Surveying covers the measurement and mapping of the earth's surface — from classification and chain surveying fundamentals, through compass surveying, levelling, theodolite traversing, and tacheometry, to field astronomy, GPS, map preparation, photogrammetry, remote sensing, layout surveys for civil structures, and setting out of curves. Every design formula, IS/IRC code reference, diagram, solved example and exam-pattern table is included.

Learning Objectives

After studying this chapter you will be able to:

• Classify surveys and apply chain-surveying corrections (slope, temperature, pull, sag, wrong chain length)
• Convert bearings, detect and correct local attraction, and adjust a closed traverse using Bowditch's and Transit rules
• Reduce levels using the HI and Rise-and-Fall methods, and apply curvature and refraction corrections
• Measure horizontal and vertical angles with a theodolite and compute traverse closure
• Apply stadia and tangential tacheometric formulae to determine distance and elevation
• Apply the astronomical triangle to find azimuth/altitude, and understand GPS positioning and error sources
• Interpret contour maps and select an appropriate method of plane-table surveying or contouring
• Compute photogrammetric scale, relief displacement, and parallax-based heights
• Apply NDVI and other spectral indices, and distinguish active from passive remote sensing
• Design and set out simple circular curves, transition curves, and vertical curves for roads and railways

Exam Relevance

Concept Map

Where This Fits

Prerequisite: Environmental Engineering (network layout and pipeline alignment there rely on the same field measurement principles developed here). Leads to: Highway Engineering, which directly builds on curve setting-out, gradient, and alignment survey concepts covered in this chapter.

1.1 Classification of Surveys

1.2 Principles of Surveying

• Working from whole to part: Establish large framework first, then fill in detail — prevents error accumulation
• Locating a point by two measurements: Any point fixed by minimum 2 independent measurements (angles, distances, offsets)
• Check: Always check measurements; use redundant observations

1.3 Chain Surveying — Instruments

1.4 Errors in Chain Surveying and Corrections

Chain and tape measurements require several systematic corrections before being used for design: a correction for the chain/tape's actual length differing from its nominal length, a slope correction to convert measured slope distance to horizontal distance, a temperature correction (the tape expands or contracts relative to the temperature it was standardised at), a pull (tension) correction (a stretched tape reads slightly long), and a sag correction (an unsupported tape hanging in a catenary between supports always measures longer than the true straight-line distance, so this correction is always negative). When the applied pull is increased specifically to make the pull correction exactly cancel the sag correction, the result is called the normal tension.

1.5 Offsets — Perpendicular and Oblique

Offsets are short measurements taken perpendicular (or oblique) from the chain line to locate detail points. An angular error in laying off a perpendicular offset produces a positional error that grows with the square of the offset length, which is why accurate work limits how long an offset may be. Oblique offsets (measured from two chain points to a single object) are located using the cosine rule once both distances and the included angle are known.

1.6 Triangulation and Trilateration

Triangulation establishes control by measuring all the angles of a network of triangles along with just one baseline distance; trilateration instead measures all the side lengths directly (feasible today with EDM/GPS) without needing angle observations at all. Most modern control surveys combine both techniques. A well-conditioned ("strong") triangle keeps every angle between 30° and 120°, with 60° (equilateral) as the ideal — angles below 30° are elongated and let small angular errors propagate into large position errors.

2.1 Bearings

A Whole Circle Bearing (WCB) is measured clockwise from north through the full 0°–360° range. A Quadrantal (Reduced) Bearing is instead measured from north or south toward east or west, always within 0°–90°, and is converted from WCB using a quadrant-dependent rule. The back bearing of a line is obtained from its forward bearing by adding or subtracting 180°. Declination is the angle between magnetic north and true north at a location, and must be added to or subtracted from a magnetic bearing (depending on whether declination is east or west) to obtain the true bearing.

2.2 Local Attraction

Local attraction is a disturbance of the magnetic needle caused by nearby magnetic influences (iron objects, electric cables, steel structures), and it affects every bearing read at that particular station equally. It is detected whenever the forward bearing and back bearing of a line do not differ by exactly 180°. Correction proceeds by first finding a station free from local attraction (where FB−BB=180° exactly), then propagating the error found at each subsequent affected station back through the traverse.

2.3 Traversing with Compass

Each traverse line is resolved into its latitude (north-south component) and departure (east-west component) using its bearing and length. For a properly closed traverse, the algebraic sum of all latitudes and the algebraic sum of all departures should each be zero; any non-zero result is the closing error, whose magnitude (as a fraction of the total traverse length) expresses the survey's precision.

2.4 Adjustments of Traverse

Bowditch's (Compass) Rule distributes the closing error to each line's latitude and departure in proportion to that line's own length — appropriate when angular and linear measurement accuracy are comparable. The Transit Rule instead distributes the error in proportion to the magnitude of each line's latitude/departure component — used when angular accuracy is much better than linear accuracy (typical for a theodolite traverse), since it avoids over-correcting lines that are already well-oriented.

2.5 Gale's Traverse Table and Area Calculation

After adjustment, the independent coordinates (northing and easting) of each traverse station are computed by successively accumulating the corrected latitudes and departures from a starting point. The enclosed area is then found directly from these coordinates by the coordinate (or "double meridian distance") method — no separate area survey is needed. Where the boundary is irregular rather than a closed traverse, area under an offset curve can instead be approximated by the trapezoidal rule, or more accurately by Simpson's 1/3 rule (which requires an even number of equal divisions).

3.1 Basic Terms

3.2 Methods of Levelling Reduction

The Height of Instrument (HI) Method computes the elevation of the horizontal line of sight at each instrument setup, then subtracts each staff reading from it to get that point's RL — simple and fast, but its arithmetic check (ΣBS−ΣFS = Last RL−First RL) does not catch errors made in intermediate sight readings, a significant limitation when many IS readings are taken. The Rise and Fall Method instead computes the rise or fall between every consecutive pair of readings and accumulates RL directly; its three-way arithmetic check (ΣBS−ΣFS = ΣRise−ΣFall = Last RL−First RL) does catch IS errors, making it the preferred method for precise work despite requiring more calculation.

3.3 Types of Levelling

3.4 Curvature and Refraction Corrections

Over long sight distances, the earth's curvature makes a level line appear to fall away below the true horizontal, causing a distant staff reading to appear too large — the curvature correction accounts for this. Atmospheric refraction bends the line of sight downward (toward denser air near the ground), partially compensating for curvature by roughly one-seventh of its magnitude. Since curvature and refraction act in opposite directions but curvature dominates, their combined net effect still makes distant readings appear too high, so the combined correction must still be subtracted from the observed staff reading.

3.5 Errors and Permanent Adjustments of Level

The two-peg test detects an error in the line of collimation (the line of sight not being exactly horizontal when the bubble is centred). By first setting the instrument exactly midway between two staff points — where a collimation error affects both readings equally and cancels out — the true elevation difference is found. Moving the instrument next to one of the points reveals the collimation error directly, since any discrepancy from the true difference must now come from the tilted line of sight (no longer cancelled by symmetry).

3.6 Precise Levelling and Trigonometric Levelling

Precise (geodetic) levelling uses an invar staff, a parallel-plate micrometer, short balanced sight lengths, and careful field procedure to achieve millimetre-level accuracy over long distances — essential for establishing a national vertical control network. Trigonometric levelling instead computes elevation difference from a measured vertical angle and either a horizontal or slope distance; for short sights the simple trigonometric relation suffices, but for longer sights the same curvature-and-refraction correction used in direct levelling must be added.

4.1 Theodolite — Components and Types

A theodolite is a precision optical instrument for measuring horizontal and vertical angles. The vernier theodolite (least count ~20″) has largely been superseded by the optical/micrometer theodolite (least count ~1″), and both are increasingly replaced by the electronic total station, which combines a theodolite with an EDM (electronic distance measurement) in a single unit and stores data digitally for later download — eliminating manual booking errors.

4.2 Temporary Adjustments of Theodolite

Every time a theodolite is set up, three temporary adjustments must be carried out in sequence: centring (bringing the instrument exactly over the survey station, using a plumb bob or optical plummet), levelling (using the plate bubble and the standard two-screw-then-rotate-90°-then-third-screw procedure to bring the vertical axis truly vertical), and elimination of parallax (focusing the eyepiece on the cross-hairs and the objective on the target, so the image and cross-hairs appear to move together when the eye shifts slightly).

4.3 Measurement of Horizontal Angles

The method of repetition accumulates an angle over several repeated turns before taking a single final reading, dividing by the number of repetitions to get the average — this averages out graduation errors and increases precision roughly in proportion to the square root of the number of repetitions. The method of reiteration (directions method) instead takes direct and reverse (Face Left / Face Right) readings on all signals at a station; averaging FL and FR readings eliminates line-of-collimation error, trunnion-axis error, and eccentricity error simultaneously, making it the most systematic method for triangulation work.

4.4 Measurement of Vertical Angles

A vertical angle is likewise measured on both Face Left and Face Right and averaged to eliminate index error — the small residual error remaining when the vertical circle does not read exactly 0° (or 90° for zenith angle) when the line of sight is truly horizontal. The zenith angle is measured from the vertical (rather than the horizontal), so it equals 90° minus the vertical angle.

4.5 Theodolite Traversing

In the included angle method, the interior (or exterior) angle is measured directly at each traverse station; the sum of interior angles of a closed polygon traverse must equal \((n-2)\times180°\), giving a direct arithmetic check on the whole traverse. The deflection angle method instead measures the angle between the prolongation of the back line and the forward line at each station — common in route surveys (roads, railways) where the traverse follows a long, roughly straight alignment rather than closing into a polygon. If one side's length or bearing is accidentally omitted from the field measurements, it can be recovered afterward from the traverse's closing-error geometry.

4.6 Electronic Distance Measurement (EDM)

EDM instruments send an electromagnetic wave (microwave, infrared, or laser) to a retroreflector and measure the phase difference between the outgoing and returning signal to compute distance. Two systematic error types must be calibrated out: an additive (prism) constant — a fixed offset introduced by the reflector — and a scale error, which grows proportionally with distance and is caused mainly by atmospheric refraction (varying with pressure, temperature, and humidity along the sight path), correctable via an atmospheric correction formula.

5.1 Principle of Tacheometry

Tacheometry allows simultaneous determination of horizontal distance and elevation difference using a theodolite and graduated staff, without any tape measurement. It is rapid but less accurate than chaining — well suited to topographic surveys, contour mapping, and detail surveys where speed matters more than millimetre precision.

5.2 Stadia Method

The stadia method uses two additional horizontal cross-hairs (stadia hairs) in the telescope, symmetrically placed above and below the main cross-hair. The vertical intercept these hairs subtend on a graduated staff — the staff intercept — is directly proportional to the horizontal distance to the staff, via a simple multiplying constant fixed by the telescope's optical design (internal-focusing telescopes have essentially zero additive constant, simplifying the formula considerably).

5.3 Stadia Formulae

For a horizontal line of sight, distance is simply the staff intercept multiplied by the stadia (multiplying) constant. When the line of sight is inclined at angle α (as is usually the case when sighting up or down sloping ground), both the horizontal distance and the vertical component must be corrected by trigonometric factors of \(\cos\alpha\) — because the staff, held vertically, is no longer perpendicular to the inclined line of sight, and the effective staff intercept the formula needs is the intercept projected perpendicular to the sight line.

5.4 Tangential Method (No Stadia Hairs)

When stadia hairs are unavailable (or an ordinary theodolite without a stadia diaphragm is used), distance can still be found tacheometrically by observing two separate vertical angles — one to the top and one to the foot of a staff of known height — and applying trigonometry. This tangential method requires more careful angle observation than the stadia method since the accuracy depends entirely on precise angular measurement rather than an optical intercept.

5.5 Subtense Bar

A subtense bar is a fixed-length horizontal bar (typically 2 m) set up at the point whose distance is to be found; the theodolite at the observing station measures the small horizontal angle the bar subtends, from which distance follows directly (for small angles, distance is inversely proportional to the subtended angle). This method achieves good accuracy (about 1 in 1000) for moderate distances without needing any staff reading at all.

6.1 Celestial Sphere Concepts

6.2 Astronomical Triangle and Formulae

The PZS (astronomical) triangle is formed on the celestial sphere by the pole (P), the observer's zenith (Z), and the observed star (S). Its three sides — co-latitude, zenith distance, and co-declination — and three angles — hour angle, azimuth, and parallactic angle — are related by spherical trigonometry (the cosine rule for spherical triangles), giving the fundamental altitude formula that connects a star's observed altitude to the observer's latitude, the star's declination, and the hour angle. Rearranging the same relationship gives the azimuth of the body, which is what a surveyor actually needs to establish true north.

6.3 Determination of True North

• By observation of Polaris (North Star): Most accurate in the N hemisphere; observed at elongation (greatest E or W) when the azimuth correction is maximum but its rate of change with time is minimum — reducing timing errors
• By solar observation: Measure altitude and time; compute azimuth from the astronomical formulae
• By equal altitudes of the sun: Sun observed in the morning and afternoon at equal altitudes; the bisector of the two observed directions gives true north directly, without needing precise time or the astronomical formula at all
• By magnetic bearing plus declination: Least accurate; relies on a compass and the local declination value

6.4 Global Positioning System (GPS)

GPS (along with GLONASS, Galileo, and India's own NavIC) is a satellite-based positioning system relying on trilateration (not triangulation) — the receiver computes its position from measured ranges to multiple satellites, not from angles. Because the receiver's own clock is not synchronised to the same precision as the atomic clocks onboard the satellites, a fourth satellite is needed in addition to the three needed for basic 3D position, specifically to solve for the receiver clock error as a fourth unknown.

6.5 GPS Error Sources

The Dilution of Precision (DOP) quantifies how the satellite geometry amplifies these ranging errors into a position error — a low PDOP (well-spread-out satellites) gives a much more reliable fix than a high PDOP (satellites clustered together), even with identical individual ranging accuracy.

6.6 NavIC (Navigation with Indian Constellation)

NavIC is India's own regional (not global) navigation satellite system, covering the Indian subcontinent and roughly 1500 km of surrounding region using a constellation of geostationary and geosynchronous satellites, providing accuracy comparable to (and in some regional respects better than) standard GPS.

7.1 Plane Table Surveying

Plane table surveying is a graphical method in which the map is drawn directly in the field simultaneously with the survey, eliminating separate office plotting and calculation. Directions are sighted with the alidade, and distances are measured by tacheometry or tape. Four principal methods are used depending on the field situation — radiation (rays drawn from one station to every detail point), intersection (rays from two known stations intersect to fix an inaccessible point), traversing (the table itself is moved and re-oriented station to station), and resection (the instrument station itself is located from rays back to points already plotted on the sheet — the classic "three-point problem").

7.2 Contour Lines

A contour line joins all ground points of equal elevation, giving a two-dimensional representation of three-dimensional terrain. The contour interval is chosen based on the map scale and the required level of engineering detail — smaller-scale maps use a larger interval, while detailed engineering plans use a much finer interval (0.5–2 m). The horizontal distance between successive contours (the horizontal equivalent) directly indicates ground slope: closely spaced contours mean steep terrain, widely spaced contours mean gentle terrain.

Characteristics of Contours

• Contour lines never cross or branch (except for overhanging cliffs)
• Every contour line closes on itself, either within or outside the map
• Closely spaced contours → steep slope; widely spaced → gentle slope
• Contours crossing a valley form V-shapes pointing upstream (uphill)
• Contours crossing a ridge form V-shapes pointing downhill
• Uniformly spaced = uniform slope; concave up = slope flattens upward
• Circular contours with decreasing values outward → hill; increasing → depression

7.3 Methods of Contouring

The direct method physically locates points of exactly known elevation on the ground and plots them directly — the most accurate but slowest approach, generally reserved for high-precision work. The indirect (cross-section/grid) method instead takes spot heights at a regular grid or along cross-sections, and interpolates the contour lines afterward — much faster and the most commonly used method for open ground. For large-area mapping, aerial photogrammetry is fastest, deriving contours from stereo photograph pairs.

7.4 Uses of Contour Maps

• Determine slope of land at any point
• Draw profile/section along any line
• Locate a route of a given gradient between two points
• Estimate volume of earthwork (for roads, embankments, reservoirs)
• Delineate catchment area of a watershed
• Determine intervisibility of two points (line of sight)

8.1 Introduction and Types

Photogrammetry is the science and art of obtaining reliable measurements from photographs, extracting geometric and spatial information from images without direct physical contact with the object. It ranges from terrestrial (close-range) photogrammetry for architectural or industrial measurement, through aerial photogrammetry for topographic mapping, to UAV/drone and satellite photogrammetry for site surveys and global-scale mapping respectively.

8.2 Aerial Camera and Photo Geometry

The scale of a vertical aerial photograph is simply the ratio of the camera's focal length to the flying height above the ground — for undulating terrain, the local scale actually varies point to point depending on each point's own elevation above the datum, since flying height is effectively reduced for higher ground. A critical distortion unique to aerial photographs is relief displacement: any object with height above the datum appears radially displaced outward from the photograph's centre (principal point), an effect that grows with both the object's height and its distance from the centre — this is precisely what allows building or tree heights to be measured from a photograph, but it also means aerial photos (unlike orthophotos) are not true planimetric maps.

8.3 Overlap and Coverage

Successive photographs along a flight line must overlap substantially (typically 60% forward overlap) so that every ground point appears on at least two photographs, which is essential for stereoscopic (3D) viewing and measurement. Adjacent flight lines also need a smaller side overlap (typically 30%) to ensure full coverage without gaps. The base-to-height ratio (B/H) — the air base (distance between successive exposure stations) relative to flying height — governs stereoscopic measurement precision, with an optimal range of about 0.5–0.6.

8.4 Stereoscopy and DEM Generation

When the same point appears on two overlapping photographs taken from different exposure stations, its image position differs slightly between the two photos — this parallax is directly related to the point's elevation. A stereoscopic instrument (or software) measures this parallax difference between a point of unknown elevation and a reference datum point, from which the elevation follows directly — this is the fundamental principle behind generating a Digital Elevation Model (DEM) from a stereo pair.

8.5 Orthophoto and Digital Elevation Model (DEM)

• Orthophoto: Aerial photo corrected for relief displacement and tilt — like a true planimetric map; scale is uniform throughout
• DEM (Digital Elevation Model): Grid of elevation values; generated from stereo pairs
• DSM (Digital Surface Model): Elevations include buildings, trees (first return lidar)
• DTM (Digital Terrain Model): Bare earth elevations (filtered lidar ground returns)
• LiDAR: Light Detection and Ranging; airborne laser scanning; point cloud; density 1–30 pts/m²; accuracy ±5–15 cm

9.1 Principles of Remote Sensing

Remote sensing acquires information about an object or area from a distance — typically from a satellite or aircraft sensor — without physical contact, by exploiting electromagnetic radiation (EMR) reflected or emitted from the Earth's surface. Different EMR wavelength bands reveal different surface properties: visible and near-infrared for general reflectance and vegetation, thermal infrared for temperature, and microwave for all-weather imaging capable of penetrating cloud cover.

9.2 Types of Remote Sensing Systems

9.3 Image Resolution Types

9.4 Indian Remote Sensing Satellites

9.5 Image Interpretation and Spectral Indices

Spectral indices combine reflectance from specific bands to isolate a particular land-cover property. The NDVI (Normalized Difference Vegetation Index) exploits the fact that healthy vegetation strongly reflects near-infrared while absorbing red light for photosynthesis, giving a high positive index value for dense healthy vegetation and low values for bare soil, water, or snow. Land-cover classification is done either by supervised classification (an analyst defines training sites of known class, and the algorithm classifies the rest) or unsupervised classification (clustering algorithms like ISODATA or k-means group pixels automatically, with classes assigned meaning only after the fact).

9.6 GIS (Geographic Information System)

GIS integrates spatial data (maps, satellite imagery) with attribute data (population, soil type, land use) to enable overlay analysis, buffering, spatial querying, network analysis, and 3D modelling. Spatial data is stored either as vector data (points, lines, polygons — precise boundaries, compact file size) or raster data (a grid of cells, each with one value — the natural format for satellite imagery and DEMs). Since India's older Survey of India toposheets used the Everest 1830 datum with a Polyconic projection, while GPS and modern systems use WGS-84 with UTM, a datum/projection transformation is often needed when combining old and new spatial data.

10.1 Road / Highway Alignment Survey

A road alignment is finalised through progressively more detailed stages: a reconnaissance survey walks the route to identify obstacles and alternatives; a preliminary survey runs a traverse and levels along each alternative to draw a longitudinal section; and a location survey then works out the detailed cross-sections and volume estimates for the chosen alignment before pegging the centre line at regular intervals. The ruling gradient is the maximum gradient normally permitted for the terrain and road category — steeper gradients require special justification, and gradients on horizontal curves must additionally be reduced (compensated) since the combined effect of curve resistance and steep grade would otherwise strain vehicles beyond the intended design limit.

10.2 Railway Alignment Survey

Railway ruling gradients are generally much flatter than road gradients (since trains cannot climb steep grades), and — like roads — must be compensated on curves to offset the extra resistance a curve imposes on a train. Track geometry standards fix the gauge (rail spacing) and the maximum superelevation permitted on curves, with a required transition length over which the cant (superelevation) is introduced gradually, governed by the rate of change of cant that passengers can comfortably tolerate.

10.3 Canal Alignment Survey

Canal alignment typically follows a contour or the natural watershed so that water flows by gravity throughout. The Full Supply Level (FSL) must follow the required hydraulic gradient along the entire route, while the bed level (below FSL by the canal's design depth) must never rise above the FSL — a canal physically cannot flow uphill. Earthwork along the alignment is estimated from cross-sections at regular intervals, with cut sections (where FSL sits below existing ground) and fill sections (embankments, where FSL sits above ground) computed separately.

10.4 Bridge Survey

A bridge survey requires a hydrological survey of the stream (cross-sections, flood levels, and bed profile both upstream and downstream) to establish the High Flood Level (HFL) and estimate afflux — the rise in backwater level caused by the bridge piers obstructing flow, which must be kept small to avoid upstream flooding. Pier positions are set out from control points on each bank, typically using the intersection method with two theodolites (or a single total station) when direct measurement across the water is impractical.

10.5 Building Layout

Setting out a building begins from an established baseline (a road edge or property boundary), from which the building's corners are located by offset and then squared using a right-angle method — either the classical 3-4-5 triangle or an optical square (which uses two mirrors at 45° to sight a perpendicular direction without any calculation at all). The layout is verified by checking that the diagonals of the resulting rectangle are equal, confirming all four corners are correctly square.

10.6 Culvert Survey and Layout

A culvert site is ideally chosen where a road or railway crosses a drainage channel at a right angle (a "square" culvert), since a skewed crossing is structurally and hydraulically more complex, though sometimes unavoidable if the culvert must follow the natural drainage direction. The minimum road level at the culvert must clear the stream's full supply level plus afflux plus freeboard plus the pavement thickness — an important check to avoid the road itself being overtopped during a flood event.

11.1 Types of Curves

11.2 Elements of Simple Circular Curve

A simple circular curve connects two straight tangents deflecting by angle Δ at their point of intersection (PI). Its key elements — tangent length, arc length, long chord, mid-ordinate, and external distance — are all functions of just the radius R and the deflection angle Δ, following directly from the geometry of a circular arc. The degree of curve re-expresses the same curve sharpness as an angle (the angle subtended by a fixed chord length, typically 20 or 30 m) rather than as a radius — a convention still used in some route-survey practice.

11.3 Methods of Setting Out Circular Curves

The offsets-from-long-chord method needs no theodolite and is fast for small curves, computing perpendicular offsets from the chord at measured intervals. The successive offsets from chord produced (Rankine) method instead works peg-by-peg along the curve itself. The deflection angle method (Rankine's method) is the most widely used in practice: a theodolite at the point of curvature is oriented toward the PI (zero reading), and each successive peg is located by turning a computed deflection angle and chaining the fixed chord length from the previous peg — a running check confirms the cumulative deflection at the point of tangency equals exactly half the total intersection angle. Where chaining across the curve is impossible (a ravine or river), the two-theodolite method locates each point purely by the intersection of two simultaneously-observed rays, with no distance measurement at all.

11.4 Transition Curves

A transition curve gradually introduces curvature — from zero at the tangent point up to the full curvature \(1/R_0\) of the following circular arc — so that centrifugal force builds up smoothly rather than abruptly, essential for railways (where it is mandatory) and strongly recommended for highways. The ideal transition shape is the clothoid (Euler's spiral), in which the product of the local radius and distance travelled along the curve is held constant. Introducing a transition curve requires the main circular curve to be shifted slightly inward (the "shift") to accommodate the transition geometry at each end. Superelevation (cant) — banking the road or track cross-section inward on a curve — offsets centrifugal force and is subject to a maximum permitted value for safety and passenger comfort.

11.5 Vertical Curves

A vertical curve is a parabolic arc smoothly connecting two different grades at their vertical point of intersection (VPI), chosen as a parabola because it gives a constant rate of change of gradient along its length — a mathematically convenient and comfortable profile. A summit (crest) curve must be long enough to provide adequate sight distance for a driver to see and stop for an obstacle before reaching it, while a valley (sag) curve is instead governed at night by the reach of the vehicle's headlight beam, since the driver's own eye height doesn't help over a dip the way it does over a crest.

Engineering geology applies the study of rocks, minerals and earth structures to civil projects — choosing safe sites and sound foundations for dams, tunnels, bridges and buildings, and anticipating hazards such as landslides and seepage.

13.1 Rocks & Minerals

A mineral is a naturally occurring inorganic solid of definite composition; a rock is an aggregate of minerals. Rocks are classified by origin:

13.2 Geological Structures

The attitude of a rock bed is given by its dip (angle of maximum slope below horizontal) and strike (horizontal line on the bed, perpendicular to the dip). Deformation produces:

• Folds — bends in strata (upfold = anticline, downfold = syncline).
• Faults — fractures with relative displacement of the two sides (normal, reverse, strike-slip).
• Joints — fractures without displacement, important for rock strength and seepage.
• Unconformity — a buried erosion surface marking a gap in the geological record.

13.3 Weathering & Site Applications

Weathering (physical, chemical, biological) breaks rock down in place, weakening foundations and forming soil. Geological knowledge guides site selection: a dam needs a watertight, strong foundation free of adverse faults and permeable joints; a tunnel alignment avoids weak, water-bearing or heavily jointed rock; a reservoir site must not leak through porous strata.

A. Key Formulae at a Glance

B. Exam-Angle Summary

C. Mnemonics & Memory Aids

Practice Questions

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