NAVIGATIONAL CHART OF THE NORTH ATLANTIC
USING AN OBLIQUE CONFORMAL MAP PROJECTION
fay
M r. A ndre
G
o u g e n h e im
In g e n ie u r H y dro g raph e General de l re classe (Retd.)
F o rm e r D irector of the F re nc h N av al H y d ro g ra p h ic Office
The F re n c h N aval H y d ro g ra p h ic Office has recently p u b lis h e d a
n a v ig a tio n a l ch art (No. 6504) o f the n o rth e rn p o rtio n of the A tla n tic Ocean
u s in g a n o b liq u e c o n fo rm a l m a p p ro je ctio n . T his chart, o n a scale of about
1/6 000 000, is in te n de d specifically for ships ru n n in g betw een N o rth
A m e rica (fro m New Y o rk to G reenland) a n d the E u ra fric a n c o n tin e n ta l
fro n t (fro m Bergen to the C an ary Islan d s), so th a t they m a y m ore easily
m a k e use of great circle tracks (figure 1).
I n the present article it is
by the p lo ttin g of great circle
p rin c ip a l characteristics of the
the c o m p u ta tio n process used
proposed, after recalling the p ro b le m posed
tracks on a n av ig a tio n a l chart, to give the
H y d ro g ra p h ic O ffice’s new chart, as w ell as
in its c o m p ila tio n .
I. — THE C A R T O G R A P H IC REPRESEN T A T IO N O F O RT H O D RO M ES
F o r a lo n g w hile the M ercator pro je ctio n w as a necessity in m a ritim e
n a v ig a tio n because constant h e a d in g tracks are here represented by straight
lines. M oreover this pro je ctio n has two im p o r ta n t advantages : on one h a n d
the angles are retained in a ll areas of the chart, a n d on the other the
re cta n g u lar g rid representing the geographic m e rid ia n s a n d p a ra lle ls m akes
the scaling o f a po sitio n ’s coordinates very easy, a n d likew ise the p lo ttin g
of a po s itio n by m eans of its coordinates.
I n h ig h a n d m id d le latitud e s this system, due to the re latively ra p id
v a ria tio n in local scale, w h ic h is n early p ro p o rtio n a l to the secant of the
latitud e , has the disadvantage th a t great circles on the sphere are represented
o n the c h a rt by arcs o f rather great curvature.
N avigators have m a n y sim ple m ethods at their disposal for c o m p u tin g
or p lo ttin g o rth o d ro m ic arcs corresponding to the shortest pa th s or to
ra d io bearings. I n fact, g n o m o n ic pro je ctio n charts of the oceans have been
d ra w n u p fo r this purpose. I n the g n o m o n ic system great circles of the
spherical e arth are sho w n o n the c h a rt as straight lines w h ic h are
s ub seq u en tly plotted p o in t by p o in t onto the M ercator projection n a v ig a ­
tio n a l chart. I n the case of o rth o d ro m ic n a v ig a tio n these lines are easily
p lo tte d by jo in in g the po ints of d ep arture a n d of arrival, b u t this is n o t the
case fo r radio bearings fro m a tra n s m ittin g station in a given a z im u th ,
fo r here the m a p p ro je ctio n used noticeably distorts angles, a n d it is
necessary to show o n the chart a ro u n d each tra n s m ittin g station a rose
g ra d u a te d in a z im u th , thu s su p p ly in g a second p o in t for rectilinear p lo ttin g
o f bearings. The non- conform ity of g n o m o nic pro je ctio n also has the d is­
adv antag e th a t it p ro h ib its p ra ctically a n y a n g u la r m easurem ent o n the
c h a rt, a n d in p a rtic u la r th a t of an o rthod rom e ’s a z im u th at any p o in t on
its p a th . S im ila r ly the significance o f line a r d isto rtio n prevents a n y
d istance m e asu re m e nt on the chart. T hus this c arto grap hic solutio n is not
as attractive as m ig h t be im a g in e d at first sight. W h ile it is strictly
an n li(*flh ]p to t h p n r n h l p m n f r p p filin p sjr r p n r p s p n t a t in n o f
ix — - ** -- r ------ -- -------- — i------■
-- ~r>irolo ***q■f p c
it how ever lends itse lf very ill to the even a ppro x im ate solution of any
o ther problem s co ncerning angles a n d distances.
T hus, for a lo ng w hile, it w as th o u g h t th a t a com prom ise solutio n
s h o u ld be sought, a n d th a t a plane representation system should be used
w h ic h , th o u g h s u p p ly in g o n ly an approx im ate rectilinear representation of
great circles, w o u ld keep the advantages of being c o n fo rm a l a n d p e rm it
distance m e asu re m e nt w ith a good degree of accuracy. These properties
exist in the M ercator pro je ctio n in the v ic in ity of the equator. Indeed this
system is a true c o n fo rm a l one and the scale v a ria tio n is o n ly of the second
order w ith respect to latitud e , so long as this last re m a in s low.
It suffices th e n to choose a n “ a u x ilia ry ” equ ator across the region
bein g considered, a n d to take this lin e as a basis for a M ercator proje ction.
In th is new system, an oblique c y lin d ric a l c o n fo rm a l projection, the
geographic m e rid ia n s a n d parallels obviously no longer fo rm a re ctangu lar
re ctilin e a r g rid as in the case of the M ercator proje ctio n, b u t nevertheless
a ll the a n g u la r a n d m etric properties of the M ercator p roje ction are
re tain ed . In practice, for the scaling o f geographic coordinates of a p o sitio n
or fo r the p lo ttin g o f a position by its coordinates, it is necessary to g r a d u ­
ate a ll the m e rid ian s a n d parallels of the g rid as in the case of the g n o m o n ic
pro je ctio n , w hereas in the M ercator pro je ctio n it is sufficient th a t the
borders of the c h a rt carry a scale. A n oblique c y lin d ric a l c o n fo rm a l
p ro je c tio n for a spherical earth has been used, by M r. Louis K a h n in
p a rtic u la r, to d ra w u p small-scale charts o f the p rin c ip a l routes for longrange aerial n av ig a tio n .
n . — THE FREN C H H Y D R O G R A P H IC O F F IC E O B LIQ U E
C O N FO R M A L CH ART
I n spite o f the advantages of this system of representation it w o u ld
appe ar not to have been yet advocated fo r m a ritim e n avig atio n . Therefore,
as a n experim ent, we decided to m ake a first a p p lic a tio n o f this system to
a region sufficien tly far fro m the equator in order th a t its value m ig h t
become evident. F ro m this p o in t o f view the n o rth e rn p a rt of the A tla n tic
O cean between 40° a n d 50° N orth la titu d e w here there are busy s h ip p in g
routes w as p a r tic u la rly suitable.
Choice of in itia l data
M u c h e x p e rim e nta tio n was necessary before fix in g the lim its of the
chart, because in the firs t place we w ished to keep to the 96 X 65 cm
fo rm a t used fo r F re nc h n a u tic a l charts, a n d then w ith in this fram e w o rk
to give the c h a rt the largest possible scale, ta k in g in New Y o rk a n d the
B e rm u d a s to the west over to the in n e r p a rt of the B ay of Biscay to the
east, the C a n ary Islan d s to the south a n d fin a lly the s ou the rn parts of
Ice lan d a n d G reenland to the n o rth . In fact a m a x im u m scale corresponds
to as s m all as possible an extent on each side o f the a u x ilia ry equator,
th a t is to say, to “ a u x ilia ry ” latitudes as low as possible for the a u x ilia ry
parallels fo rm in g the borders of the chart. As the scale on these extreme
parallels is e q u a l to the equato rial scale m u ltip lie d by the secant o f th e ir
a u x ilia ry la titu d e the scale v a ria tio n in the area represented is thus reduced
to the m in im u m t1'.
F in a lly , the fo llo w in g basic d ata were adopted as the characteristics
of the system :
C entral p o in t : L a titu d e 45°0 2 'N , L o n g itu d e 3 7 ° 4 3 'W
M a jo r axis (a u x ilia ry equator) : A z im u th at the cen tral p o in t 78°52:5
: T otal range 53°36'
Conforms! representation of the ellipsoid on the sphere
As the use o f an a u x ilia ry equator is o n ly convenient on a sphere it
w as first of a ll necessary to m ake a c o n fo rm a l representation of the
terrestrial ellipsoid on a n a u x ilia ry sphere. B y reason of the sm allness of
the c h a rt’s scale, a n d also o f the fact th a t the direction of the chosen m a jo r
axis is separated by h a rd ly m ore th a n 10° fro m th a t of the central p o in t’s
parallel, we adopted the sim plest system o f p roje ction w h ic h consists of
representing the ellipsoid on a sphere transversally osculatory at the central
p o in t, th a t is to say tan g e n t to the ellipsoid along the p a ra lle l of this p o in t,
a n d h a v in g as ra d iu s the great n o rm a l to the ellipsoid at this latitud e . In
this system the spherical longitudes are equal to the ellipso idal longitu de s;
the la titu d e o f the central p o in t is also retained, a n d for a ll other points,
the difference in m e rid io n a l parts between a given p o in t a n d the central
p o in t has the same value on the sphere as on the ellipsoid. A m o n g
c o n fo rm a l representations o f an ellipsoid on a sphere it is n ot the best fro m
(1) In fact when designing the chart it was possible to use a slightly larger format
— 97.5 x 66.3 cm. Nevertheless the in itial data used for the 96 x 65 cm format have been
retained, in particular the scale deduced from this format. Thus the chart covers an
expanse a little larger than that in itially planned. All the computations given in the
present article refer to the original format 96 x 65 cm.
However, in order to meet a demand expressed after the chart had already been
compiled, a window was inserted to the left side of the frame so as to extend the chart
as far as the port of Norfolk on the east coast of the United States of America.
the p o in t o f view of lin e a r distortions, b u t these are m u c h s m alle r in this
system th a n those encountered later on in the co nform al representation of
the sphere on a plane, a n d there are no practical objections to their
acceptance. In the m ost u n fa v o u ra b le case of chart coverage, the len gth
d isto rtio n is in fact still below 1/2 000; a n d this is the d isto rtio n of the
M ercator pro je ctio n 1°46' fro m the equator.
C hart scale
In fact the basic d ata adopted for the ch art do n ot define a geodetic
line o n the ellipsoid, w h ic h w ou ld be useless, since this lin e is not
represented by a great circle on the transversally osculatory sphere. These
^ATTToiror
n ro
r p n r p « p n t o t iv p
o f ' th*>
m
iv ilia rv ennator
on the
---••
*
u a a ic
u a ttt)
i i v **v» v* f mi ■
*
- * '“i'*
*'*■-- >—
sphere. T his conside ration allow s us a s im p lific a tio n w h ic h o n ly involves
a s lig h t m o d ific a tio n of the lim its of the ch art coverage a n d it in no w ay
interferes w ith the a n g u la r a n d m etric properties of the p lan e representa­
tion.
Given th a t a n arc of 53° 36' on the a u x ilia ry equator of the sphere
whose ra d iu s is N 0 is represented on the chart by a length of 96 cm , it is
easy to deduce the scale of the plane representation of the equato r, i.e.
1/6 226 000, a n d this in practice can be ta k e n as the scale on the m a jo r axis
because, as we have seen, it is perm issible to o m it the change in scale
in tro d u c e d by the c o n fo rm a l representation of the ellipsoid o n the sphere.
In these c o nd itio ns h a lf the length of the shorter border o f the chart,
i.e. 32.5 cm , represents the a u x ilia ry m e rid io n a l parts of the c h a rt’s borders;
the ir a u x ilia ry latitud e , som ew hat less th a n 18°, is easily deduced there­
fro m . In a zone extending a little less th a n 18° on each side of the equator
the c h a r t has, the n, the properties of the M ercator projection. The ra tio of
the scale on the a u x ilia ry parallels b o u n d in g the chart to the e qu atorial
scale is 1.05057, so th a t the v a ria tio n in scale w ith in the c h a rt’s borders
does n o t exceed 5 % , or o n ly p lu s or m in u s 2.5 % in re la tio n to the two
a u x ilia r y parallels of 12“40, latitu d e w here the scale is exactly in te rm e d iate
betw een the m in im u m scale on the c h a rt’s m a jo r axis a n d the m a x im u m
scale o n the lo n g itu d in a l borders.
C om putation of the geographical grid
To pass fro m geographic coordinates on the sphere to a u x ilia ry
coordinates serving as a basis fo r the plane representation, i.e. a u x ilia ry
latitud e s a n d longitudes counted respectively fro m the a u x ilia ry equator
a n d fr o m the a u x ilia ry m e rid ia n of the central p o in t, one m u s t resolve the
spherical trian g le fo rm e d by the sphere’s geographic pole, the a u x ilia ry
e q u a to r’s pole a n d the p o in t u n d e r consideration. The ch art is afterw ards
c o m p ile d w ith the help o f these a u x ilia ry coordinates as in the case of a
n o r m a l M ercator p ro je ctio n .
However, since it is the a c tu a l geographical grid, and n ot the a u x ilia ry
grid, th a t is to be show n o n the ch art it is necessary both for the w o rk in g
o ut a n d for the use o f this g rid to calculate the p lan e re ctang ular
coordinates o f the intersections of the geographic m e rid ia n s a n d parallels
after h a v in g d e te rm in e d their a u x ilia ry coordinates.
F o r the conversion o f coordinates o n the sphere, the longitudes on the
ellipsoid are n a tu r a lly used as these are retained in the representation of
th is surface on the sphere. In the case of the latitud e s, however, the
representation of the ellipsoid on the sphere is take n in to account by using
the spherical la titu d e s deduced fro m the e llipsoidal latitud e s u n d e r
consideration.
The c o m p u ta tio n was m ade for geographic m e rid ia n s a n d p a ra llels at
2° 30' intervals, the distance between the corresponding intersections on
the ch art being 45-47 m illim e tre s in la titu d e a n d considerably sm aller in
longitude.
Discrepancies between charted orthodromes and their chords
It is n o w a p p ro p ria te to fin d o u t if the chart answ ers its purpose w ell,
th a t is to say h o w m u c h the orthodrom es on the ch art deviate fro m straight
lines, n o t o n ly fro m the p o in t of view o f angles, b u t also fro m the p o in t of
view of the difference in len g th betw een the arc of great circle a n d the
corresponding stra ig h t line on the chart. To s im p lify the c alcu lation s these
m a y be m ade fro m the sphere and n o t the terrestrial ellipsoid, th u s in v o lv in g
a negligible discrepancy.
O b v io u s ly the m ost u n fa v o u rab le case concerns the great circle jo in in g
tw o corners o f the ch art both situated on the same side of its great axis.
Its plane representation falls outside the chart a n d at its extrem ities m akes
an angle of 8°48' w ith the c h a rt’s border, its d ip in re la tio n to the border
being very n ea rly 2°, i.e. 119 n a u tic a l m iles. However the error in distance
is very s m a ll; the border of the ch art measures 3 061 m iles, whereas the
o rthod rom e is 3 050 m iles lo ng (fig u re 2).
I f we consider a slig h tly less extreme case, one nearer th a t encountered
in actu al practice, the charted o rth o d ro m ic arc comes appre ciab ly nearer
its chord. F o r exam ple, ta k in g tw o po in ts whose difference in a u x ilia ry
lo n g itu d e is 36°, i.e. about 2/3rds of the lo n g itu d e range of the chart, a n d
whose a u x ilia ry la titu d e , 12“, is e q u a lly abo u t 2 /3 rd s th a t of the c h a r t’s
borders, we f in d th a t the o rtho d ro m e o n ly m akes an angle of 3° 53' w ith
its chord at its extrem ities, th a t it has a d ip of 36 n a u tic a l m iles a n d th a t
the len gth of the arc is 2 111 m iles as against 2 113 m easured on its chord.
These figures show that, in the n o r m a l use of the chart, w hose m ost
useful p a rt is situ ate d in the v ic in ity of its m a jo r axis, the a u x ilia ry equator,
there w ill in general be no disadvantage in u sing the p lo ttin g of stra ig ht
lines instead of the charted great circles.
T his s u b s titu tio n allow s a considerable benefit to be gained in
parison w ith s im ila r su b stitu tio ns m ade on the u su a l M ercator chart,
com pariso n to the use o f loxodrom es (fig ure 3). The fo llo w in g table
u p the c o m pariso n for the tw o extreme cases, the u p p e r a n d the
borders of the o b liq u e chart.
com ­
i.e. in
sum s
lower
at the
right hand
border
Length
in
nautical
miles
38" 57'
122° 35'
3 050
straight lin e p lo tte d on
the o b liq u e c h a r t .........
47” 45'
113” 47'
3 061
119
lo x o d r o m e on the
, Mercator c h a r t ................
77° 49'
77° 49'
3 320
584
' orthodrom e
' o r th o d r o m e
N o r th e r n
border
S o u th e r n
border
Max. distance
to the
orthodrome
in nautical
miles
at the
left hand
border
Azimuths
....................
.....................
70° 58'
93” 56'
3 050
stra igh t lin e p lo tt e d on
the o b liq u e c h a r t .........
62” 10'
102° 45'
3 061
119
lo x o d r o m e on the
Mercator c h a r t ................
81” 49'
81» 49'
3 069
156
F o r the lower border of the chart, in v o lv in g geographic latitud e s
betw een 19° a n d 27”, the loxodrom e w ith reference to the o rthod ro m e
presents deviations w h ic h o n ly s lig h tly exceed those existing between the
II
l §
51
§■£
II
o2
‘S
fi
c©
vi
i
orth o d ro m e a n d the border of the ob liq ue chart, since this border is
e q u iv a le n t to a loxodrom e plotted at 18° of a u x ilia ry latitude.
B u t as regards the upp er border, fo r w h ic h the arc of a great circle
ru n s betw een 49° a n d 66" of geographic latitud e , the deviations between
the o rth o d ro m e a n d the loxodrom e become considerable, whereas those
betw een the o rtho d ro m e and the u p p e r border are identical w ith those
ex isting on the lower border.
The a n g u la r error in a d o p tin g a rectilinear course on the ch art in ste ad
of the true c u rv ilin e a r p lo ttin g of the ortho d rom e can easily be tak e n in to
acco u nt. In fact this error, w h ich is the angle o f contact at one e nd o f the
p a th , is nearly equal to h a lf the convergency of the m e rid ian s betw een the
tw o extrem ities, w h ic h at the first a p p ro x im a tio n is equal to the range in
a u x ilia r y lo n g itu d e m u ltip lie d by the sine of the m e an a u x ilia ry latitud e .
S in c c
t i l l s la titu d e is r s t h c r
lo w
- a lv / s y s
le s s
tl1 3 .1 1 18 ° —
w s
m ay
consider th a t the convergency of the m e rid ian s, i.e. the a n g u lar error, is
p ro p o rtio n a l to the area in clu de d between the p a th a n d the a u x ilia ry
equato r. W e have previously established th a t in the extreme case of the
border of the chart, th a t is to say for an area equal to h a lf th a t of the chart,
the a n g u la r error between the o rthodrom e a n d its chord is 8?8. It suffices
therefore to com pute ro u g h ly the ra tio of the area u n d e r consideration to
th a t of h a lf the chart to o b ta in im m e d ia te ly the size of the expected a n g u la r
error. I f the tw o extrem ities of the p a th are situate d on either side o f the
c h a r t’s lo n g itu d in a l axis, the area to be com pu ted is the difference of the
areas of the two triangles in c lu d e d between the p a th a n d the a u x ilia ry
e quato r.
U I. — CO M PU TA TION AL PROCESS
The c o m p u ta tio n s required for the estab lish m e nt of the c h a rt’s grid
h a v in g been m ade by an electronic com puter, we shall develop in d e ta il the
fo rm u la e used on this occasion a n d ob tain ed by the conversion of the u s u a l
spherical trig o n o m e tric al form ulae. P a rtic u la rly for the deduction of arcs
d e fin e d by a trig o n o m e tric line it is convenient th a t this line s h o u ld be
a c irc u la r tangent or a hyperbolic sine; fu rth e rm o re the c om putation s were
m a d e by expressing the angles in degrees a n d decim al fractions of a degree.
The ellipsoid adopted is the in te rn a tio n a l e llipsoid (M adrid,
characterized by the values o f its fla tte n in g a = 1/297 a n d of its
m a jo r axis a = 6 378 388 m . E x ce n tricity e, w h ic h fre q u e n tly appears
fo r m u la e is given by e = (2a — a 2)£ •
It sh o u ld be rem em bered th a t the great n o rm a l N to the ellipsoid
p o in t w ith la titu d e L is expressed by :
a n d th a t the
e llip s o id by :
N = a (1 — e2 sin 2 L) ~ i
m e rid io n a l part, or M ercator variable, is defined
C = L o g tan
/ tt
L\
,
<:
1 — e sin L
^ L og —
— —
a re la tio n in w h ic h £. is expressed in ra d ia n s.
1924)
sem i­
in the
at the
on
the
C onform al representation of the ellipsoid on the sphere
In the system used the re latio n between the spherical latitud e s cp a n d
the e llip s o id a l latitud e s L is :
$> —
= £ — £o or 0) = £ + (£„ — £ 0
$ being the m e rid io n a l p a rt on the sphere a n d the subscript 0 being assigned
to elem ents re la tin g to the central po int.
Since m oreover <p0 = L 0, it follow s th a t :
e
$ = J? — —
2
1 — e sin L n
Log — — --- :-- —
1 + e sin L 0
or :
Log ta n ( — + —
(f +
1
) = Log lan ( t + 1 )
e
1 — e sin L
e
1 — e sin L 0
-I--- Log s i n ------------------ L o g ------------2
1 + e sin L
2
1 -f- e sin L 0
F o r m a c h in e co m pu ting , we th e n have :
e
1 — e sin L
1 -I- e sin L 0 \2
----------------IL------- °.\
K
1 + e sin L
1 — e sin L 0 /
F or m a n u a l c o m p u ta tio n it is possible to em ploy
a p p ro x im a tio n the restricted developm ent :
/
L \“|
ta n ( 4 5 ° H--- )
\
w ith
2/J
sufficient
cp = L -)- e2 cos L (sin L 0 — sin L )
The s im ila rity ratio at a n y p o in t on the c h a rt is strictly d e fin e d by :
Sphere
N 0 cos cp
E llip s o id
N cos L
A t the firs t a p p ro x im atio n it m a y be w ritte n :
1 -|--- (sin L 0 — sin L )
2
Scale and extent of the chart
(a)
L et p be the c h a rt’s le n g th a n d q its w id th , expressed in metres,
p, the range of the a u x ilia ry equ ato r in degrees. The value of th is range
in ra d ia n s is :
Its le n g th in metres is consequently N 0 A a n d the scale on the a u x ilia ry
p
equator is w ritte n E = ----, neglecting the alte ratio n in tro d u c e d by the
N0 A
c o n fo rm a l representation of the ellipsoid on the sphere.
q
(b ) H a lf the sm aller border o f the chart, w hose length is — , has an
2
ap e rtu re of
2 N„ E
2
/>
Since the c h a rt is in M ercator pro je ctio n referred to the a u x ilia ry
equ ato r, this q u a n tity represents the a u x ilia ry m e rid io n a l part \e o f the
c h a rt borders; th e n th e ir a u x ilia ry la titu d e le is represented by :
Log
ta n
=
le = 2 t a n -1
-
X, =
A
2
JL
p
K)
(c)
The scale on the borders of the chart, w h ic h is the m a x im u m sca
encountered in ine w hoie area of the chart, is :
E /cos le
or
E ch Xe
It is inte re sting to determ ine the a u x ilia ry parallels on w h ic h the
scale is exactly half-w ay between the scale on the equator a n d the scale
on the borders. T heir latitu d e is given by :
cos lm -- V cos l6
or a g a in by :
ta n (tf) F o r the H y d ro g ra p h ic O ffice’s c h a rt :
p = 0.96
q = 0.65
= 53?6
we fin d :
e q u a to ria l scale : E = 1/6 226 069 ;
a u x ilia ry la titu d e of the borders of the chart : le = 17?8498 = 17°50:9
scale on the c h a rt borders : 1/5 926 364 ;
parallels of the inte rm e d iate scale : lm = 12?6738 = 12°40:43 ;
scale on these parallels : 1/6 074 368 .
Com putation of spherical auxiliary coordinates
(a) In w h a t follow s we assum e th a t the longitudes are coun te d
po sitiv ely tow ards the east. W e shall call M 0 the lo n g itu d e of the cen tral
p o in t, a n d
a n d M the geographic la titu d e a n d lo ng itude of a p o in t A
on the sphere h a v in g geographic pole P (figure 4).
Q
T o o b ta in a u x ilia ry la titu d e I a n d a u x ilia ry lo ng itude m of this point,
referred to the a u x ilia ry equator a n d to the a u x ilia ry m e rid ia n of the central
p o in t C, it is necessary to determ ine b eforehand, on the one h a n d the
geographic coordinates tp! a n d
of the pole Q of the a u x ilia ry equator,
a n d on the other h a n d the a u x ilia ry lo n g itu d e m 0 of the geographic pole,
a lo n g itu d e w h ic h is moreover e qual to the distance fro m the cen tral p o in t
to the geographic vertex o f the a u x ilia ry equ ator (figure 5).
Since the a u x ilia ry pole Q is situ ate d on the m e rid ia n o f th is vertex
a n d at 90° fro m this p o in t, the coordinates of the vertex are expressed
by 90° — cpt a n d M i — 180°, so th a t we m a y im m e d ia te ly o b ta in :
m 0 = t a n - 1 (cos Z 0 cot L 0)
/ cot Z„ "\
M a = M 0 -f t a n - 1 I ------ ) -f 180°
\sin L„ j
cpi = t a n -1 (sin m 0 ta n Z 0)
(/n0 =
10?9087 =
10°54{52)
(M i = 157?8158 = 157°48:95)
(cpi =
43?9010 =
43°54?06)
(b ) The conversion to a u x ilia ry coordinates is then m ade by resolving
the spherical triangle Q P A for w h ic h angle P a n d the tw o a d ja c e n t sides
are k n o w n . The stand ard fo rm u la e of spherical trigonom etry give :
sin I = sin cp sin <px + cos cp cos ^ cos (M x — M)
cos <p sin (M x — M)
sin (m — m 0) = -------------------cos I
B u t to o b ta in I a n d m by trig o n o m e tric tangents the c o m p u ta tio n is
a little m ore co m plicated; it is necessary to have recourse to the Neper
fo rm u la e ;
sin 9 — 9 i
.'M i l
tan
A + Q
cot
cos
ta n
A + Q
cp +
91
2
cos 9 — 91
2
s in 9 + 9 i
cot
M — Mx
A n d , since Q = m 0 — m , by e lim in a tin g A we o b tain for the a u x ilia ry
lo n g itu d e :
[
m = m 0 + tan
<f>i -— T
cp
SPl
cot
2
coslL+»
<fi
tp
[
cos ----- -
]
2 J-i
I
2
______
*
— in co.
The a u x ilia ry la titu d e I is com
pu ted
a s im ila r w ay by using the
fo rm u la e :
si„Vi +S
2 J
sin — [(/n0 — m ) -- (M — M x)]
< ------- CD
"
ta n ' ~ ± = --- 2 ~
---- -----------—
t a n { 45- -
s in - i- [ (m 0 — m ) + (M — M x)]
1
cos — [(m 0 — m ) — (M — M x) ]
I — cc
2,
c o t ---- — = ---------------------------- ta n
2
1
cos — [(m 0 — m ) + (M — M x)]
( « - f)
W h e n ce , by the e lim atio n o f cp :
r c o s — [(Mx — M ) — ( m — m0)]
/ = 90° — t a n - 1
_ ? -------------- :---------[ _ c o s — [(M i — M) + (m — Jn0)]
ta n - 1
I—
1
I sin —
[(Mx — M ) — ( m — m 0)]
H ------------------
I sin — [(M j — M) + (m — m n)]
*
2J
, a n ( 45 " - i )
Com putation of plane rectangular coordinates
The p lan e re cta n g u lar coordinates X a n d Y are expressed in m illim etre s
a n d are referred to two axes of re ctangu lar coordinates having p o in t O,
co rrespo n d ing to the c h a rt’s central p o in t C, as origin. The axis of the
abscissae O X represents the a u x ilia ry equator w h ic h at C m akes an
angle Z 0 w ith the m e rid ian . Abscissae are counted parallel to this axis
in the directio n o f increasing lo ngitude, th a t is a ppro x im ate ly towards the
east. A t p o in t O the O Y axis is p e rp e n d icu lar to O X a n d the ordinates
are counted fro m the a u x ilia ry equator in the direction of increasing
latitud e s, th a t is a p p ro x im ate ly tow ards the north.
W e expressed a, consequently N, in metres a n d m
consequence the M ercator pro je ctio n fo rm u la e give :
X =
1 000 N 0 E —
m
1ol)
in
degrees.
In
Y =
1 000 N 0 E L og ta n ((445*
5* + i - )
B u t earlier on we w rote :
p
E =
N0 A
w hence :
X =
p
180
N0
I* 71
m
1 000 p —
V180 000
/
I \
Y = -------- L og ta n I 45° + -— )
H 7i
V
2/
I f the c h a r t’s length p is also expressed in m illim e tre s :
X = p —
180
/
I \
Y = p ------ L o g ta n ( 45° + —-)
[1.71
\
2)
The accuracy of 0.1 m illim e tre is sufficien t in the d e te rm in a tio n o f X
a n d Y.
Specific points
F o r the construction of the ch art it is use ful to determ ine the
coordinates o f a certain n u m b e r of specific po in ts; this is easily done by
the s ta n d a rd c o m pu tatio n s of spherical trig o n o m e try w h ic h need not be
developed here.
W e have already m e n tio n e d the geographical vertex of the a u x ilia ry
eq u a to r; its geographic coordinates on the sphere are :
cpr = 90" — cp,
M„
— 180°
The places of m a x im u m la titu d e on the upp er a n d lower borders of
the c h a rt are situated on the same m e r id ia n as the vertex a n d have as
respective latitud e s :
cp, -f- I,, a n d
cpr — le
The sim plest w ay to o b ta in each of the east a n d west extrem ities
of the cen tral axis is to resolve the re c ta n g u la r spherical trian g le w h ic h
has as its vertices the geographic pole, the aforem entioned vertex a n d the
e x tre m ity u n d e r consideration, this latter being separated fro m the vertex
n
by an arc e q u a l to — ± m 0 . The a z im u th of the central axis at these points
is obtained fro m these same triangles.
The geographic coordinates o f the corners of the chart are d eterm ined
w ith the help of spherical triangles defined by the geographic pole, the
corner u n d e r consideration a n d the extrem ity of the central axis nearer to
this corner. The value o f the arc of great circle jo in in g these is l e. T his
arc is o rtho g o nal to the cen tral axis.