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Cahill-Keyes Octant Graticule:
Principles and Specifications

with Perl programs and OpenOffice.org 2.0 macros
 for 1/1,000,000 Megamap

Gene Keyes
2010-08-20

The creation of a master plan of the world is a matter of design first and foremost, the design of a frame within which intensive mathematical details shall be afterwards subordinated.

©1975, 1980, 2010 by Gene Keyes
cc logo
For fair use, with source credit, by anyone:
copy, share, adapt, whatever.
(Optional: send me your link!)
For commercial use, contact gene.keyes~~at~~gmail~~dot~~com


Abstract
How a complete one-degree graticule of an eight-octant world map at 1/1,000,000 is made with Perl and macros in a free OpenOffice.org 2.0 vector-drawing program on a $300 Asus netbook. This is another installment of my drafting notes for the Cahill-Keyes "Real-World" map. It describes only the graticule, in intricate detail, both as hand or computer drawn archetypes. Inputting GIS data is the next challenge.

Update 2010-11-28: I have added an appendix with additional programming explanations and guidelines by Mary Jo Graça, about how land data might be converted into this graticule, elaborated with six diagrams of constructing a single half-octant, then duplicating, transposing, and assembling them into the full eight-octant M-profile. This is still a work in progress (non-profit); suggestions and participation are invited.

Update 2012-02-09: The first Cahill-Keyes Multi-scale Megamap Beta-1 has now been published on this website in considerable detail:
http://www.genekeyes.com/MEGAMAP-BETA-1/Megamap-Beta-1.html

See also my previous works:
Notes on Re-designing B.J.S. Cahill's Butterfly World Map

Critique of Dymaxion Map as compared to Cahill's

Notes on Scaling Cahill and Cahill-Keyes Maps

Photos of the Cahill-Keyes Megamap Prototypes

Geocells and the Megamap

10 Principles for a Coherent World Map System

B.J.S. Cahill Resource Page


Contents

Page
§
Subtitle



1
1
Fundamental considerations

2
X-Y coordinates by hand or computer

3
Orientation, hand-drawn

4
Orientation, computer-drawn

5
Coordinates for hand-drawn version of octant perimeter

6
Constructing meridians and parallels in the
Cahill-Keyes Octant Graticule

7
Further explanation of the "supple zones"

8
Latitude lengths per 1 and 5 degrees in
Cahill-Keyes Octant Graticule
(central and outer meridians)




2
9
Constructing the Cahill-Keyes Octant Graticule
and Megamap with Perl programming
and OpenOffice.org Draw macros

10
Perl Program "HalfOctant8" by Mary Jo Graça
to calculate one-degree coordinates
for Cahill-Keyes Octant Graticule Template
(including scaffold triangle and octant itself)



3
11
Complete x-y coordinates for
Cahill-Keyes Octant Graticule Template
(and 1/1,000,000 Megamap)



4
12
Perl program "OOmacroMaker4" by Mary Jo Graça
to produce OpenOffice.org Draw macros
for Cahill-Keyes Octant Graticule




5
13
OpenOffice.org (OOo) macros via Perl
for Cahill-Keyes Octant Graticule and Megamap




6
14
Cahill-Keyes Octant Graticule
and Megamap Grid as
illustrated from OOo macros




7
(new)
15
Appendix: Further programming details and guidelines by Mary Jo Graça, with 6 illustrations



1) Fundamental considerations

We now get into what Cahill called the "intensive mathematical details", which must fulfill my four essential design criteria for the Cahill-Keyes master map:
1) Visual fidelity to a globe in each and all of its eight equal octants, as well as showing each and all of its 64,800 one-degree geocells (except for the sliver-like 3,600 at the poles between 90 and 85°, drawn with 1° latitudes, but 5° longitudes );

2) Proportional geocells in all adjacent ranks and files, none being grossly out of sync with its neighbors, despite calculated shrinkage toward the octants' centers, and other minor adjustments along the octants' outer meridians.

3) 10,000 km lengths for each of its octants' three main jointed edges, the (rounded) metric distance from pole to equator, being one-quarter of the earth's circumference, ca. 40,000 km, and:

4) An M-shape Master-Map profile whose eight octants fit exactly in a km-gridded rectangle whose span is 40,000 km, representing that very circumference.

Caveats:
1) "Fidelity" is not the same as "accuracy" in the geodetic, equal-area, or equal distance sense, but rather, an overall likeness to a globeto its continents, and to its graticulebetter than any other entire world map.

2) "10,000 km" in principle is the metric distance from pole to equator, but the earth is not a perfect metric sphere, nor are these map's edges a perfect metric match, being 10,042 km, due to design constraints explained below.

3) A computer monitor cannot show you maps at their proper scale and dimensions, in both width and height. My hard copy map originals and printouts are the right size each way; but not on a monitor. (See demonstration here.) I wish monitors were otherwise, but if you want to get the scales and appearance just right, you'll have to print one or more of the pdf files linked on p. 6 . Note well that while computer-drawings are extremely accurate, monitors are merely loosey-goosey.

4) Until the digital graticule presented below, the Cahill-Keyes map has existed only as hand-drawn prototypes at scales of 1/20,000,000 and smaller, plus a one-degree octant at 1/10,000,000; plus some 17 square-meter test panels (out of 496) for the 1/1,000,000 Megamap graticule (see photos here), plus two 1/1,000,000 test panels, of Denmark, and Canada's Atlantic provinces. (Plus earlier offset-print specimens, shown on this website as jpegs.)

5) My academic field is world politics; I am not a mathematician. While this map does not embody geodetic precision nor cartographic formulas, I had to do a lot of trigonometry in my quest for proportional geocells across the entire face of the earth on the length and breadth of the Cahill-Keyes world map. (And Mary Jo Graça's Perl programming for it, next page, adds a further degree of mathematic nicety.)

Those design criteria in turn imposed some constraints, and indicate why the map is done in a certain way, and not some other way. (Cahill himself had evolved at least five different projection variations within his basic framework, but for reasons explained elsewhere, I started from scratch on a different path.)

Besides the four main criteria stated above, three other considerations had to be met; again, differing in detail from Cahill:
5) That the octants be entirely symmetrical: all six end segments be equal to each other, and all three main mid-segments likewise equal one another.

6) That the end segment polar incision be no longer than 17° of latitude;

7) That the polar zone parallels from latitudes 89 to 75 be drawn as circular arcs.
My design had to simultaneously satisfy all seven desiderata, none of which were present in Cahill's versions. It is their combination which establishes the character of the Cahill-Keyes — and entailed such difficulty in doing it. As Cahill himself wrote, "simplicity in most things is achieved by a circuitous route through all manner of complications." (1912, p. 164)
____________

Example of a design constraint

Note that a 10,000 mm scaffold-triangle altitude is required to produce an M-profile Cahill-Keyes map with a span of 40,000 mm [km]. (See Notes on Scaling Cahill and Cahill-Keyes Maps.) But the inscribed octant's altitude (scroll down to Fig. 2) is ipso facto shorter than 10,000 km, while the triangle sides are longer than 10,000 km here in Fig. 1.

Fig. 1: Scaffold triangles and M-profile of Cahill-Keyes map. Original drawing by GK; reproduced in ClarisWorks 5 by Mary Jo Graça.


For example: what happened when I tried to attain a 3-segment side that was exactly 10,000 mm within a scaffold triangle whose altitude is 10,000 mm? Well, I was able to do that after a lot of trial-and-error calculation, and found that the octant's altitude must be 9,033.3625 mm, instead of my chosen 9,060 mm. But it resulted in the end segments being somewhat too long, requiring an unacceptable stretch at the octants' outer meridians in two of the polar latitude degrees between 75° and 73°: namely, 167 mm per degree, or 150% of any latitude's nominal 111.111 mm, instead of my already excessive 130 mm there. Likewise, the one-degree latitude-lengths on the outer meridians from 0° to 15° would have been stretched to 124.5 mm, i.e., 112%, instead of my already excessive 121 mm. In addition, the shorter altitude of the "exactly 10,000 km" octant imposes still greater compression on the central portions of the graticule.

Therefore, I reluctantly decided to let the "proportional geocell" principle prevail over the "exactly 10,000 km" principle. Nevertheless, the quarter-circumference octant edge is only 0.4% larger than ideal, while I thus avoid grossly stretching some geocells up to 150%. (More details are in the "supple zone" notes below.)
__________

2) X-Y coordinates by hand or computer

From 1980 to 1984, I computed all the x-y coordinates of all 8,100 geocells of an entire Cahill-Keyes Octant Graticule (minus the 450 polar spiky ones, drawn @ 1 x 5°), copying the 5-decimal-place numbers by hand into tables, from a TI-30 scientific calculator, and doing others with a BASIC program into a fanfold printout.

This was well before the Internet. Now that I have revived this project for my website, after a 25-year dormancy, I boggled at keyboarding so much data.

However, my companion Mary Jo Graça has patiently written a Perl program (p. 2 below) to re-calculate all those x-y coordinates and output them in HTML tables here (p. 3). Next, she exerted her coding skills to prepare macros (p. 4 and 5) using the Perl data, which could plot the map graticule via a vector drawing program. In doing these digital versions, I have been the architect, as it were, and Mary Jo the engineer.

At first we were using an outdated 1993 Mac version of TurboCAD, but its exported .dxf files proved unusable, and it could not do very large scales. So we switched to OpenOffice.org Draw 2.0 (OOo for short; it is cross-platform, and free.). Though more difficult for Mary Jo to program its macros, it could output image files in either .pdf or .odg form.*
__________

* Impetus to try a computer version of the Cahill-Keyes map came from Clay Cole in February 2010, who started one in AutoCAD. But that is a pricey program for PCs, and our machines are an old Mac G3 Biege OS 9, and a pair of cheap Asus eee 701 netbooks with Linux (Xandros).

3) Orientation, hand-drawn

If you are doing this map in a vector-draw or CAD program, then one size fits all, depending on how big your monitor is, and how much you can zoom in or out. But in 1980, when I began hand-drafting the 1/1,000,000 megamap on separate square-meters of millimeter-grid cross-section paper, I compiled tables that were adaptable two ways: 1) for a single octant template (upright and tilted 30° to the left.) at 1/10,000,000, in a grid of 1,000 x 1,200 millimeters; i.e., just over one square-meter; 2) and for the scaled up Megamap itself, whose single-octant template would occupy 62 square meter sheets within a grid of 10,000 x 12,000 millimeters: (See diagrams below, and Photos of the Cahill-Keyes Megamap Prototypes.) That in turn is the prelude to a hoped-for 20 x 40 meter exhibition piece (65 x 130').

Therefore, numbers in tables 1 and 2 below on this page (for the hand-drawn version) are stated as coordinates of a millimeter grid that is 10,000 x 12,000 mm for the 62-sheet 1/1,000,000 template. (Note that x-line 0 starts 3,000 mm above the bottom of the grid, so as to intersect point G, which is the octant's central meridian, at the Equator's midpoint.) When I was simultaneously drafting the ten-times-smaller octant at 1/10,000,000, I would move the decimal one place to the left; i.e., I would read only the first column, [before the (*) digit]. (That would still yield a pretty big wall map, 2 meters high by 4 meters long, if the octant template is reproduced eight times.)

The "Panel" column in table 1 refers to coding for the 62-square-meter version, shown as dark red numbers in Fig. 3 below. There are ten inverted-duplicate panels, to which I have appended letters A through J, in front or back, depending on whether they belong to the "up" or "down" portion of the wedges GHIJ or GHQR. (That latter "reciprocal wedge" GHQR is seen in Fig. 2, but not in Fig. 3.)


4) Orientation, computer-drawn

To repeat, these diagrams here on p. 1 show how I constructed one complete octant on paper, the 1/10,000,000 version, upright and tilted left. The specifications and tables on this page establish the basic shape, orientation and segment lengths of the complete Cahill-Keyes Octant Graticule.

However, when it came to doing the same thing with Perl and computer graphics, Mary Jo's approach was to draft only half of an octant, lying horizontally, then copy, flip, and rotate it successively, via macros, to attain a full octant and then the complete eight-octant M-profile in its 40,000 km (mm) grid.

Much easier said then done; but she did it, on her little Asus netbook. The specifications on pages 2 and 3 describe the exact graticule of a one-half octant, to be replicated 16 times (8 left, 8 right) for a complete world map. The macros shown on pages 4 and 5 draw that graticule. Though her coordinates occupy a lengthy and unabridged table for every single geocell, there are only half as many as in my original set. Despite her different orientation, the outcome is still the same eight-octant graticule for a 1/1,000,000 Cahill-Keyes Megamap — or any other specified scale or size.

My hope is that it can then absorb all kinds of global data, and be useful to anyone who cares to work with it — in hard copy, or digital display, or both. (A related wish of mine is for counterpart one-degree and five degree globes, from small to giant, marked with the Cahill-Keyes octant. See FDR globe, etc., here, esp. figs. 9.6.6 and 9.6.12.)

Fig. 2: Cahill-Keyes Octant Perimeter points. (Note that unlike an earlier version of this on my "Scaling..." page, midpoints "C" and "K" are no longer needed. Also, point "G" is now set on the zero y-axis, rather than y 200 as before.)
Cahill-Keyes Octant Perimeter



Fig. 3:
An assembled composite of two Cahill-Keyes half-octants via Mary Jo Graça's Perl program & macros
Cahill-Keyes Octant Graticule



5) Coordinates for hand-drawn version of octant perimeter

Deferring to Mary Jo's automated re-orientation of the graticule's x-y coordinates (p. 2 and 3 below), I have omitted most of my original whole-octant tables — except the ones given here, which specify the original design framework and segment lengths (as shown above). They remain the same in the subsequent computer-drafted version.

For a 1/10,000,000 map, read first column of digits only. For a 1/1,000,000 map, append (*) column's digit. Other decimals are for optional precision: e.g., if a 1/100,000 map, append first digit from four-place decimal column; for a 1/10,000 map, append the second; etc. Decimal places are for more exact computation, not visible to naked eye, until they start to cumulate.


Table 1:
Scaffold Triangle and Octant Perimeter: x-y coordinates and segment lengths in millimeters

Point
Panel
X
*
decimal
Y
*
decimal
Scaffold Triangle








Base midpoint (altitude foot)

G
57
0500
0
0000
0


Vertexes:









M
n.a.
0


866
0
2541

N
42-J
1000
0
0000
0288
6
7513

P
J-42
0


-0288
6
7513









Octant Perimeter Itself








Vertexes








Starting point
A
1
0047
0
0000
0784
6
1902

E
42-J
0906
0
0000
0288
6
7513

I
J-42
0047
0
0000
-0207
2
6873

Q
42-J
0953
0
0000
0207
2
6875
Joints









B
3
0222
4
0639
0737
6
1902

D
23
0777
5
9361
0417
0
8152

F
50-H
0777
5
9361
0160
2
6875

H
H-50
0222
4
0639
-0160
2
6875

J
A-62
0


-0031
8
6236

L
4
0


0609
2
1263

R
62-A
1000
0
0000
0031
8
6326










Segment Lengths, in mm


mm
*
decimal




Scaffold Triangle









MN, etc.: sides


1154
7
0053



MG, etc.: altitude(s)


1000
0
0000



MB, DN, etc.: vertex sides


256
8
1278



EN, etc.: altitude indent


94
0
0000



MD, etc.: vertex to 2nd indent


897
8
8773




Octant Perimeter Itself









AG etc.: internal altitude


906
0
0000



AB, etc.: each end-segment


181
5
9406



BD, etc.: each mid-segment


641
0
7498



ABCDE, etc.
each 3-segment side



1004
2
6307





6) Constructing meridians and parallels
in the Cahill-Keyes Octant Graticule

(These specifications apply both to hand drawn and computer-drawn versions)

Meridians

Except where noted, parallels are mainly a function of equisecting the meridians, so we begin with meridians.

Note: Below are construction numbers for meridians in an octant, not their actual globe numbers, which are compiled in a separate table.
M 0 [meridian zero, the central one] is the altitude AG, and is straight; all the others have two joints.

Meridians are numbered from 0 to 45 (the east edge of the octant, ABCDE); and from 0 to negative (-45) on the west edge.(ALKJI).
Latitude lengths are the same for each respective plus-or-minus-numbered meridian, (e.g., anything on meridian 37 equals meridian (-37)., but they will have different x-y coordinates, as well as a specified variance (from earth itself) for each separate meridian pair-per-octant..

(All terrestrial degrees of latitude are ca. 111.111 km, but on this flat map, shrinkage is proportionally shared toward the octant's center, as adapted from Buckminster Fuller's Dymaxion map. Also, some other tweaks are made in the "supple zones" explained below.)


Meridians are constructed in three zones:
Polar (ALB in diagram above),

Middle-Sector (LBDJ), and

Equatorial (JDEI).
All meridian angles change from one meridian to the next at a constant rate:
In the Polar zone, each meridian's angle is 1° greater or less than its neighbor, within a 90° spread.

In the Middle-Sector, each meridian's angle is 2/3° greater or less than its neighbor, within a 60° spread.

In the Equatorial zone each meridian's angle is 1/3° greater or less than its neighbor, within a 30° spread.
The three swathes of meridians bend at respective joints which are not tangent to a parallel. [My earliest design bent meridians from given parallels of 73° and 17°, but this resulted in disparate meridian angles, and longitude widths, throughout.]
Polar zone meridians start at vertex A; all 90 (or 91, including both outer sides) are drawn as one-quarter of a standard 360° polar grid, i.e., 90°, spaced 1° apart. (From the Pole to parallel 85, meridians are spaced 5° apart, but all 1° parallels are drawn.)

Middle-Sector meridians start at vertex M (not shown before they meet their polar counterparts); all 90 are likewise drawn, except that their angles are 2/3° apart, i.e., 90°, 89 1/3°, 88 2/3°, 88 1/3°, etc., reaching the outer angle of 30° relative to the x-axis.

Equatorial zone meridians do not have a common vertex, but start at equisected points along the Equator, changing their angles @ 1/3° from 75° at edge I J to 45° at edge DE, and meet their Middle-Sector counterparts at joints calculated by trigonometry.
(Clay Cole suggested that the Equatorial zone meridians could meet at point "X" of an overlapping (unshown) 30° isosceles triangle, whose vertex is far beyond the main scaffold triangle. with x-y coordinates (at 1/1,000,000) of -4489.4388 mm and 16,436.1902 mm, outside the given drafting grid of 10,000 x 12,000 mm. However, Mary Jo Graça recalculated and confirmed with CAD my original conclusion that there is no common meridian vertex if the Equator's longitudes are all identical. Using a point "X" origin for the Equatorial meridians would result in somewhat differing longitude widths, contrary to the design principles of this map.)


Parallels

Parallels, too, are constructed in the same three zones:

1) Polar, P 89 to 75: has concentric circular arcs centered at the pole, A, with radii cumulating at 104 mm per degree of latitude (104, 208, 312 mm etc.)
1a) Between M 30 and M 45 there is a polar "supple zone" (illustrated and further explained below). P 74 and 73 are circular arcs, but their radius increases by 100 mm per degree, instead of 104 mm. As well, P 74 and 73 are circular arcs only from M 0 to M 30.
2) The predominant Middle-sector, P 73 to P 15, is where all meridians are equisected, and those points are joined to form gradually flattening curves as they approach the Equator. Here, all latitude lengths along the outer meridians (M 45) are 110.4278 mm. Along M 0, they are all 100 mm, and all meridians in between are interpolated proportionately.

3) The Equatorial zone, P 15 to 0, is where meridians are equisected with the same latitude lengths as above, but only as far as M 30 on either side.
3a) Another “supple zone” is in the equatorial right-angle corners of the octant (Likewise illustrated and further explained below). It is an area comprising 15° each of latitude and longitude (M 30 to 45. and P 15 to 0), where the equisection is redistributed proportionately beginning from a longer latitude length along M 45: 121.0627 mm per degree instead of 110.4278 mm as in the Middle-sector.

7) Further explanation of the "supple zones"

To attain smooth continuity of parallels across octants in the predominant Middle-sector, avoiding cusps and sags, the Cahill-Keyes graticule adds a little extra stretch along the outer edge (M 45), at two relatively small "supple zones" (on each side of the octant, or four in all). The supple zones' excess decreases evenly within the longitudes of M 45 to 30, between the latitudes of P 75 to 73, and likewise within M 45 to 30 between P 15 to 0 respectively. The stretching is compensated by a small reduction on M 45 in each Middle sector geocell height to 110.4078 mm, instead of an ideal 111.1111, along M 45.

Polar supple zone:
From P 75 to 74, and 74 to 73, on M 45, there is a maximum stretch in each latitude to 130.9883 mm, diminishing proportionately to 100 mm at M 30 through 0 (vs. that ideal nominal length of 111.111 per degree of latitude). In addition, P 73 is drawn as a line perpendicular from M 45, until it becomes a circular arc at M 30.

P 74 then splits the difference between the arc of P 75, and the straight line of P 73 between M 45 and M 30. The circled portion below shows the one sector of the octant which slightly deviates from the proportional-geocell principle of avoiding disparity among adjoining geocells.

Fig. 4: Diagram by Gene Keyes in OpenOffice.org Draw 2.0; coding by Mary Jo Graça.

Polar supple zone


Equatorial supple zone:

From P 0 to 15 on M 45, there is a maximum stretch to 121.0627 mm per degree (vs. the ideal 111.111 mm), diminishing proportionately to the same latitude lengths on P 30 to 0 as in the Middle Sector.

Note: Parallel 15 meets joints J and D, of the middle and equatorial segments along M 45, but P 73 is 6.0360 mm below joints L and B of the middle and polar segments along M 45.

Fig. 5: Diagram by Gene Keyes in OpenOffice.org Draw 2.0; coding by Mary Jo Graça.

Equatorial supple zone


Parallel 15 was a special (and difficult) case in the design of the map, a kind of a fiat or "skyhook" construct to knit the graticule between its upper and lower portions. When hand-drawing it, I had equisected the meridians, as stated, but had connected the points between M 35 through M 20 with a spline, to smooth out P 15 and lower parallels, so that they were unlike the angular joints on the Equator (P-0). However, when it came to doing the same parallel with a computer, Mary Jo devised a circular arc (replicated at either end of the parallel) centered at 2786.8887, 1609.0110 in MJ's half-octant coordinates, with a radius of 5760,8557 mm. See further explanation, p. 2 below.

Summary of supple zones:
Overall effect on the three Great-Circle outer edges of the
Cahill-Keyes Octant Graticule

Besides geocell proportionality, the Cahill-Keyes design emphasizes the 10,000 km distance on each main side of the octant, embodying the metric principle of 1/4 of a great circle from pole to equator, on a perfect metric sphere where each degree of latitude (and longitude only at the Equator), is ca. 111.111 km.

However, the earth is not a perfect metric sphere, nor is this map. When I say 10,000 km, i.e., mm, as the octant edge, I am rounding from 10,042.7 mm, a design constraint as explained above.

The equator-longitude widths on Cahill-Keyes are equal across the board, and around the world: 111.5847 mm (vs. 111.321 in reality, and 111.111 in principle).

But the two Great Circle meridians bounding all the Cahill-Keyes Octants, (the same 10,000 km as the Equator), are finessed somewhat by the two "supple zones" along M 45. While most of the lattitude lengths (i.e., the Middle sector, P 73 to 15), each 110.4278 mm, are just a bit under 111.1111, they are a little too long in the supple zones at P 75 to 73 (130.9883 mm per degree), andP 15 to 0(121.0627 mm), and somewhat too small in the Polar zone itself, P 90 to 75 (104 mm).

Nonetheless, the overarching aim is always to minimize adjoining geocell disparity across the map as a whole, while keeping all eight parts visually attuned to their counterparts on globes, and to the metric principle of 10,000 kilometers along any edge of an octant.


8) Latitude lengths per 1 and 5 degrees
in Cahill-Keyes Octant Graticule
(central and outer meridians)


(These specifications apply both to hand drawn and computer-drawn versions)

For a 1/10,000,000 map, read first column of digits only. For a 1/1,000,000 map, append (*) column's digit. Other decimals are for optional precision: e.g., if a 1/100,000 map, append first digit from four-place decimal column; for a 1/10,000 map, append the second; etc. Decimal places are for more exact computation, not visible to naked eye, until they start to cumulate.


Table 2: Latitude lengths per one degree along M-0 and M-45



mm
*
decimal












1-degree Latitude lengths
along M-0








P-90 to P-75


10
4
0000



P-75 to P-0


10
0
0000












1-degree Latitude lengths along
M-45









P-90 to P-75


10
4
0000



supple zone P-75 to P-74


13
0
9883



" P-74 to joint


12
4
9523



" joint to P-73
(These 2 combined are also 130.9883)



6
0360



P-73 to P-15


11
0
4278



supple zone P-15 to P-0


12
1
0627





Table 3: Cumulative latitude lengths per five degrees along M-45 and M-0 from pole
5-degree cumulative lengths
from pole onward



M-45

mm


*


decimal
M-0

mm


*


decimal

90-85


52
0
0000
52
0
0000
85-80


104
0
0000
104
0
0000
80-75


156
0
0000
156
0
0000
[supple zone, 75° to 73°]








75-74


169
0
9883
na


74-73 (as far as joint)


181
5
9406
na


P-74-73 (beyond joint


181
9
9766
na











75-70
(including supple zone & joint)


215
3
2600
206
0
0000









70-65


270
5
3990
256
0
0000
65-60


325
7
5380
306
0
0000
60-55


380
9
0770
356
0
0000
55-50


436
1
8160
406
0
0000
50-45


491
3
9550
456
0
0000
45-40


546
6
0940
506
0
0000
40-35


601
8
2330
556
0
0000
35-30


657
0
3720
606
0
0000
30-25


712
2
5110
656
0
0000
25-20


767
4
6500
706
0
0000
20-15


822
6
7890
756
0
0000
Equatorial zone
with adjustments








15-10


878
5
9785
806
0
0000
10-5


941
4
3535
856
0
0000
5-0


1004
2
7285
906
0
0000


Table 4: Cumulative latitude lengths per five degrees along M-0 and M-45 from equator,
and from P-15.

5-degree cumulative lengths
from equator onward:
equatorial segment only,
P-0 to P-15




M-45




mm





*





decimal

M-0




mm





*





decimal

0-5


62
8
3750
50
0
0000
5-10


125
6
7500
100
0
0000
10-15


181
5
9395
150
0
0000
M-45
5-degree re-cumulating lengths
for Middle-Sector only
from P-15 to P-73









15-20


55
2
1390
50
0
0000
20-25


110
4
2780
100
0
0000
25-30


165
6
4170
150
0
0000
30-35


220
8
5560
200
0
0000
35-40


276
0
6950
250
0
0000
40-45


331
2
8340
300
0
0000
45-50


386
4 9730
350
0
0000
50-55


441
7
1120
400
0
0000
55-60


496
9
2510
450
0
0000
60-65


552
1
3900
500
0
0000
65-70


607
3
5290
550
0
0000
supple zone @ 1°








70-71


618
3
9568
560
0
0000
71-72


629
4
3846
570
0
0000
72-73


640
4
8124
580
0
0000

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