measuring
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The latitude and longitude scales in the borders of a map can
be measured to provide some insight to the map's projection. This
has only been done crudely for a few maps in the Old Hampshire
Mapped project. Measurements are made in pixels from a fairly
large digitised image of the map: as we use a flat bed scanner and
keep careful control of the scanning resolution, our digitised
images can be relied on for measurement. Remember (for Hampshire
maps using a prime meridian at London: latitude increases east to
west, right to left; longitude increases to the north, up the page;
x,y pixels run from an origin top left, x increases left to right,
y increases top to bottom. This can be confusing.
The measurements made are:-
Four outer corners of the lat and long scales:-
B (xb,yb), C (xc,yc)
A (xa,ya), D (xd,yd)
Longitude marks on top and bottom borders:-
xMaxLongTop, x1degLongTop, xMinLongTop
xMaxLongBtm, x1degLongBtm, xMinLongBtm
Latitude marks on left and right borders:-
yMaxLatLt, y51dLatLt, yMinLatLt
yMaxLatRt, y51dLatRt, yMinLatRt
If marks are obscured their positions might be interpolated.
Other marks might be measured, but are not used in this
exercise.
Calculations made are:-
1 degree longitude, top = (xMinLongTop-xMaxLongTop)
/(MaxLongTop-MinLongTop) pixels
1 degree longitude, bottom = (xMinLongBtm - xMaxLongBtm)
/(MaxLongBtm - MinLongBtm) pixels
If the two values are within 5 percent they are taken to be
equal within errors, and an average is found:-
1 degree longitude in pixels
Otherwise the lat and long grid, graticule, could be
trapezoid. It should be narrower at the top. Calculating
the latitude of the top and bottom borders first, it is
possible to compare the ratio of the longitude scales to
what it should be, the ratio of the cosines of the latitudes.
1 degree latitude, left = (xMinLatLt-xMaxLatLt)
/(MaxLatLt-MinLatLt) pixels
1 degree latitude, right = (xMinLatRt - xMaxLatRt)
/(MaxLatRt - MinLatRt) pixels
If the two values are within 5 percent they are taken to be
equal within errors, and an average is found:-
1 degree latitude in pixels
Otherwise the scales are 'wrong'.
The ratio of latitude to longitude scale is found:-
1 degree latitude / 1 degree longitude
If the lat and long scales are much rotated there should be
a correction involving the cosines of the rotation of the
meridians and the parallels (which can be different).
At 51 degrees the ration lat/long should be 1/cos(51) = 1.59
(for the oblate spheroid that the Earth really is the figure
is nearer 1.585). If the figure found is different, outwith
errors, then the map is stretched N-S or E-W.
A little trigonometry is used to find any rotation of the
scales.
angle of 1 degree meridian =
tan-1 ((x1degLongTop - xb) - (x1degLongBtm - xa) / (ya - yb))
If the angle is small it might be assumed to be zero, no
rotation.
angle of 51 degree parallel =
tan-1 ((y51degLatLt - ya) - (y51degLatRt - yd) / (xd - xa))
If the angle is small it might be assumed to be zero, no
rotation.
The two angles can be compared to see the shape of a lat long
'cell'. This might be a rectangle or sheared into a
parallelogram. If the map is a trapezoid this comparison is
void.
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