Tacheometry, Plane Table & Curves - Geomatics Engineering (Surveying) -

Tacheometric Surveying

Tacheometry (from Greek, meaning "quick measure") or telemetry is a branch of angular surveying in which horizontal and vertical distances of points are obtained by optical methods instead of by direct measurement with tape or chain. The method is rapid and convenient and is particularly suitable for rough, steep, broken, or otherwise inaccessible ground such as deep ravines, water stretches, swampy tracts and similar obstacles where chaining is difficult or impossible.

Uses of Tacheometry

• Preparation of topographic maps that require both elevations and horizontal distances.
• Surveying in difficult terrain where direct methods are inconvenient.
• Detail-filling for existing surveys.
• Reconnaissance surveys for highways, railways and alignments.
• Checking previously measured distances.
• Hydrographic surveys (for near-shore work) and establishing secondary control.

Instrument

• An ordinary transit theodolite fitted with a stadia diaphragm (stadia hairs) is commonly used for tacheometric work.
• The stadia diaphragm contains two horizontal stadia hairs, one above and one below the central cross-hair, placed equidistant from the central hair on the same vertical plane as the horizontal and vertical cross-hairs.

Different types of stadia hairs

Principle of Tacheometry

All tacheometric methods use the principle that the horizontal distance between an instrument station A and a staff station B depends on the angle subtended at A by a known distance on the staff at B and on the vertical angle of sight. This principle is applied in different ways to obtain both horizontal distance and elevation with fewer observations than chaining and separate levelling.

Main methods of tacheometry

• Stadia system
• Tangential (or tangency) system

1. Stadia System of Tacheometry

In the stadia system, a single telescope observation gives both the horizontal distance to the staff station and the elevation of the staff station with respect to the instrument line of sight. Two principal variants are used:

• Fixed-hair method (stadia hairs fixed in the diaphragm)
• Movable-hair method (subtense method; stadia hairs adjustable)

(i) Fixed-hair method

• The telescope is fitted with two additional horizontal cross-hairs, one above and one below the central hair; these are the stadia hairs.
• When a staff is sighted, the stadia hairs intercept a length of the graduated staff. This intercepted length varies directly with the distance between the instrument and the staff.
• Because the stadia hairs are fixed relative to the central hair, this arrangement is called the fixed-hair method.

(a) Principle of the stadia hair method

The stadia method is based on similarity of isosceles triangles formed by the rays through the stadia hairs and the central ray. For similar configurations along the sight line the ratio of perpendicular to base remains constant, leading to a simple linear relation between staff intercept and distance.

• Let two stadia rays OA and OB be equally inclined to central ray OC.
• Let A₂B₂, A₁B₁, and AB be intercepts on staffs at different distances. Then the ratio OC/A₂B₂ = OC/AB = constant (k).
• For a given instrument this constant k depends on the angle between the extreme rays and is equal to ½ cot(β/2) in the geometric model.
• In practice, observations may be made with a horizontal line of sight or with an inclined line of sight; the staff may be held vertical or normal to the line of sight. The distance-elevation formulae change correspondingly.

(b) Horizontal line of sight - distance equation

Notation:

• ab = i - interval between the stadia hairs (stadia interval)
• AB = s - staff intercept between upper and lower stadia hairs
• f - focal length of the objective
• f₁ - horizontal distance of the staff from the optical centre O of the objective
• f₂ - horizontal distance of the cross-wires from O
• d - distance of the vertical axis of the instrument from O
• D - horizontal distance of the staff from the instrument vertical axis
• M - instrument centre (vertical axis)

Distance equation (horizontal sight):

D = (f/i) · s + (f + d)

• The constant k = f/i is called the multiplying constant or stadia interval factor.
• The constant C = (f + d) is called the additive constant of the instrument.
• Thus, to obtain the horizontal distance D from the instrument axis to the staff, find the staff intercept s and substitute into the distance equation D = k·s + C.

(c) Inclined line of sight - staff held vertical

Let the slant distance measured along line of sight be L. Then, for a line of sight inclined at angle θ to the horizontal:

L = k · s + C

Horizontal distance, D = L · cos θ

Vertical distance (elevation difference with respect to line of sight), V = L · sin θ

Elevation of staff station when line of sight is at an angle of elevation θ:

Elevation of staff = Elevation of instrument + h + V - r

Elevation of staff station when line of sight is at an angle of depression:

Elevation of staff = Elevation of instrument + h - V - r

where h = instrument height above instrument station benchmark and r = staff reading at appropriate hair (axial hair reading as required).

(d) Inclined line of sight - staff held normal to the sight

Case (a): Line of sight at an angle of elevation Θ

• Let AB = s = staff intercept
• Let CQ = r = axial hair reading
• MC = L = k·s + C
• Horizontal distance, D = MC' + C'Q' = L cos Θ + r sin Θ = (k·s + C) cos Θ + r sin Θ
• Vertical component, V = L sin Θ = (k·s + C) sin Θ

Case (b): Line of sight at an angle of depression Θ

• MC = L = k·s + C
• Horizontal distance, D = MQ' = MC' - Q'C' = L cos Θ - r sin Θ = (k·s + C) cos Θ - r sin Θ
• Vertical component, V = L sin Θ = (k·s + C) sin Θ
• Elevation of staff Q = Elevation of instrument + h - V - r cos Θ

(ii) Movable-hair method (Subtense method)

In the movable-hair method the stadia hairs are adjustable. The staff used has two fixed targets (vanes) set at a known distance apart (for example, 3.4 m or other convenient separation). The instrument operator varies the stadia hairs until the two targets are seen between the stadia hairs and measures the stadia interval (intercept) i. The horizontal distance is then computed from the known separation and the measured stadia interval. The movable-hair (subtense) method is less commonly used than the fixed-hair method but is useful in certain specialised situations.

Note: Between the two stadia methods, the fixed-hair method is the more widely employed in normal surveying practice.

2. Tangential System of Tacheometric Surveying

• In the tangential system two observations (two vertical angles) are necessary from the instrument station to the staff station to determine horizontal distance and elevation difference between the line of collimation and the staff station.
• The advantage of the tangential system is that it can be performed with an ordinary transit theodolite without stadia diaphragm, making the instrument cheaper and the equipment more economical.
• However, the method requires measurement of two vertical angles, which may introduce additional error if the instrument is disturbed between observations, and it requires more observations than the stadia system, reducing speed. Atmospheric variations also affect accuracy.
• The staff used is similar to that used in the movable-hair stadia method, with two targets or vanes separated by a known distance (often 3-4 m).

Mathematical relations - typical cases

Case 1: Both angles above the horizontal line of sight

Let D be the horizontal distance from instrument point P to staff point Q, let V be the vertical distance from line of collimation to the lower vane, and let s be the separation between vanes.

V = D · tan θ₁

V + s = D · tan θ₂

Eliminating V gives

s = D (tan θ₂ - tan θ₁)

• Therefore, D = s / (tan θ₂ - tan θ₁)
• Reduced level of Q = (Reduced level of P + h) + V - r, where h is instrument height and r is staff reading at the lower vane.

Case 2: Both angles below the horizontal line of sight

• V = D · tan θ₁
• V - s = D · tan θ₂

Thus,

s = D (tan θ₁ - tan θ₂)

or

D = s / (tan θ₁ - tan θ₂)

and

V = D · tan θ₁

Therefore reduced level of Q = (Reduced level of P + h) - V - r.

Case 3: One angle above and one below the horizontal line of sight

• V = D · tan θ₁ (for the vane above)
• s - V = D · tan θ₂ (for the vane below)

Therefore

s = D (tan θ₂ + tan θ₁)

or

D = s / (tan θ₂ + tan θ₁)

and

V = (s / (tan θ₂ + tan θ₁)) · tan θ₁

Reduced level of Q = (Reduced level of P + h) - V - r.

Advantages and Limitations of Tacheometry

• Advantages: Rapid data collection, fewer instruments required, useful on difficult terrain, simultaneous determination of horizontal distances and vertical differences in many cases.
• Limitations: Accuracy is limited compared with precise chaining and levelling; dependent on instrument constants (k and C) which must be known; inclined sights and atmospheric refraction affect results; tangential system requires two angle readings which may introduce errors if disturbed.

Plane Table Surveying

Plane table surveying is a graphical method of surveying in which field observations and plotting are performed simultaneously. The plane table consists of a drawing board mounted on a tripod, an alidade for sighting and plotting, a spirit level, and drawing paper. It is best suited for small and medium-scale surveys where immediate plotting in the field is desirable.

Equipments and accessories

• Plane table (drawing board) - usually square or rectangular; made rigid and flat.
• Alidade - a straightedge with a sighting device (plain alidade or telescope alidade) used for drawing lines and measuring directions.
• Spirit level - to level the table accurately.
• Drawing paper, pins or clamps to fix paper on the board, pencils, ink, scales and protractor or minicircle as needed.

Methods of plane table surveying

• Radiation - plotting points by drawing rays from a plane table station to detail points and measuring distances along those rays.
• Intersection (resection by sighting from two stations) - plotting a point by intersecting two rays drawn from two different plane table stations.
• Traversing - plotting a traverse by successive stationing of the plane table and drawing the connecting lines directly on the sheet.
• Resection (three-point problem) - determining the plane table position by sighting to three known plotted points and drawing back-rays to find the instrument position.
• Comparative methods - plane table combined with few measurements for exact control.

Procedure - general notes

• Set up the plane table over a station and level it precisely.
• Orient the table correctly to the previously plotted portion (by back-sighting, orientating with a known direction, or by resection methods).
• Use the alidade to sight detail points, draw the rays and measure distances if required; mark elevations and detail features on the drawing sheet.
• Move the table to successive stations as per the chosen method and continue plotting until the survey is complete.

Advantages and limitations

• Advantages: Immediate field plotting and visual control of work, simple and quick for small-scale surveys, useful for topographical and engineering surveys where rapid depiction is needed.
• Limitations: Not suitable for very large areas or where high precision is required; table must be carefully levelled and oriented; wind and weather can disturb the sheet and affect accuracy.

Applications

• Topographic surveys for small areas.
• Site surveys for buildings, roads, and minor engineering works.
• Reconnaissance mapping where rapid graphical output is valuable.

Curves in Surveying (Horizontal Alignment)

Curves are introduced in the alignment of roads, railways and canals to provide a gradual change of direction between straight tangents. The common curves used are simple circular curves, compound curves, reverse curves and transition (spiral) curves. The material below summarises the elements and setting-out of a simple circular curve, which is the fundamental type.

Elements of a simple circular curve

• Radius (R) - radius of the circular arc.
• Central angle (Δ) - angle subtended at the centre by the arc (in degrees).
• Length of curve (L) - length of the arc.
• Tangent length (T) - distance from the point of intersection of tangents (PI) to the tangent point on the curve.
• External distance (E) - distance from PI to the midpoint of the arc measured perpendicular to the bisector.
• Mid-ordinate (M) - maximum offset from the chord to the arc at midpoint of the curve.
• Chord - straight-line segment joining two points on the curve; often chords of equal length are used in setting out.
• Degree of curve (D) - in Indian practice the degree of curve is commonly defined as the central angle subtended by a 30 m arc. Then R ≈ 5729.58 / D.

Key formulae for a simple circular curve

Let Δ be in degrees and R in metres.

• Length of curve, L = (π · R · Δ) / 180
• Tangent length, T = R · tan(Δ / 2)
• External distance, E = R · sec(Δ / 2) - R
• Mid-ordinate, M = R - R · cos(Δ / 2) = R (1 - cos(Δ / 2))
• Degree of curve, D (for 30 m arc): R = (30 · 180) / (π · D) ≈ 5729.58 / D

Setting out a simple circular curve - common methods

• Tangent-chord (deflection) method - set out successive chords by deflection angles measured from the tangent line; deflection angles are computed from the radius and chord length.
• Chord and offset method - set out chords of fixed length from the tangent point and measure perpendicular offsets to locate intermediate points on the arc.
• Rankine's method - numerical method using precomputed tables or calculations of chord lengths and offsets for equal chord stations.
• Two-ordinate method - uses measured ordinates (offsets) at given distances from the tangent point to define the curve.

Deflection angle computation (example for equal chord length)

When setting out with equal chords of length c, the deflection angle δ from the tangent for the nth chord is computed from the chord length and radius. For small chords:

δ ≈ (c / R) in radians, or δ (degrees) ≈ (c · 180) / (π · R).

Using deflection angles, successive bearings from the tangent are obtained and chained along the chord lengths to lay out the curve.

Transition (spiral) curves - brief note

Transition curves (such as clothoids) provide gradual change of curvature between tangent and circular arc, reducing lateral acceleration and providing comfort and safety on highways and railways. Transition curves are essential where superelevation and comfortable change of direction are required. Setting out transition curves requires computed offset tables or use of specialised instruments and is taught as a separate, detailed topic in geometric design.

Applications of curve setting

• Alignment design and construction for roads, railways and canals.
• Field staking out of horizontal alignments for civil engineering works.
• Survey control when converting design into field staking coordinates.

Worked example - stadia distance equation (derivation)

The following gives the derivation of the stadia distance equation for a horizontal sight. Equations are shown step-wise for clarity.

Consider an external focusing telescope with optical centre O and three rays through the central and the two stadia hairs intersecting the staff at points giving intercept AB = s. The interval between stadia hairs on the diaphragm is i.

Using similar triangles we obtain proportionality between the distance from the objective to the cross-hairs and the intercept on the staff.

Thus, f / i = f₁ / s

Therefore, f₁ = (f / i) · s

But horizontal distance from instrument axis to staff D = f₁ + (f + d)

Substituting f₁ gives

D = (f / i) · s + (f + d)

Hence, writing k = f / i and C = f + d, the distance equation becomes

D = k · s + C

Summary

Tacheometry provides rapid methods to obtain horizontal distances and elevations by optical observation, with the stadia (fixed-hair) method being the most common. The tangential system is useful when only an ordinary theodolite is available. Plane table surveying offers immediate field plotting for small-to-medium surveys. Curves are essential in horizontal alignment design and several standard formulae and setting-out methods exist for simple circular curves and transitions. Together these topics form important practical techniques in geomatics and civil engineering fieldwork.