GATE Geomatics Engineering Section I Syllabus

What is a conformal (orthomorphic) projection?

A projection that preserves local angles/shapes (small features keep their true shape) so that the scale is the same in all directions at a point. Example: Mercator and the Universal Transverse Mercator (UTM).

What is an equal-area (equivalent) projection?

A projection that preserves true relative areas of regions, at the expense of distorting shapes. It is useful for thematic and statistical mapping where area comparison matters.

Describe the Universal Transverse Mercator (UTM) projection.

UTM is a conformal transverse cylindrical projection dividing the world into $6^\circ$-wide longitudinal zones (60 zones). Each zone has a central meridian with a scale factor of $0.9996$, giving Easting/Northing in metres with low distortion within the zone.

Why can no map projection be both conformal and equal-area simultaneously?

Because the Earth's curved surface cannot be flattened without distortion; preserving shape (conformality) requires equal scale in all directions at a point, while preserving area requires compensating scale changes — the two conditions are mutually exclusive except on a sphere of zero distortion.

What is a map datum?

A reference framework (a defined ellipsoid plus its orientation/position relative to the Earth) used as the basis for computing coordinates and elevations. It defines the origin and orientation of the coordinate system (horizontal datum) and the zero of heights (vertical datum).

What is Mean Sea Level (MSL) and why is it used as a datum?

MSL is the average level of the sea's surface over a long period (e.g. 19 years) measured at tide gauges, averaging out tides and waves. It serves as the conventional zero (vertical datum) for orthometric heights/elevations.

Define the geoid.

The geoid is the equipotential surface of the Earth's gravity field that best approximates global mean sea level, extended continuously under the continents. It is an undulating, irregular surface perpendicular everywhere to the direction of gravity (the plumb line).

Define a spheroid (reference ellipsoid) as used in geodesy.

A spheroid/reference ellipsoid is a smooth mathematical surface (an ellipse rotated about its minor axis) that approximates the geoid for computation, defined by its semi-major axis $a$ and flattening $f = \frac{a-b}{a}$, where $b$ is the semi-minor axis.

What is the difference between the geoid and the ellipsoid?

The geoid is the physical, irregular equipotential (gravity) surface; the ellipsoid is a regular mathematical surface fitted to it. Their vertical separation is the geoid undulation $N$, where ellipsoidal height $h = H + N$ ($H$ = orthometric height).

What is WGS-84?

The World Geodetic System 1984 is a global geocentric datum and reference ellipsoid used by GPS. It has semi-major axis $a = 6378137\,\text{m}$ and flattening $f = \frac{1}{298.257223563}$, with its origin at the Earth's centre of mass.

Relate ellipsoidal height $h$, orthometric height $H$, and geoid undulation $N$.

$$h = H + N$$ where $h$ is the height above the ellipsoid (from GPS), $H$ is the orthometric height above the geoid/MSL, and $N$ is the geoid–ellipsoid separation (geoid undulation).

In levelling, what is a 'level surface' versus a 'horizontal plane'?

A level surface is a curved surface everywhere perpendicular to the direction of gravity (parallel to MSL). A horizontal plane is tangent to the level surface at a point and is perpendicular to the plumb line only at that point.

Define datum, benchmark (BM), and reduced level (RL) in levelling.

Datum is the reference surface from which elevations are measured (often MSL). A benchmark is a fixed point of known elevation. Reduced Level is the elevation of a point above (or below) the chosen datum.

What is meant by Back Sight (BS), Fore Sight (FS), and Intermediate Sight (IS)?

BS is the first staff reading taken on a point of known RL after setting the instrument (used to find Height of Instrument). FS is the last reading before shifting the instrument. IS is any reading between BS and FS on points of unknown RL.

State the Height of Instrument (HI) and Rise-and-Fall methods of reducing levels.

HI method: $HI = RL_{\text{known}} + BS$ and $RL = HI - (\text{IS or FS})$. Rise-and-Fall method: compute rise/fall between consecutive points (previous reading minus next reading) and add algebraically to the previous RL. Rise-and-Fall provides a full arithmetic check.

State the arithmetic checks for the HI and Rise-and-Fall methods of levelling.

HI method: $\sum BS - \sum FS = \text{Last RL} - \text{First RL}$. Rise-and-Fall method: $\sum BS - \sum FS = \sum \text{Rise} - \sum \text{Fall} = \text{Last RL} - \text{First RL}$.

What is differential (or fly) levelling?

A levelling method to determine the difference in elevation between two points by a series of instrument setups and turning points, used when the points are far apart or have large height difference.

What is reciprocal levelling and when is it used?

A method used to determine accurate level difference across an obstacle (e.g. a river) where the instrument cannot be set midway. Readings are taken from both banks and averaged to eliminate errors due to curvature, refraction, and collimation.

Give the combined correction for curvature and refraction in levelling over a distance $d$.

$$C = 0.0673\,d^{2}$$ where $C$ is in metres and $d$ in kilometres. Curvature alone is $0.0785\,d^{2}$ (lowers the reading) and refraction is about $\frac{1}{7}$ of curvature (opposes it).

What is a prismatic compass and what does it measure?

A magnetic compass with a prism that allows simultaneous sighting of an object and reading of the graduated ring, used to measure the magnetic bearing (Whole Circle Bearing) of a line directly, $0^\circ$ to $360^\circ$.

Distinguish Whole Circle Bearing (WCB) from Reduced Bearing (RB).

WCB is measured clockwise from magnetic north, $0^\circ$ to $360^\circ$ (given by a prismatic compass). RB (Quadrantal Bearing) is measured from north or south, eastward or westward, $0^\circ$ to $90^\circ$ (given by a surveyor's compass), expressed e.g. as $N\,40^\circ E$.

Define magnetic declination.

The horizontal angle between the magnetic meridian (direction of the compass needle) and the true (geographic) meridian at a place. It is termed east (positive) or west (negative) and varies with location and time.

What is a theodolite used for?

A precise surveying instrument used to measure horizontal and vertical angles. It is used in traversing, triangulation, setting out works, measuring deflection angles, and (with a stadia setup) tacheometric distance/height determination.

What is a Total Station and what are its main uses?

An electronic instrument combining a theodolite (angle measurement) with an Electronic Distance Meter (EDM) and an onboard processor/data logger. It measures horizontal/vertical angles and slope distances, and computes coordinates, elevations, and areas, used for detailed topographic and setting-out surveys.

Compare a theodolite and a Total Station.

A theodolite measures only angles and requires separate distance measurement and manual computation. A Total Station measures angles AND distances electronically, stores data digitally, and computes 3D coordinates directly, making it faster and more accurate.

What is a tacheometer (tachometer) and its principle?

A transit theodolite fitted with a stadia diaphragm (three horizontal cross-hairs). It rapidly determines horizontal distance and elevation by observing the staff intercept, based on the stadia principle that the staff intercept is proportional to the distance.

Give the stadia (tacheometric) distance formula for a horizontal sight.

$$D = K\,s + C$$ where $D$ is horizontal distance, $s$ is the staff intercept (top minus bottom stadia readings), $K$ is the multiplying constant (usually $100$), and $C$ is the additive constant (often $\approx 0$ for internal-focusing instruments).

Give the tacheometric formulas for horizontal distance and vertical height with an inclined sight (angle $\theta$, staff held vertical).

$$D = K\,s\cos^{2}\theta + C\cos\theta$$ $$V = K\,s\,\frac{\sin 2\theta}{2} + C\sin\theta$$ where $D$ is horizontal distance, $V$ is vertical height between instrument axis and central hair, $s$ is staff intercept, and $\theta$ is the vertical angle.

What is trigonometric levelling?

An indirect method of determining the difference in elevation between points using measured vertical (zenith/altitude) angles and horizontal or slope distances, applying trigonometry, used in hilly terrain where direct levelling is impractical.

Give the basic formula for height difference in trigonometric levelling for a vertical angle $\alpha$ over horizontal distance $D$.

$$V = D\tan\alpha$$ where $V$ is the height of the object above the instrument axis. The RL of the point is then $RL = RL_{\text{instrument axis}} + V$ (plus curvature/refraction correction over long sights).

What is traversing in surveying?

A method of control surveying consisting of a series of connected lines (a traverse) whose lengths and directions (bearings/angles) are measured, to establish the relative positions (coordinates) of the traverse stations.

Differentiate a closed traverse from an open traverse.

A closed traverse starts and ends at the same point (loop) or between two known points, allowing checks on accuracy (angular and linear closure). An open traverse does not return or close on a known point, so it cannot be checked and is used only when unavoidable (e.g. roads, rivers).

Define latitude and departure of a traverse line.

For a line of length $L$ and WCB $\theta$: Latitude (N–S component) $= L\cos\theta$ and Departure (E–W component) $= L\sin\theta$. North latitudes and East departures are positive; South and West are negative.

What is the condition for a closed traverse to be geometrically closed, and what is closing error?

For a closed traverse $\sum \text{Latitude} = 0$ and $\sum \text{Departure} = 0$. The closing error $e = \sqrt{(\sum L)^{2} + (\sum D)^{2}}$, where $\sum L$ and $\sum D$ are the residual sums of latitudes and departures.

State Bowditch's (compass) rule for traverse adjustment.

Correction to a line's latitude (or departure) is proportional to its length: $$\text{Correction to latitude of a line} = -\sum L \times \frac{l}{\sum l}$$ where $l$ is the line length and $\sum l$ the perimeter. Used when angular and linear measurements are of equal precision.

What is triangulation?

A control survey method in which the area is covered by a network of connected triangles. One side (baseline) and all angles are measured precisely; the remaining sides are computed using the sine rule. It provides horizontal control over large areas.

What is trilateration?

A control survey method in which the lengths of all sides of a network of triangles are measured (using EDM) and the angles are computed trigonometrically. It is the distance-based counterpart of triangulation.

Compare triangulation and trilateration.

Triangulation measures angles (plus a baseline) and computes sides; trilateration measures all sides (distances) and computes angles. Trilateration became practical with EDM. Modern practice often uses a combined triangulation-trilateration network for strong, well-checked control.

In triangulation, how is an unknown side computed from the baseline using the sine rule?

With baseline $a$ opposite angle $A$, an unknown side $b$ opposite angle $B$ is $$b = a\,\frac{\sin B}{\sin A}.$$ All angles of each triangle are measured and the sine rule propagates lengths through the network.

What are the orders/classes of triangulation based on precision?

First (primary/highest precision, longest sides for national/geodetic control), Second (secondary), and Third (tertiary, for local detail) order. Precision is judged by average triangular closure and baseline accuracy, decreasing from first to third order.

Classify aerial photographs based on the camera axis orientation.

Vertical photographs (optical axis truly vertical, tilt $< 3^\circ$), tilted photographs (small unintentional tilt), and oblique photographs (axis intentionally inclined). Obliques are further divided into low oblique (horizon not shown) and high oblique (horizon shown).

What is the difference between a vertical and an oblique aerial photograph?

A vertical photo is taken with the camera axis pointing (nearly) straight down, approximating a map and useful for measurement. An oblique photo is taken with the axis tilted, covering more area but with variable scale and perspective distortion.

What is the difference between a low oblique and a high oblique photograph?

A low oblique photograph does not show the horizon (camera tilted modestly), while a high oblique photograph shows the horizon and a large area. Both have scale that varies across the image.

Define flying height in aerial photography.

Flying height ($H$) is the vertical height of the aircraft/camera above the chosen datum (often MSL). The height above the ground/terrain is $H - h$, where $h$ is the average terrain elevation above datum.

Give the scale of a vertical aerial photograph over flat terrain.

$$\text{Scale} = \frac{f}{H - h}$$ where $f$ is the camera focal length, $H$ is the flying height above datum, and $h$ is the terrain elevation above datum. Equivalently scale $= \frac{\text{photo distance}}{\text{ground distance}}$.

How does terrain elevation affect the scale of a vertical aerial photograph?

Scale increases for higher ground (smaller $H-h$) and decreases for lower ground (larger $H-h$), since scale $=\frac{f}{H-h}$. Thus a single vertical photo has variable scale unless the terrain is perfectly flat.

If focal length $f$ and photo/ground distances are known, how is flying height $H$ found over flat ground at elevation $h$?

From scale $=\frac{f}{H-h}=\frac{\text{photo distance}}{\text{ground distance}}$, rearrange to $$H = h + f\,\frac{\text{ground distance}}{\text{photo distance}}.$$

What is relief displacement in a vertical aerial photograph?

The radial shift of an image point caused by the object's elevation above datum; tall objects lean outward from the principal point (nadir). It is given by $$d = \frac{r\,h}{H}$$ where $r$ is the radial distance of the image from the centre, $h$ the object height, and $H$ the flying height above the base.

What is the contour interval and how is it related to map scale?

Contour interval is the constant vertical distance between successive contour lines. It is generally smaller for large-scale maps and flat ground, and larger for small-scale maps and steep/hilly terrain, balancing detail against clarity.

What does a graphical (linear) scale on a map provide that an RF does not?

A graphical scale is a line subdivided to show ground distances directly, and it remains valid even if the map is photographically enlarged or reduced (it scales with the map), whereas the printed RF would become incorrect after such resizing.