Old Hampshire Mapped


Hampshire Lat and Long Scales


measuring 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.