D

'1

=

y >

°

fory=O, or y=q

2 cos (OM-lXi)

The positive roots '1i obtained in this way are supplemented bymeans of (3)with the

appropriate Çi values and substituted in the conversion formula (5b), for the lines

x=constant, or (5a),for the linesy=constant.

The result of this latter phase is tested against the second criterion giving 0,2,4 or 6 values for

ç

i,'1iwhich results in points on the edge of the sheet. They must now be

sorted into pairs byorder of magnitude for '1i or Çi; the smallest value for each pair shows the beginning of the line to be drawn and the largest value the end of the Iine, Itis therefore necessary to plot 0,1,2 or 3 arcs for which beginning and end, but not the intermediate, points have been ascertained.

3.3. Selecting the position lines on a chart sheet

We are now in a position to determine the methad for selecting the position lines

which must appear on a particular chart sheet. Formula (I) is used to calculate lane number Lm e.g.for the centre of the chart sheet. As the initial value Ls we take the highest multiple of AL included in Lm.After processing L; we go on to lane Ls

+

AL;

if Lswas calculated with'1as the parameter, lane Ls

+

AL is calculated with

ç

as the

parameter etc.We now add AL to the lane number again and again until either the

maximum lane number of the pattern is exceeded or an examination of the points of intersection shows na more intersections.

We then turn to Ls-AL; we substract AL from the previous lane number until

either the minimum lane number for the pattern is passed or an examination of the

points of intersection shows na more intersections.

3.4. Calculating intermediate points on thepostdon lines to be plotted

The problem th at remains is to determine the values'1or

ç

for the points between A and B of the lane L, to be plotted. That in each case the distance between the chord and the corresponding are may not exceed the plotting accuracy t,for example t=0.1 mm, is taken as a criterion. This distance represents a distance ofs, tin the terrain if the chart scale is 1 : s.

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I I I I I

"1

I à" / Id: ~ <, <, <, <, <, <,

P

~

o

,

I I I I I / I I / / / / I /

.. SI

Figure 7. Points to beplotted in alane

Figure 7shows an extreme example ofthe first chord from Ato point 1. The requiremenr is to determine 111 from 1/A in such a way that the above toleranee is not exceeded. At point Rwhere the tangent to the are runs parallel with the chord from A to point 1,

the maximum distance between the chord and the are is reached. Chord and tangent

are extended to the asymptotes, which they reach at points Pkand Qk,and P, and Q" respectively. The area ofthe trapezoid PkQkQ,P, isdetermined by the formula:

b2 sin

rx

cos

rx

(

1/1 1/A )

A= -+--2

16 1/A 1/1 (8)

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Ifwe assu me this area to be equal to

A

we obtain a complicated equation for '11.

The minimum variabIe distance which could occur in the latter expression is b sinrY.,

i.e. for P'rQ'" being the tangent perpendicular to the base MS!, with the point of contact R' on the position line.

The area A in equation (8) is therefore certainly larger than stb sinrY.. If we introduce

this value in the left-hand term, '11will always be too small as an unknown value, i.e.

it will always fall within the specified tolerance. We now take the expression v'

= ~

'1/1

as the unknown. Equation (8) now becomes:

b2 sinClcosrY. ( 1 )

v'

+

-

2 =stb sinCl 16 v' or: 1 16st v+--2- =0 v' b cos rY. Hence ( 8 st)

J

(

8 st

)

2

v'

=

1

+

1

+

-

1 b cos Cl - b cos Cl

The product ofthe two roots is 1 and in our casev' > 1,so that:

( 8st)

J

(

8 st

)2

v' = 1

+

1

+

-

1

b cos rY. b cos Cl

(9)

where v' is the factor bywhich'1Amust be multipled to give'11.

This factor is a constant for lane L, so that for the following intermediate point 2:

For the final intermediate point nwe obtain: '1n=v'x'1n-l =(v'r x'1A

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Itis,however, more elegant to divide the wholeare from A to B with n intermediate points in such a waythat:

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We then obtain in the opposite direction :

(Wa)

so that the same number of intermediate points n isreached. The multiplicative con-stant v' must be decreased to vin such a way that equation (10)issatisfied.

For this purpose v', calculated in accordance with formula (9), is considered as an approximation of v.

If we now assume

it followsfor n

+

1 that:

10g17B -log17A

n+I=-----

-logv' (11)

In general(n

+

I) is an improper fraction; the entier of which (i.e.the greatest integer not exceeding n+1)must be introduced asn.

After determining n, vfollows from:

10g17B -log17A

logv= ---

-n+1 (12)

Finally, the point parameters 17from A to B or

ç

from B to A form a geometrical series with a ratio v > 1.

3.5. Summary of input and formulae to be programmed on tbe basis of tbe system analysis

3.5.1. Input constants

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Chart area: Xl' Yl, 1.4,p,q.

Plotting: s, L1L, t.

3.5.2. Preliminary caiculations

Transformation :

x=(X-X1) cos l.4-(Y- Yl) sin1.4)

y=(Y- Y1) cos1.4+(X-X1) sin 1.4

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for the points Mand SI.

Middle of the base:

YM

+

YSI

Yo

=

2

Chart centre:

p Xe = -2 q Ye = -2 Distances: InitialIane number: . (l(b+MC-SIC)+k) Ls=AL entier L1L (3)

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Constant c: b2 c= --16 Direction OM: -OM-= arctan

(X

_M

__

X

_

O) YM Yo

3.5.3. Limitationof lane calculationLs Constant ex: bi - (Ls-k) cos ex

=

(ex<2008) bi (4) a. Points of intersection with 1,2 :Y=0 Discriminant Dy: 2 -_

-Dy=(y- Yo) -4c cos (OM

+

ex)cos (OM -ex) (7b) General condition: Dy > 0 Roots: (y-Yo)

±

vi

n;

1]= 2 cos (OM -ex) Condition :1]>0 Corresponding

ç:

c

ç

=

-1] Corresponding x: (5a) Condition: O<X<p 27

(29)

b. Points of intersection with 3,4 :y=q

as under a but withy=q.

c. Points of intersection with 1,4 : x=O Discriminant

o..

(7a) General condition : Roots: (x-xo)

+

yDx 1'/=

2 sin(OM -a) Condition: 1'/> 0 Corresponding

ç

:

c

ç

=

-1'/ Corresponding y:

y= Yo+

ç

cos(OM +a)+1'/ cos(OM -a) (5b)

Condition:

O<y<q

d. Points of intersection with 2,3x =p

as under c but with x=p e. Results of limitation:

Sorting the permissible 1'/roots in order of magnitude, in pairs I'/A, 1'/8'

3.5.4. Calculating intermediatepoints for plotting

Formulae:

(5a)

(30)

Determination of v':

( 8st)

J

(

8st)2

V'

=

1

+

1

+

-1

b cos Cf. b cos (X (9)

Number of intermediate points: . (lOg'1B - log'1A) n= entier logv' (11) Determination of v: log '1B - log '1A log v = n+l (12)

To be substituted in formulae 5a and 5b:

c starting point: '1A

,

eA

=

'1A c 1st intermediate point: '11=V'1A , el V'1A c 2nd intermediate point: '12=V'11 , e2 V'11 nth intermediate point: '1n=V'1n- 1 , en = ---c V'1n-l end point:

3.5.5. Calculating lst sequenceof lanes Start:

L=Ls+L1L

Calculation asin 3.5.3.but sorting mentioned in 3.5.3") in accordance with order of magnitude ofpairs eA, eB, then as in 3.5.4.

General:

(31)

End:

Lk=Lk-l

+

LIL, if the results e) for LHl =Lk+LlL gives no value for 1],

ç

.

Sorting is carried out alternately in accordance with 3.5.3.e) and 3.5.5.e).

3.5.6. Calculating 2nd sequence of lanes Start:

L=Ls-LiL

Calculation: as in 3.5.5.

General:

End:

(32)

4. Elements of system analysis for plotting the specific circular and arc of circle patterns

The are of circle pattern must be considered as a variant of thecircular pattern. The principle of the circular pattern system isthat concentric circlesare plotted which are split up into arcs of circle at the points where the circles interseet the edges of the chart. In areofcircle patterns onlyarcsofcircle are plotted, the eentresof whichvary.

The edgesof the chart often make it necessaryto split these arcs upagain. Here too it is possible tostart byplotting one circular position line provided that it issituated within thearea ofthe chart.

4.1. Theparameter tjJ for pointson a circle (on arc of circle)

It isobvious to introduce a system of polar coordinates, r, tjJ,to calculate the coor -dinates of points on a circle with centre M, indicated by the coordinates XM, YM'

The circle isthen determined bythe choice of the radius r.; for which, inaccordance with 1.2,

L, - k

r,= (13)

, 2/

A point j on this circle is determined by the azimuth tjJj of the radius vector rio This azimuth tjJtherefore serves asa point parameter.

We then obtain:

Xi,j=xM+ri sin tjJj

I

(14)

Yi,j=YM+ri cos tjJj

4.2. Calculating the point parameters for the points of intersection of a circlewith the edgeof thechart

The equation for acircle with radius r in theX,y chart coordinate system is asfollows:

(33)

The coordinates for the points of intersection with the lines x=0, x=P, Y=0 and

y=q are to be calculated. lt is useful to observe a specific sequence for this purpose. This sequence, together with the conditions with which the results must comply, is given in the following formulae.

Sl:Xl=0,Yl=YM+j,2 -XM2 S2: x2 =XM-j,2_(q-YM)2, Yz=0 S3: X3=XM+j,2_(q-YM)2,Y3=0 S4: X4=P,Y4=YM+j,2_(p-XM)2 Ss:Xs=P,YS=YM-j,2_(p-XM)2 S6: X6=XM+j,2_yM2,Y6=0 S7: X7=XM-j,2_yM---Z:Y7=0 Ss: xs=0,YS=YM+j,2_XM2

[Q

54 \ I 5 57---5 5 _--- <; 6 /' "

//

D

"

/

"

/ "-I \ / \ I \ I \ I \ I • I \ M

!

\ I \ I \ / \ /

"

/ / <, /' <,..._---,,/ l st test

,

2

> (q-YM)2

,

2

> (q-YM)2

,2

> (P-XM)2 ,2 > (p-XM)2

,2

> YM2

,2

> YM2

,2

> XM2 2nd test q ;?Yl ?0 P? X2? 0 q;?Y4?0

s

>

Ys ?0 q;?Ys ?0

(34)

Sinee the eirc1eisa closed curve, 8,6,4, 2or 0 roots, eorresponding toan equivalent number of real points of intersection, satisfy the requirements. Figure 8shows the various possibilities.

As regards theeireular pattern, the aresto be plotted are indicated by the roots which satisfy both tests, arranged in pairs aecording tothe numerical order ofthe point S eoneerned, and beginning with the lowest odd number.

If there are no real points of intersection the circ1econcerned may be situated inside or outside the chart. In the former caseitwiJl be plotted but in the lattercase(outside the ehart) this isunnecessary.

This means that if there are no valid roots, a further test must be earried out to determine whether the condition XM

+

r ~ p is fulfilled to enable plotting of this position line.

Thevalid points of intersection are expressed in point parameters

!/J

aceording to the formula: ( XS !/JMS = arctan Ys (15)

4.3. Calculating thepoint parametersfor the starting and end pointsof theposition lines to beplotted inthecaseof an are of circlepattern

In the arc of circle pattern XMandYMare not given; these eoordinates are calculated from: LR (6) r=-- -2 sin ')' (17)

The result for

!/J

obtained byformula (15) in aecordance with 4.2,serve as a basis for selecting the part ofthe eirc1eto be plotted, running from !/JMRto !/JML· If !/JMR < !/JML>

the point parameters will increase from

!/J

MR to

!/J

ML· If

!/J

MR >

!/J

ML>inereasing point

parameters are obtained bysplitting up the are from !/JMR to 2nand from 0 to !/JML.

The criterion for plotting is that the arcs or sections of arcs, found in aceordance with 4.2, will only be aecepted if they are located within the range indicated. This means that the point L may serve as the end point and the point R as the starting point of an are whieh is to be plotted; this is iIIustrated in figure 9.

(35)

R

L

Figure 9. Ares of eirclesto be plotted on the ehart

_---_

In this wayeaeh are of eircle to be plotted is therefore fixed by starting point Ahaving parameter !/JMAandby the end pointBhaving parameter !/JMB'

If plotting is effeeted in a cloekwise direction, a positive LI!/Jis introdueed for the intermediate points. If plotting is earried out in an anti-cloekwise direction, LI!/Jis negative and we start at point Band finish at point A.Analternating plotting direction ean be programmed.

4.4. Caleulation of intermediatepoints Kon thc aresof eirclcto beplotted

Variations in the azimuth !/JMK followan interval LI!/J.This produces a distanee on the ehart for whieh atoleranee t ean be introdueed.

In figure 10 this distanee is indieated by the geographieal distanee S.t. A eonservative value for LI!/J'applieable to LI!/Jis derived as follows.

PK PQ 2r S.t PK PK PK=J2r. s. t while PK :::;r. LI!/J' 2 LI!/J'=

J

8 ;

t

From we obtain so that:

In order to divide the are of eircle AB into equal parts for plotting purposes, the number of intermediate points n ean be taken as:

(36)

p K st

%

~~-""1-

K

=-

-

---

p

---L-C)

"

,

\

I

//

',

\ 1 / 1 , \ I / 1 -. / / \ \ I / 1 \ \ ~I

'Y / /

\

\~/ I \

~!"""1

1 \ \ I / 1 \ \\1/ / I

W'

1 I Mil I I 1 / I 1 I 1 1 I

l

iIJ

'If

/

I 4

1

I

1--./

/

1// , . / I I /

11 ,,/

11 .// I/ ,,"" JI »>

Q"'--Figure 10. Approximation of plotting arcs

(

I/IMB

-

I/I

M

A

)

n = entier

J

8;t

(18)

so that the interval is determined by:

The coordinates are calculated in accordance with formulae 14.

4.5. Selectiog tbe position lines on a cbart sbeet

We begin by calculating the line parameter for the cent re of the chart, obtaining a

(37)

value L; for the circular pattern and a valueYcfor the are of circlepattern. Plotting

begins either with the initial valueL., which is the largest multiple of,dL included in

Lc, or aty., which is thelargest multiple of Ay included inYc.

Further processing is effected as described in 3.3.

4.6. Summaryof tbe input and formulae to beprogrammed on the basis of system analysisin the caseof a circular pattern

4.6.1. Inputconstants Pattern: XM' YM'I,k Chart area: Xl' Yb 1.4,p, q Plotting: s,AL, t 4.6.2. Preliminarycalculations Conversion : Chart centres: p Xe = -2 q Ye = -2

Initial line parameter:

Initial radius:

rs =

L, - k

(38)

4.6.3. Limitation of fine ca/cu/ation Ls

Test: frompoints ofintersection SI to Ss in accordance with 4.2. with r=rs and with no points ofintersection forXM

+

rs~ p

Sorting of the permissible intersection points S: in pairs, odd numbered points S=A with immediately following even numbered point S=B.

Result of limitation: express coordinates for sorted points A and B in their point parameter

t/I

bysubstitution in the formula

(

X - XM)

t/I

= arctan

Y - YM (15)

and in the case of a complete circle introduce :

4.6.4. Calculating intermediatepoints for plotting Formulae:

x=xM+rs sin

t/I )

Y= YM+rscos

t/I

(] 4)

N umber of intermediate points:

(

t/lMB - t/lM

A

)

n = entier

J

8st r, (18) Determination of L1

t/I :

Substitute in formulae (14): starting point:

lst intermediate point: 2nd intermediate point: nth intermediate point: end point:

t/l

MA

t/l

MA

+L1t/1

v

«

,

+2L1t/I

t/l

MA

+nL1t/1

t/l

M

B

37

(39)

4.6.5. Calculating the 1st sequence of position lines Start: L=Ls+AL

Calculation as for Ls, but sorting in sequence BA General: Lj+l =Lj+AL

End: Lk=Lk-l +AL if the results in accordance with 4.6.3. for: LHl =Lk+AL yield

no values for

!/J

.

Sorting is effected alternately in sequence AB and BA.

4.6.6. Calculating the 2nd sequence of position lines

Start: L=Ls-AL

Calculation: as in4.6.5.

General: Lj+l =Lj-AL

End: as in 4.6.5.

4.7. Summary of input and formulae to be programmed on the basis of system analysisin thecaseof an are of circlepattern

4.7.1. Input constants Pattern: XL> YL> XR, YR Chart area: Xl' Yl,1.4,p, q Plotting: s, Ay, t 4.7.2. Preliminary calculations Conversion :

for points Land R Distance LR:

(40)

Azimuth LR: ( XR I/ILR = arctan YR Chart eentres: p 2 Ye q 2 Initial line parameter:

Ys Lly entier

Initial radius: LR

2sinYs

Coordinates of initial centre point:

4.7.3. Limitation of fine calculation

Output data: resuIts of limitation in accordance with 4.6.3. Calculation of I/IMR and I/IML:

(41)

Determining the range of the position line:

lst

l/tMR

<

l/tML

range of point parameters

l/tMR

--+

l/tML

2nd

l/tMR

>

l/tML

range of point parameters

l/tMR

--+ 2n--+ 0--+

l/tML

Selecting of the Spoints within this range:

Compare

l/t

MS with this range and check whether a discarded A point can be replaced

by point R or a discarded B point by point L.

Results of limitation: indicate the starting point of an are of circle to be plotted with

l/t

MAand the end point with

l/t

MB·

4.7.4. Calculating intermediate points for plotting

See 4.6.4.

4.7.5. Calculating Ist and 2nd sequence of position lines

(42)

5.

Influence

on the hyperbolic chart pattern,

if

the earth is considered as a sphere or as an ellipsoid

5.1. General considerations

In the case of the plane, hyperbolic pattern the distances ai ,b and Ciare calculated

from formula (1) according to the location of the points in the plane of the chart

n

projection. Ifn is the number oflanes in the pattern, then/=-. 2b We now write lane number L,according to this pattern as:

_ n(iii

+ 5 -

ei)

L·= -s-k:

• 25 (19)

The mathematical model for the pattern is refinedby locating it on an earth considered as a sphere or ellipsoid. We then obtain a spherical or spheroidal pattern, for which

,.-..,

the formula for the lane number L, is: ,-., ,.-..,

'"'

,-., nta,

+

b - C.)

Li = • ~ •

-s

k

2b

(20)

inwhich the distancesai' band c, are taken over the surface ofthe sphere or ellipsoid.

The general constants which define this pattern are the geographical coordinates of

,-., n

the Master and Slave stations: ÀM' qJM' Às!> qJSI'the multiplicative constant / = ~ 2b and the additive constant k.

,.-..,

Figure 11 shows the difference between Land L for points in the Ymuiden Ri-Fix

pattern and figure12forpoints in theFrisianIslandsDeccapattern, Lbeing calculated ,-.,

inaccordance with formula (19) and Lin accordance with formula (20) for distances on the ellipsoid.

(43)

Cl Cl Cl Cl Cl Cl Cl _Cl C7l ₂₀₀ 'I 100 I +60000 o o (') ro 8 o en N +40000 +20000 Cl Cl Cl Cl C7l I Cl Cl Cl Cl t-I

(44)

,-._ Point LJL=L-L

---{LN

=

175.568 LN

=

175.572 -0.004 1. ,-._ Lz

=

333.235 Lz

=

333.235 0.000 ,.-.. {LN

=

248.112 LN

=

248.114 -0.002 2. Lz

=

283.855 Lz

=

283.855 0.000 ,-._ {~N

=

312.825 LN

=

312.826 -0.001 3. ,-._ Lz

=

226.803 Lz

=

226.804 -0.001

----{~N

=

125.631 LN

=

125.633 -0.002 4. Lz

=

341.910 Lz

=

341.910 -0.000 ,.-.. {IN

=

235.515 LN

=

235.516 -0.001 5. ,-._

\.

L

z

=

243.453 Lz

=

243.454 -0.001 ,-._ {~N

=

338.403 LN

=

338.403 0.000 6. ,-._ Lz

=

168.951 Lz

=

168.952 -0.001 ,-... {IN

=

34.989 LN = 34.990 -0.001 7, _'""'

I

z =

222.071 Lz

=

222.072 -0.001 ,-._ {~N

=

158,908 LN

=

158.909 -0.001 8. ,-._ Lz

=

88.907 Lz = 88.908 -0.001

---{~N

=

346.319 LN

=

346.320 -0.001 9.

---Lz

=

79.209 Lz = 79.210 -0.001 43

(45)

o 55 o 54 o 53

Figure 12. Decca patterns of the Frisian Islands Chain, U.T.M. projection

o 55 o 54 o 53

(46)

,-., ,--. Point L1L=L-L ,-., {LR

=

77.135 LR = 77.079 -0.056 À=3° q1

=

530 ,-., LG

=

277.987 LG = 277.984 -0.003 ,.-.., {:R = 145.479 LR =145.474 -0.005 À=3° q1=540 ,.-.., LG =227.008 LG =226.980 -0.028 ,.-.., {:R

=

226.088 LR =226.154 +0.066 À=3° q1=55° ,.-.., LG =186.080 LG =186.024 -0.056 ,.-.., {LR = 63.336 LR = 63.281 -0.055 À=4° q1=530 ,.-.., LG

=

270.743 LG

=

270.738 -0.005 {LR = 144.730 LR = 144.727 -0.003 },=4° q1=540 ,-., ,LG

=

199.982 La =199.946 -0.036 ,.-.., {:R

=

242.360 LR =242.434 +0.074 À=4° q1=550 ,.-.., LG

=

159.663 LG

=

159.596 -0.067 ,.-.., {LR

=

45.295 LR = 45.246 -0.049 À=5° q1= 530 -e-, LG =229.232 LG

=

229.215 -0.017 ,.-.., {:R = 145.905 LR

=

145.908 +0.003 À=5° q1=540 LG

=

155.075 LG= 155.029 -0.046 ,.-.., {:R

=

267.283 LR = 267.365 +0.082 À=5° q1=55° ,-.. LG

=

126.630 LG =126.553 -0.077 ,-.. {LR = 21.404 LR = 21.372 -0.032 À=6° q1=530 ,.-.., LG

=

131.860 LG= 131.832 -0.028 ,.-.., {:R

=

153.340 LR

=

153.353 +0.013 À=6° q1= 540 ,.-.., LG = 94.513 LG= 94.466 -0.047 ,--. {:R

=

306.061 LR

=

306.149 +0.088 À=6° q1=55° ,.-.., LG

=

90.367 LG= 90.288 -0.079 45

(47)

In the area of 40x40 km for which the Y muiden Chain was designed, these differences,

arranged according to the point location, amount to the following values in thousandths

of a lane:

North pattern South pattern

-1 0 -1 -1 -1 -1

-2 -1 -1 0 -1 -1

-4 -2 -1 0 0 -1

These differences are of no real practical significance, because the scale divisions on

the Decometers used rep re sent 5 thousandths of a lane.

On the other hand in the 201 x 228 km area of the Continental Shelf in accordance

with figure 12,we find the following difference:

Red pattern Green pattern

-66 -74 -82 -88 +56 +67 +77 +79

+ 5 + 3 -13 -13 +28 +36 +46 +47

+56 +55 +49 +32 + 3 + 5 +17 +28

In the latter case the conclusion can be drawn that the spheroidal pattern gives a

better mathematical model in practice than the plane pattern in the U.T. M. projection.

In the introduetion we saw that the chart pattern is superimposed on a grid or graticule

in order to make graphic position fixing possible. The chart patterns for the chain can

now be used either in the form of plane, hyperbolic patterns and the graticule modified,

or alternatively the plane, hyperbolic pattern can be rnodified and the grid or graticule

used in the chart projection chosen is maintained

The first method has the advantage that the programmes already discussed remain

unchanged. The graticule can be easily adjusted as we shall see below. In fact we are

now introducing an entirely new chart projection, the principle of which is that

spherical or spheroidal hyperbolae are represented as plane hyperbolae.

The second method has the advantage that the chart projection chosen allows

un-distorted additions to be made to the fixing charts, using data from other charts of the

same projection. The modification of plane hyperbolae to obtain position lines in

accordance with the projection chosen does, however, require additional

program-ming, which will be discussed below.

For both methods it is necessary to determine the length of the geodetic on the

ellip-soid between two points of which the latitude and longitude Al, IfJl and A2 and 1fJ2are

given (the second principal problem of geodesy).J.c.P. DE KRUIFhas found an elegant

solution to this problem using an iterative programme. The basic principles of his method and the corresponding programme in Algol '60 were published in the

(48)

-e-,

it is simple to calculate by automatic computer lane number L,using formula (20) for apoint i, given in geographic coordinates Àj, (/)j in relation to a pattern defined

,..._ in accordance with the generalpattern constants )'M, (/)M' ÀSI' (/)SI' land k.

Before elaborating the methods indicated above (to be discussed in sections 5.2 and 5.3), it must bepointed out that the plane pattern can be superimposed on an un-modified grid or graticule in the chart projection used, provided that a correction

,..._

----

,-...

Al:

=

L - L, valid for thatparticular place, is applied to each reading L.Because of the principles on which they are based,these corrections will show a regularpattern in the area under consideration. Itwill be necessary to analyze a sufficient number of points inthe area covered (as in figure 12) to compile a generalchart on which the density of these corrections is sufficient so that interpolation forpractical purposes is justified.

5.2. Modification of the graticule

In the case of theplane hyperbolic pattern, the relativeposition of the transmitters to each other can be determined in accordance with a traditional chart projection. To plot the graticule, the intersections of the meridians and parallels which are to be shown must first be calculated in spheroidal-hyperbolic coordinates in accordance

----

,-..

with formula (20)in lanes L, and L2' after which the rectangular coordinates of the

----point of intersection of lanes LI=E; and L2=L2 are calculated and plotted in the system ofthe planepattern.

This problem of determining Xs Ys, of which lane number L, and L2 for point S

are given, has already been described on pages 44, 45 and 46 of Rijkswaterstaat Communications No.2 (seebibl. 1), whereaniterative solutionis arrived at.VERSTEL -LEpublished a direct solution in the Supplement to the International Hydrographic Review of December 1963(seebib!. 3). Ris method has been programmed by the SurveyDepartment of the'Rijkswaterstaat'. In actual factitisBALLARIN'S method for aplane surface.

Modification of the graticule by the method indicated above was usedfor the prov-isional pattern charts of the Continental Shelf on a scale of 1:50000. Since 1970, however, the method described in 5.3 has been used. For practical purposes it is perfectly feasible to use corresponding charts plotted by these two systems in con-junction withone another.

However, the method described in this section only results in a one-to-one corres-pondence if the relativepositions of the transmitters to eachother are determined in the followingmanner insteadofby traditional chartprojection.

----

,-... ... In M Sl, SI2(figure 13)the distances over the ellipsoid are given bybi b2 and s. For

thepoint SI2we obtain:

(49)

Figure 13. The transmitter positions of Master and Slaves

.---

.---.--- bj

+

b2 - S

L, = nj

+

kj

bi

For a one-to-one correspondence projection InSl2 the following condition must

therefore be met:

And for a one-to-one correspondence projection in Sl, :

This means that for the relative positions in the plane, the following condition must be met:

.---

.---bi :5j = b2 :52 = s :oS

A further consequence of this is that discrepancies will occur when working out a system with three slaves. As long as these discrepancies remain less than five-thousandths of a lane, they can be disregarded for practical purposes.

(50)

5.3. Modification of the plane hyperbolic pattern

In the International Hydrographic Review, vol. XXVII-2-1950, Prof. BALLARIN

published an artiele entitled 'Geometrie properties of position lines in hyperbolic navigation and their lay-out on the reference ellipsoid'. (see bibl. 4). In it the author

,,-., ,,-.,

described a solution to the problem of converting intersections of L1, L2 into the

corresponding geographical position <p, J.. If the results tp ; À were to be converted intoX, Y bymeans of the formulae for the chart projection used, this would in prin-ciple represent a solution to the problem of plotting points in the spheroidal chart pattern.

In the publication referred to earlier,VERSTELLE advanced certain practicalobjections

toBALLARIN'S method. For the purpose of preparing charts for the Continental Shelf a new method has been worked outand programmed in accordance with ideas deve-loped by the Mathematical Department of I.T.C. The theory of this method is

described byKUBIK and others in Rijkswaterstaat Communications No. 12(see bibl. 7). In the following a short outline of this method and its practical consequences are given.

If we base the method described in 5.2. and 5.3. on the same system of X and Y coordinates, then the transmitters acq uire the sameX, Ycoordinates in both systems

,-., ,,-.,

of calculation. For a given point S with readings LI' L2 and coordinates X, Y a different pair of coordinates, X', Y',applicable to point S'in the system of method 5.3

,,-., ,,-.,

is calculated in the plane pattern, in accordance withL1= I1andL2= I2, using the

in-tersection oflanes II and I2• Figure 14is a diagram showing both points.

This means that corrections L1X' and L1Y'must be made to the result as calculated in the programme under discussion, in order to correct the location of S'based on

,

6 X

/ CV

-...J

Figure 14. Discrepancy between plane chart pattern and spherical, or spheroidal, chart pattern 49

(51)

plane hyperbolae to give the corresponding position in asystem of spheroidal hy per-bolae.

For the area covered by the Frisian Islands Chain, shown in figure 12, the position of the given 12 points S in X and Y coordinates wasobtained from U.T.M. tables,

while the position of the corresponding points S' was determined according to the method described.

These points results in the following L1X' and Y' corrections in metres:

X': -104 -118 -132 -143 - 95 - 70 55 48 -245 -146 57 33 L1y': 63 47 23

₊

11

+

4

₊

2 2 4

+

24

₊

36

₊

50

₊

32

The necessary corrections will show a regular pattern over the area concerned because of the fundamental law on which they are based. This law can be approximated by means of adaptation formulae forL1X' and L1Y'.

We have found that formulae ofthe following type:

L1=a

+

bx

+

cY

+

dx2

+

exy

+

f

y2

+

gx3

+

hx2 Y

+

ix

l

+

j y3

+

k x3Y

+

1x2 y2

+

mx y3

+

n x3y2

+

0 x2 y3

+

px3

l

are very suitable for this purpose. The 16 coefficientsatop must then be accurately determined from redundant data of L1,x and y of a large number of reference points distributed regularly over the whole area. The limits of an area for which a given combination of coefficientsatopis valid, are shown by the remaining discrepancies in the reference points.

These discrepancies must be acceptable for plotting on a scale of 1:s and must not,

for example exceed a toleranee t=sx0.1 mmoThe size of the area of application of the methad is consequently also dependent on the scale of the chart, The platting of a spheroidal pattern must therefore be preceded by analysis of a division of the area into blocks in order to establish the coefficients of the adaptation formulae necessary for determining the corrections L1X' and L1Y' for each sub-division of the area.These coefficients are constants for the formulae that are used for an extension of the pro-gramme for plane hyperbolae. Certain additional modifications must also be made to the programme to adapt the recalculated part of the pattern.

The mathematical department ofI.T.c. developed these adaptation formulae in the form of third order polynomials to derive a spherical or spheroidal pattern from a plane hyperbolic pattern. This is a special application of the general problem to make a complex relationship between the dependent variables cand dand the independent

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variables aandbmore manageable in terms of computation technologyin accordance with e=f(a,b) and d=g(a,b). Approximation c' and d' for eand daccording to less complex relations are taken as the starting point: e'=f'(a,b) and d'=g'(a,b) so that e=e' +L1e'and d=d' +L1d' with L1e'and L1d'as the newunknowns; the relationships between L1e'and L1d'and c' and d' respectivelycao in practice be assumed to be in accordance with third order polynomials. A precondition however isthat exactvalues for L1e' and L1d'cao be calculated for n chosen values for a and b. The adaptation formulae obtained can then be used for incidental valuesofaandb.An artiele will be published on this subject,describing experience with applications in a wide range of fields.The first application is the one described in this article.

Systems for automatic computation and plotting of position fixing patterns

SYSTEMS

FOR

AUTOMATIC

COMPUTATIONS

AND

PLOTTING

OF

POSITION

FIXING

PATTERNS

t

Contents

1.

Definition of the

specific position fixing pattern and chart sheet

+

+

+

®

+y

x-

®

®

plane h

y

perbolic chart

pattern

ç

ç

ç

ç

ç,

=

I

ç,

4

ç

ç

°

°

ç

n,

ç

.

+

+

v

D;

°

v

=

°

ç

+

+

ç

ç

"1

P

~

o

,

.. SI

rx

rx

(

= ~

+

-

-

J

(

)

2

=

+

+

+

-