Maps, Projections and Datums

Map scale and digitizing accuracy

The representative fraction (RF) of a map is the ratio of a distance on the map to the same distance on the ground, written as 1/N or 1:N, where N is the scale denominator. A larger scale means a smaller denominator, and maps are classed as large or small by this ratio, not by the size of the sheet or the area shown. Large scales such as 1:1000 and 1:5000 show fine detail, while small scales such as 1:50,000 and 1:100,000 show wide regions.

When a graphical scale is read, both lengths must be converted to the same unit before forming the ratio. For example, if 2 cm on the map represents 500 m on the ground, then \text{RF} = 2/50000 = 1/25000.

The scale that suits a purpose depends on how much detail must be shown. Urban planning needs individual plots, streets and buildings, so the largest of the usual scales, such as 1:10,000, is preferred over 1:25,000, 1:50,000 or 1:100,000.

The smallest distance that can be plotted on paper is the graphical limit, about 0.25 mm. Its ground equivalent is found by multiplying by the scale denominator, 0.25\ \text{mm} \times N. A 1:100,000 map therefore cannot show ground detail finer than 25 m, and a 1:250,000 sheet cannot show ground detail finer than 62.5 m.

Errors in digitizing or plotting on the map are converted to ground errors in the same way, namely ground error = map error \times N. A digitizing error of \pm 0.2 mm on a 1:25,000 map thus corresponds to about \pm 5 m on the ground.

For maps of the same physical sheet size, the scale depends on the ground area represented: the smaller the area covered, the larger the scale. A sheet showing a 4 km^2 estate and a sheet of the same size showing 4000 km^2 therefore have different scales, and the estate map is the one at the larger scale.

A grid distance on a UTM map differs from the true geodetic distance by a scale factor, here about one part in 4000. The geodetic length is recovered by dividing the grid length by this factor, so the correction is small but not negligible for precise work.

Horizontal control networks that fix ground positions for mapping are built either by triangulation or by trilateration. Triangulation measures all angles of a chain of triangles and is suited to hilly or rugged ground where long sides are hard to measure directly. Trilateration measures all side lengths, usually with EDM, and is suited to flat terrain. Triangulation gives higher accuracy in angles, while trilateration gives higher accuracy in sides.

Map projection properties

A map projection is a systematic method of representing the curved surface of the Earth, including its graticule of latitude and longitude lines, on a plane. Because a curved surface cannot be flattened without distortion, every projection preserves some properties and distorts others, and the choice depends on the extent and location of the area mapped.

Latitude is the angular distance north or south of the equator and ranges from 0^\circ to 90^\circ. Longitude is the angular distance east or west of the Greenwich meridian and ranges from 0^\circ to 180^\circ. The longitude of a point is the angle between the Greenwich meridian and the meridian through that point. Under the conventional definition, coordinates are referred to a fixed datum, and slow crustal motion is treated as a separate correction.

A conformal projection preserves angles locally, so the shapes of small areas are retained. The transverse Mercator, stereographic and Lambert conformal conic projections are conformal. The sinusoidal projection is equal-area and distorts angles, so it is not conformal.

The Mercator projection is a conformal cylindrical projection. Shape and true direction are preserved at every point, not only along the standard parallel, and lines of constant true bearing appear as straight lines. Areas are progressively enlarged toward the poles, so equal area is not preserved.

The Universal Transverse Mercator system is based on the transverse Mercator projection. In that projection a cylinder is set with its axis in the equatorial plane, which makes UTM a cylindrical projection.

Survey of India sheet numbering

Survey of India topographic sheets follow a hierarchical numbering scheme. A 1:250,000 sheet covers 1^\circ \times 1^\circ and is divided into 16 sheets of 1:50,000 scale in a 4 by 4 arrangement. Each 1:50,000 sheet covers 15' \times 15', and it is further divided into four 1:25,000 sheets, each covering 7.5' \times 7.5'.

To count the sheets needed for a block of given size, convert the extent to minutes and divide by the sheet size. A 4^\circ \times 4^\circ block gives 240' \div 15' = 16 sheets along each side, so 16 \times 16 = 256 sheets of 1:50,000 scale are required.

The Survey of India is the national mapping agency and publishes topographical maps as its principal series. Geological, soil and thematic maps are produced by specialist agencies and are not part of its series.

Geoid height and orthometric elevation

The orthometric height H is the height of a point above the geoid, measured along the plumb line. The geoid undulation (geoid height) N is the separation between the reference ellipsoid and the geoid, with a sign that follows the geoid's position relative to the ellipsoid. Starting from an ellipsoidal height h obtained by GNSS, the orthometric height is H = h - N.

For example, if h = 120.40 m and N = 8.15 m, then H = 120.40 - 8.15 = 112.25 m. A negative N raises the orthometric height above the ellipsoidal height.