Which of the following are parameters of the Universal Transverse Mercator system?

Scale factor 0.9996 on the central meridian

Zones numbered westward from the Greenwich meridian

Which of the following statements about datums and heights are correct?

India's traditional horizontal datum used the Everest spheroid with origin at Kalianpur

Levelling from a mean-sea-level bench mark yields orthometric heights

India's traditional horizontal datum used the Everest spheroid with origin at Kalianpur

Levelling from a mean-sea-level bench mark yields orthometric heights

The latitude and longitude of a point are the same on every datum

Which of the following statements about map projections are correct?

No projection can be both conformal and equal-area

On the Mercator projection a rhumb line (constant bearing) is a straight line

The Lambert conformal conic projection has two standard parallels

No projection can be both conformal and equal-area

On the Mercator projection a rhumb line (constant bearing) is a straight line

The Lambert conformal conic projection has two standard parallels

Adjacent polyconic sheets fit together perfectly along their edges

Maps: Types, Scales, Plotting Accuracy, Sheet Numbering, Coordinate Systems, Projections and Datums

This is a section of Part B1, Surveying and Mapping, of the Geomatics Engineering (GE) paper. Part B1, Surveying and Mapping, opens with Maps: “Importance of maps to engineering projects, Types of maps, Scales, and uses, Plotting accuracy, Map sheet numbering, Coordinate systems — Cartesian and geographical, Map projections, Map datum — MSL, Geoid, spheroid, WGS-84”. Every surveying result ends as coordinates on some datum, in some projection, at some scale, and this chapter is the frame for all of them. It covers what maps do for engineering projects and how they are classified; the representative fraction, the area scale and the plotting accuracy that fixes how fine a scale a job needs; the Survey of India and International Map of the World sheet systems; geographic and Cartesian (earth-centred) coordinates; the properties and families of projections with Mercator, Transverse Mercator and UTM, Lambert conformal conic and the polyconic; and the three reference surfaces — mean sea level, the geoid and the ellipsoid — with WGS-84 and the relation between ellipsoidal and orthometric height. The numericals are scale conversions, UTM zones and heights.

1. Maps in engineering: types, scales and plotting accuracy

An engineering project runs on maps at every stage: small-scale maps for reconnaissance and route selection, medium-scale topographic maps for preliminary alignment and catchment areas, and large-scale plans for detailed design, land acquisition, quantities and setting out. Maps are classed by purpose — topographic (relief and natural and man-made features), thematic (one subject: geology, soil, land use, population), cadastral (property boundaries, at large scale), engineering plans, navigation charts — and by scale. A planimetric map shows horizontal positions only; a topographic map adds relief by contours.

Scale is the ratio of map distance to ground distance, written as a representative fraction (RF) 1:n, as a verbal statement (1 cm = 500 m, i.e. 1:50 000) or as a graphical bar, which alone survives enlargement and shrinkage of the print. A large scale has a small n (1:1000 shows more detail than 1:50 000). Areas scale as the square: 1 cm² on a 1:50 000 map is (500 m)² = 0.25 km², so 4 cm² is 1 km² = 100 ha. A print that has shrunk uniformly has a shrunk scale = original scale × shrinkage factor, and measured distances must be divided by the factor.

Plotting accuracy: the smallest distance that can be plotted or read on paper is taken as about 0.25 mm. Multiplied by the scale denominator it gives the ground size of the smallest plottable detail, and so the scale a job requires. At 1:50 000, 0.25 mm is 12.5 m, so features smaller than that cannot be shown at true size; to show 2 m detail the scale must be at least 1:(2/0.00025) = 1:8000. The same figure sets how precisely the survey needs to be done: measuring to 1 cm for a 1:50 000 map is wasted effort.

2. Map sheet numbering

Topographic series are cut into sheets bounded by parallels and meridians, and a sheet's number encodes both its position and its scale. The Survey of India's long-standing India and Adjacent Countries series works in a hierarchy of sixteen: a 1:1 000 000 sheet covers 4° × 4° and carries a number; it is divided into sixteen degree sheets of 1° × 1° at 1:250 000, lettered A to P; each of those is divided into sixteen sheets of 15′ × 15′ at 1:50 000, numbered 1 to 16; and each 15′ sheet is quartered into 7′30″ × 7′30″ sheets at 1:25 000. So one 1:1 000 000 sheet contains 16 × 16 = 256 sheets at 1:50 000.

The sheet hierarchy
Scale	Extent	Label	Per parent sheet
1:1 000 000	4° × 4°	a number	—
1:250 000	1° × 1°	letter A–P	16
1:50 000	15′ × 15′	number 1–16	16
1:25 000	7′30″ × 7′30″	a quarter	4

The International Map of the World (IMW) at 1:1 000 000 uses sheets 4° of latitude by 6° of longitude: rows lettered from the equator toward each pole, columns numbered 1 to 60 in 6° strips from 180° — the same 6° columns as the UTM zones. Under the National Map Policy of 2005, the Survey of India issues Open Series Maps for public use on the WGS-84 datum and UTM projection, alongside Defence Series Maps.

3. Coordinate systems and map projections

Geographical coordinates give a point by latitude φ (angle from the equatorial plane), longitude λ (angle east or west of the Greenwich meridian) and height h above the ellipsoid. Cartesian (earth-centred, earth-fixed) coordinates X, Y, Z have their origin at the Earth's centre of mass, Z along the rotation axis and X through the Greenwich meridian on the equator; GNSS computes in them. For an ellipsoid of semi-major axis a and eccentricity e, with prime-vertical radius N = a/√(1 − e² sin²φ): X = (N + h) cos φ cos λ, Y = (N + h) cos φ sin λ, Z = (N(1 − e²) + h) sin φ. A degree of latitude is about 111 km everywhere; a degree of longitude is about 111.3 cos φ km, 55.7 km at 60°.

A map projection transfers the curved surface to a plane and cannot preserve everything. Properties: conformal (orthomorphic — local angles and shapes preserved; scale the same in all directions at a point), equal-area (equivalent), equidistant (true scale along certain lines) and azimuthal (true directions from a centre). No projection is both conformal and equal-area. Developable surfaces give the families: cylindrical, conical and azimuthal (planar); the lines of contact are standard lines where scale is true. Tissot's indicatrix shows the distortion as ellipses.

Projections a geomatics engineer meets
Projection	Type and property	Use and behaviour
Mercator	cylindrical, conformal, tangent at the equator	rhumb lines are straight — navigation; scale factor sec φ, 2 at 60°, infinite at the poles
Transverse Mercator and UTM	cylinder tangent (or secant) along a central meridian; conformal	UTM: 60 zones of 6°, numbered eastward from 180°; scale factor 0.9996 on the central meridian; false easting 500 000 m, false northing 0 (north) or 10 000 000 m (south); used from 80° S to 84° N
Lambert conformal conic	conic, conformal, two standard parallels	mid-latitude regions extended east–west; scale slightly small between the standard parallels and large outside them
Polyconic	every parallel is a standard parallel of its own cone; neither conformal nor equal-area	small sheets have almost no distortion, but adjacent sheets do not fit together; used for the Survey of India's older topographic sheets

UTM zone from longitude: zone = ⌊(λ + 180°)/6°⌋ + 1 with λ east positive, and the central meridian is 6 × zone − 183°. Delhi at 77.2° E: ⌊257.2/6⌋ + 1 = 42 + 1 = 43, central meridian 75° E (zone 43 spans 72°–78° E). A point at 88.4° E is in zone 45 (central meridian 87° E); one at 74° W is in zone 18 (central meridian 75° W). India spans roughly zones 42 to 47.

4. Map datum: mean sea level, the geoid, the spheroid and WGS-84

Three reference surfaces
Surface	What it is	Used for
Mean sea level (MSL)	the average height of the sea at a tide gauge over a long period, covering the 18.6-year lunar nodal cycle	the vertical datum for levelling and bench marks
Geoid	the equipotential surface of gravity that best fits global mean sea level, continued under the land; irregular, because mass is unevenly distributed	orthometric heights H; a level bubble is always tangent to it
Spheroid (reference ellipsoid)	a smooth mathematical ellipse of revolution, fixed by a and the flattening f = (a − b)/a	latitude, longitude and ellipsoidal height h; the surface projections are computed on

A geodetic datum fixes an ellipsoid and its position and orientation relative to the Earth. A local datum fits one region: India's traditional horizontal datum is the Everest spheroid with its origin at Kalianpur. A global geocentric datum puts the ellipsoid's centre at the Earth's centre of mass: WGS-84, the datum of GPS, has a = 6 378 137 m and 1/f = 298.257 223 563, so b = a(1 − f) = 6 356 752.3 m. The same point has different latitude and longitude on different datums — by up to hundreds of metres between a local and a global one — so coordinates are meaningless without their datum. Heights: GNSS gives ellipsoidal height h; levelling gives orthometric height H above the geoid; they differ by the geoid undulation N: h = H + N, so H = h − N. Where the geoid lies below the ellipsoid, N is negative and H exceeds h.

⚠️ GNSS heights are not levelled heights

A canal designed from GNSS ellipsoidal heights without a geoid model can be tens of metres out in absolute height, and worse, the geoid's slope can reverse a gentle gradient. Water flows by orthometric height; always convert with H = h − N.

Key takeaways

Scale is an RF 1:n; large scale means small n; areas scale as n²; only a bar scale survives enlargement or shrinkage.
Plotting accuracy ≈ 0.25 mm × scale denominator: 12.5 m at 1:50 000; it fixes the scale a job needs and the precision worth surveying to.
Survey of India sheets: 4° × 4° at 1:1 000 000 → 16 degree sheets A–P at 1:250 000 → 16 sheets of 15′ at 1:50 000 → 4 of 7′30″ at 1:25 000; IMW sheets are 4° × 6°.
Conformal or equal-area, never both; Mercator scale sec φ; UTM zone = ⌊(λ + 180)/6⌋ + 1, CM = 6 × zone − 183, k₀ = 0.9996, FE 500 000 m.
MSL, geoid (equipotential) and ellipsoid; WGS-84 a = 6 378 137 m, 1/f = 298.257 223 563; h = H + N.

Practice questions (15)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

A map is drawn so that 1 cm represents 500 m on the ground. Its representative fraction is 1 : ____.
Numerical answer — type the value.
Show answer
Answer: 50000
500 m = 50,000 cm, so 1 cm on the map is 50,000 cm on the ground: RF = 1:50,000. Forgetting to convert metres to centimetres gives 1:500.
A pond measures 4 cm² on a 1:50 000 map. Its area on the ground, in hectares, is ____.
Numerical answer — type the value.
Show answer
Answer: 100
1 cm = 500 m, so 1 cm² = 500 × 500 = 250,000 m². 4 cm² = 1,000,000 m² = 100 ha (1 ha = 10,000 m²). Scaling by 50,000 instead of 50,000² is the usual slip.
Taking the plotting accuracy as 0.25 mm, the smallest ground distance that can be shown on a 1:25 000 map, in metres (to two decimal places), is ____.
Numerical answer — type the value.
Show answer
Answer: 6.25
0.25 mm × 25,000 = 6250 mm = 6.25 m. Anything smaller cannot be plotted at true size, and surveying it more precisely than about this adds nothing to the map.
Details as small as 2 m must be plottable, and the plotting accuracy is 0.25 mm. The smallest scale that can be used is 1 : ____.
Numerical answer — type the value.
Show answer
Answer: 8000
The scale denominator n must satisfy 0.25 mm × n ≤ 2000 mm, so n ≤ 8000: the smallest usable scale is 1:8000, and any larger scale (1:5000, say) also works. A scale of 1:10 000 would make 0.25 mm represent 2.5 m, too coarse.
In the Survey of India's India and Adjacent Countries series, how many 1:50 000 sheets make up one 1:1 000 000 sheet?
1. 256
2. 16
3. 64
4. 400
Show answer
Answer: A — 256
The 4° × 4° sheet splits into 16 degree sheets of 1° × 1° (A–P), and each of those into 16 sheets of 15′ × 15′ (1–16): 16 × 16 = 256. Sixteen is only the first level of the hierarchy.
A map projection that preserves angles and the shapes of small areas is called
1. conformal (orthomorphic)
2. equal-area
3. equidistant
4. gnomonic
Show answer
Answer: A — conformal (orthomorphic)
A conformal projection has the same scale in every direction at a point, so small shapes and angles are kept — Mercator, Transverse Mercator and Lambert conformal conic are conformal. It cannot also be equal-area; an equidistant projection is true to scale only along certain lines.
The UTM zone number of a point at longitude 77.2° E is ____.
Numerical answer — type the value.
Show answer
Answer: 43
Zone = ⌊(λ + 180)/6⌋ + 1 = ⌊257.2/6⌋ + 1 = ⌊42.87⌋ + 1 = 43. Zone 43 spans 72° E to 78° E with its central meridian at 6 × 43 − 183 = 75° E. Forgetting the +1 gives 42.
The central meridian, in degrees east, of the UTM zone containing a point at longitude 88.4° E is ____.
Numerical answer — type the value.
Show answer
Answer: 87
Zone = ⌊(88.4 + 180)/6⌋ + 1 = ⌊44.73⌋ + 1 = 45. Central meridian = 6 × 45 − 183 = 87° E; zone 45 runs from 84° E to 90° E. Taking the zone's western edge, 84°, is the usual slip.
On the Mercator projection the scale factor at latitude φ is sec φ. The scale factor at latitude 60° is ____.
Numerical answer — type the value.
Show answer
Answer: 2
sec 60° = 1/cos 60° = 1/0.5 = 2, so lengths (in every direction, since the projection is conformal) are doubled and areas quadrupled at 60°. That is why Greenland looks the size of Africa.
A GNSS receiver gives an ellipsoidal height of 50.00 m at a point where the geoid undulation is −45.20 m. The orthometric height of the point, in metres, is ____.
Numerical answer — type the value.
Show answer
Answer: 95.2
h = H + N, so H = h − N = 50.00 − (−45.20) = 95.20 m. The geoid lies 45.20 m below the ellipsoid here, so the point is higher above the geoid than above the ellipsoid. Adding N gives 4.80 m, the sign slip.
The WGS-84 ellipsoid has a = 6 378 137 m and 1/f = 298.257223563. Its semi-minor axis b, in metres (to one decimal place), is ____.
Numerical answer — type the value.
Show answer
Answer: 6356752.3
f = (a − b)/a, so b = a(1 − f) = 6,378,137 − 6,378,137/298.257223563 = 6,378,137 − 21,384.7 = 6,356,752.3 m. The polar radius is about 21.4 km shorter than the equatorial.
The equipotential surface of the Earth's gravity field that best fits mean sea level is the
1. geoid
2. reference ellipsoid
3. topographic surface
4. map projection surface
Show answer
Answer: A — geoid
The geoid is a physical surface of constant gravity potential, irregular because the Earth's mass is uneven; a spirit level and a still lake are tangent to it. The ellipsoid is a smooth mathematical approximation to it, used for coordinates.
Which of the following are parameters of the Universal Transverse Mercator system?
1. Zones 6° of longitude wide
2. Scale factor 0.9996 on the central meridian
3. False easting of 500 000 m
4. Zones numbered westward from the Greenwich meridian
Show answer
Answer: A — Zones 6° of longitude wide; B — Scale factor 0.9996 on the central meridian; C — False easting of 500 000 m
UTM has 60 zones of 6°, a central-meridian scale factor of 0.9996 (so scale is true on two lines about 180 km either side) and a false easting of 500 000 m that keeps eastings positive. The zones are numbered eastward from 180°, so zone 1 is 180° W to 174° W.
Which of the following statements about datums and heights are correct?
1. WGS-84 is a geocentric datum
2. India's traditional horizontal datum used the Everest spheroid with origin at Kalianpur
3. Levelling from a mean-sea-level bench mark yields orthometric heights
4. The latitude and longitude of a point are the same on every datum
Show answer
Answer: A — WGS-84 is a geocentric datum; B — India's traditional horizontal datum used the Everest spheroid with origin at Kalianpur; C — Levelling from a mean-sea-level bench mark yields orthometric heights
WGS-84 is centred on the Earth's centre of mass; the Everest spheroid oriented at Kalianpur was India's classical datum; and levelling follows level surfaces, so it gives heights above the geoid (orthometric). Coordinates change with datum — by hundreds of metres between a local and a global datum — which is why a datum must always be stated.
Which of the following statements about map projections are correct?
1. No projection can be both conformal and equal-area
2. On the Mercator projection a rhumb line (constant bearing) is a straight line
3. The Lambert conformal conic projection has two standard parallels
4. Adjacent polyconic sheets fit together perfectly along their edges
Show answer
Answer: A — No projection can be both conformal and equal-area; B — On the Mercator projection a rhumb line (constant bearing) is a straight line; C — The Lambert conformal conic projection has two standard parallels
Conformality and equivalence cannot both hold on a flat map of a curved surface; Mercator was designed so that constant-bearing courses plot straight; and Lambert conformal conic is usually secant along two standard parallels. Each polyconic sheet is drawn on its own central meridian, so neighbouring sheets do not join without gaps.