GATE Civil Engineering Geomatics Engineering Syllabus

Give the formulas to convert a line's length $L$ and whole-circle bearing $\theta$ into latitude and departure.

Latitude (N–S component) $= L\cos\theta$; Departure (E–W component) $= L\sin\theta$. Latitude is positive towards North, departure positive towards East.

How do you compute the length and bearing of a line given coordinates $(E_1,N_1)$ and $(E_2,N_2)$?

$L=\sqrt{(E_2-E_1)^{2}+(N_2-N_1)^{2}}$ and bearing $\theta = \tan^{-1}\!\left(\dfrac{E_2-E_1}{N_2-N_1}\right)$ (the departure over latitude), with quadrant adjusted by signs of $\Delta E$ and $\Delta N$.

Differentiate a whole-circle bearing (WCB) from a reduced bearing (quadrantal bearing).

WCB is measured clockwise from North, ranging $0^{\circ}$ to $360^{\circ}$. Reduced bearing is measured from North or South towards East or West, ranging $0^{\circ}$ to $90^{\circ}$, and requires a quadrant designation (e.g., $N\,40^{\circ}E$).

What is the difference between a fore bearing (FB) and a back bearing (BB)?

FB and BB of a line are observed from the two ends in opposite directions. They differ by $180^{\circ}$: $\text{BB}=\text{FB}\pm180^{\circ}$ (add if FB $<180^{\circ}$, subtract if FB $>180^{\circ}$).

Define magnetic declination.

Magnetic declination is the horizontal angle between true (geographic) north and magnetic north at a place. True bearing $=$ magnetic bearing $\pm$ declination (E declination added, W declination subtracted).

Name the principal methods of measuring horizontal distance directly and an indirect (optical) method.

Direct: chaining/taping. Indirect/optical: tacheometry (stadia), and modern electronic distance measurement (EDM). Pacing and odometer are approximate methods.

State the correction for a tape that is too long (standardization/length correction).

If the tape's actual length is $L'$ and nominal length $L$, correction is added for a long tape: corrected distance $= \text{measured} \times \dfrac{L'}{L}$. A too-long tape gives a measured distance that is too short, so the correction is positive.

Give the formula for the temperature correction in taping.

$C_t = \alpha (T_m - T_0) L$, where $\alpha$ is the coefficient of thermal expansion, $T_m$ the field temperature, $T_0$ the standardization temperature, and $L$ the measured length. Positive when $T_m>T_0$.

Give the formula for the sag correction in taping (catenary).

$C_{sag} = \dfrac{w^{2}L_1^{3}}{24P^{2}}$ (subtractive), where $w$ is weight per unit length, $L_1$ the span length, and $P$ the applied pull. Sag always makes the measured distance too long, so the correction is negative.

Give the formula for the slope correction in taping.

For a measured slope length $L$ over height difference $h$: $C_{slope} = -\dfrac{h^{2}}{2L}$ (approximate), or exactly horizontal distance $= L\cos\theta$ where $\theta$ is the slope angle. Always subtractive.

State the basic tacheometric (stadia) distance formula for a horizontal sight with an external focusing telescope.

$D = K\,s + C$, where $s$ is the staff intercept (difference of stadia hair readings), $K$ is the multiplying constant (usually $100$), and $C$ is the additive constant (usually $0$ for internal/anallactic, small for external).

What is levelling, and what is its fundamental purpose?

Levelling is the branch of surveying that determines relative heights (elevations) of points. Its purpose is to establish a level surface/datum and find differences in elevation, e.g., for contours, profiles, and earthwork.

Define a level surface, a horizontal plane, and a datum.

A level surface is a curved surface everywhere perpendicular to the plumb line (parallel to mean sea level). A horizontal plane is tangent to the level surface at a point. A datum is the reference surface from which elevations are reckoned (commonly mean sea level).

Define benchmark (BM), backsight (BS), foresight (FS), and intermediate sight (IS).

BM is a fixed point of known elevation. BS is the first staff reading on a point of known RL (taken after setting up). FS is the last reading before shifting the instrument. IS is any sight between BS and FS on points of unknown RL.

State the formulas for Height of Instrument (HI) and Reduced Level (RL) in the height-of-collimation method.

$\text{HI} = \text{RL of point} + \text{BS}$; $\text{RL of next point} = \text{HI} - \text{IS or FS}$. The rise-and-fall method instead uses successive differences of readings.

State the arithmetic check for the height-of-collimation (line-of-collimation) method.

$\sum \text{BS} - \sum \text{FS} = \text{Last RL} - \text{First RL}$. (The rise-and-fall method adds the check $\sum\text{Rise}-\sum\text{Fall}=\text{Last RL}-\text{First RL}$.)

Derive the combined correction for earth's curvature and refraction in levelling.

Curvature lowers the apparent reading by $C_c=\dfrac{d^{2}}{2R}\approx0.0785\,d^{2}$ (m, $d$ in km); refraction raises it by about $\tfrac{1}{7}$ of curvature. Combined correction $= -0.0673\,d^{2}$ m (the sight reads too high, so RL correction is subtractive).

What is reciprocal levelling and when is it used?

Reciprocal levelling takes observations from both banks of an obstacle (e.g., a river) so that the effects of curvature, refraction, and collimation error cancel. It is used when the instrument cannot be set midway between two distant points.

What is trigonometric levelling?

Trigonometric levelling determines elevation differences indirectly from measured vertical (zenith/altitude) angles and horizontal or slope distances, using trigonometry instead of direct staff reading. Useful for inaccessible or elevated points.

Give the basic formula for the height of an object by trigonometric levelling when the horizontal distance $D$ and vertical angle $\alpha$ are known.

$V = D\tan\alpha$, so the elevation of the top $= \text{RL of instrument axis} + D\tan\alpha$. With curvature and refraction: $V = D\tan\alpha + 0.0673\,D^{2}$ ($D$ in km).

In trigonometric levelling, how is the height of an inaccessible object found using two instrument stations in line (different instrument-axis levels)?

With horizontal angles $\alpha_1,\alpha_2$ from two stations $d$ apart in the same vertical plane, $V=\dfrac{d\tan\alpha_1\tan\alpha_2}{\tan\alpha_2-\tan\alpha_1}$ (for equal axis levels); the object height $=$ axis RL $+V$.

Define traversing and name its two principal types.

Traversing is a method of control surveying in which a series of connected lines (legs) is established, measuring lengths and directions/angles. Types: open traverse (does not close) and closed traverse (returns to start or connects known points).

What is the check on angular closure for a closed-loop traverse with $n$ sides?

Sum of interior angles should equal $(2n-4)\times90^{\circ}$; sum of exterior angles $=(2n+4)\times90^{\circ}$. The angular misclosure is distributed equally or by judgment among the angles.

What is the closing error of a traverse and how is its relative precision expressed?

Closing error $e=\sqrt{(\sum L)^{2}+(\sum D)^{2}}$, where $\sum L$ and $\sum D$ are the algebraic sums of latitudes and departures (which should be zero). Relative precision $=\dfrac{e}{\text{perimeter }P}=1:\dfrac{P}{e}$.

State Bowditch's (compass) rule for adjusting a traverse.

Correction to the latitude/departure of a line is proportional to its length: $\text{Correction} = -(\text{total error})\times\dfrac{\text{length of line}}{\text{perimeter}}$. Used when angular and linear precision are comparable.

State the Transit rule for traverse adjustment and when it is preferred over Bowditch's rule.

Correction to latitude $=-\Delta L\dfrac{\text{latitude of line}}{\sum|\text{latitudes}|}$ and similarly for departure. Preferred when angular measurements are more precise (more reliable) than linear measurements.

What is triangulation, and what is its fundamental principle?

Triangulation is a control survey forming a network of triangles in which one side (baseline) and all angles are measured; remaining sides are computed by the sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$. It provides a framework of accurate control points.

Differentiate triangulation from trilateration.

In triangulation, angles are measured and sides are computed from a baseline. In trilateration, the sides (distances) are measured directly (typically with EDM) and angles are computed. Modern surveys often combine both (triangulateration).

What is the 'strength of figure' in triangulation?

Strength of figure is a measure (related to the distribution of angles) of the precision with which lengths are computed through a chain of triangles; it depends on the geometry and the number of conditions. Well-conditioned triangles (angles between $30^{\circ}$ and $120^{\circ}$) give greatest strength.

What is a total station and what does it integrate?

A total station is an electronic surveying instrument integrating an electronic theodolite (for horizontal and vertical angles), an EDM (for distances), and an onboard microprocessor/data logger to compute and store coordinates, elevations, and reductions.

On what physical principle does an EDM in a total station measure distance?

EDM measures distance by transmitting a modulated electromagnetic (infrared/laser) wave to a reflector and measuring the phase difference (or time of flight) of the returned signal: $D=\dfrac{1}{2}\,c\,t$ or $D=\dfrac{n\lambda+\Delta\lambda}{2}$ for phase methods.

Differentiate the resection and the free-station (free setup) method using a total station.

Resection determines the coordinates of the instrument station by observing two or more known control points. A free-station setup places the instrument at an arbitrary point and uses resection so the operator need not occupy a known point.

What is a horizontal curve and what are its two main types?

A horizontal curve connects two intersecting straights (tangents) in plan to provide a gradual change in direction. Main types: simple circular curve, compound curve, reverse curve, and transition (spiral) curve.

For a simple circular curve of radius $R$ and deflection angle $\Delta$, give the tangent length and the length of curve.

Tangent length $T=R\tan\dfrac{\Delta}{2}$; length of curve $L=\dfrac{\pi R\Delta}{180^{\circ}}=R\Delta_{\text{rad}}$. The long chord $=2R\sin\dfrac{\Delta}{2}$.

Give the formulas for the external distance and mid-ordinate (apex distance) of a simple circular curve.

External distance $E=R\left(\sec\dfrac{\Delta}{2}-1\right)$; mid-ordinate (versine) $M=R\left(1-\cos\dfrac{\Delta}{2}\right)$.

Relate the degree of curve $D$ to the radius $R$ (arc definition, $30\,\text{m}$ chord/arc).

For a $30\,\text{m}$ standard arc, $R=\dfrac{1718.9}{D}$ m (since $D$ is the central angle subtended by a $30\,\text{m}$ arc, $R=\dfrac{30\times180}{\pi D}$). The degree of curve is inversely proportional to the radius.

What is a transition curve and why is it provided?

A transition (easement) curve has a radius that gradually changes from infinity (straight) to $R$ (circular curve). It is provided to introduce centrifugal force gradually, allow gradual application of superelevation, improve comfort and safety; the ideal form is the clothoid (spiral) where $L\,R=$ constant.

What is a vertical curve and which geometric form is used?

A vertical curve connects two different grades (gradients) in elevation to provide a smooth transition. A parabola is used because it gives a constant rate of change of grade and equal horizontal spacing of offsets.

Differentiate a summit (crest) curve from a sag (valley) curve.

A summit/crest curve is convex upward, formed where the algebraic difference of grades makes the centre higher (e.g., $+$ grade meeting $-$ grade); design governed by sight distance. A sag/valley curve is concave upward; design governed by headlight sight distance and comfort/drainage.

What is remote sensing?

Remote sensing is the science of acquiring information about an object or area from a distance, without physical contact, by measuring the electromagnetic radiation reflected or emitted from it, typically using sensors on satellites or aircraft.

Differentiate active and passive remote sensing.

Passive remote sensing uses naturally available energy (e.g., reflected sunlight, emitted thermal radiation); the sensor only records. Active remote sensing supplies its own energy source (e.g., RADAR, LiDAR) and records the reflected return, enabling night/all-weather imaging.

What is spatial resolution versus spectral resolution in remote sensing?

Spatial resolution is the smallest ground area (pixel size) a sensor can distinguish. Spectral resolution is the number and width of wavelength bands the sensor can detect; finer (narrower, more) bands give higher spectral resolution. (Temporal resolution = revisit interval; radiometric = bit depth.)

What is a Geographic Information System (GIS)?

A GIS is a computer-based system to capture, store, manage, analyze, and display spatially referenced (geographic) data, integrating spatial (location) data with attribute (descriptive) data for spatial analysis and decision-making.

Differentiate the raster and vector data models in GIS.

Raster represents space as a grid of cells/pixels, each holding a value—good for continuous data (imagery, elevation). Vector represents features as points, lines, and polygons defined by coordinates—good for discrete features with sharp boundaries; it is more compact and topologically precise.

What are the basic components of a GIS?

Hardware, software, data (spatial + attribute), people (users/experts), and methods/procedures. These integrate to capture, manage, analyze, and present geographic information.

In aerial photogrammetry, give the relationship between photo scale, focal length, and flying height.

For a vertical photograph over flat terrain, scale $=\dfrac{f}{H-h}$, where $f$ is the camera focal length, $H$ is the flying height above datum, and $h$ is the average ground elevation. Equivalently scale $=\dfrac{\text{photo distance}}{\text{ground distance}}$.

How do you compute the flying height $H$ above ground for a desired photo scale $S=1:n$ with focal length $f$?

From scale $=\dfrac{f}{H'}$ where $H'=H-h$ is height above ground, $H'=f\times n=\dfrac{f}{S}$. Thus flying height above the datum $H=f\,n+h$, where $h$ is the average terrain elevation.

What is relief displacement on a vertical aerial photograph, and how is it computed?

Relief displacement is the radial shift of an image point (away from the principal point) caused by the object's height above datum: $d=\dfrac{r\,h}{H}$, where $r$ is the radial distance of the image from the principal point, $h$ the object height, and $H$ the flying height above the base.

Define overlap (forward and side lap) in aerial photography and typical values.

Forward (end) lap is the overlap between successive photos along a flight line, typically about $60\%$, required for stereoscopic (3D) coverage. Side lap is the overlap between adjacent flight strips, typically about $30\%$. These ensure complete stereo coverage.

What is parallax in stereo-photogrammetry, and how is object height obtained from it?

Parallax is the apparent displacement of a point due to change in observation position between two photos. Height $h=\dfrac{H\cdot\Delta p}{p+\Delta p}$, where $H$ is flying height, $p$ the parallax at datum, and $\Delta p$ the parallax difference; greater parallax means a higher/closer object.

Define accuracy and precision in measurement, and state how they differ.

Accuracy is closeness of a measured value to the true value (freedom from systematic error/bias). Precision is closeness of repeated measurements to one another (small random scatter). Measurements can be precise but inaccurate if a systematic error is present.