Curves, Tacheometry and Contouring

Roads and railways change direction along curves. Tacheometry finds distances and heights quickly with an optical instrument. Contours and earthwork volumes follow.

Simple circular curve

Two straights meet at the point of intersection with a deflection angle Δ; a curve of radius R joins them.

• Degree of curve D (30 m arc): R = 1718.9/D (metres, D in degrees).
• Rankine's deflection angle method (with a theodolite): δ = 1718.9 c/R minutes for a chord of length c.

Transition curves and vertical curves are covered in highway design.

Tacheometry

Stadia method (fixed hair interval, staff vertical), staff intercept s, line of sight at angle θ:

• Horizontal distance D = k s cos² θ + C cos θ
• Vertical component V = k s (sin 2θ)/2 + C sin θ
• Usually k = 100 and C = 0 (anallactic lens). On level sights: D = 100 s.

Contours

A contour joins points of equal elevation.

• Contour interval is constant on a map. Close spacing = steep slope; even spacing = uniform slope.
• Contours never cross (except overhanging cliffs) and close on themselves.
• V-shapes point uphill in valleys and downhill on ridges.

Volumes

• Trapezoidal rule: V = d[(A₁ + Aₙ)/2 + A₂ + … + Aₙ₋₁].
• Prismoidal rule: V = (d/3)[A₁ + Aₙ + 4(even areas) + 2(odd interior areas)] (needs an odd number of sections). Usually more accurate.
• Prismoidal correction = trapezoidal − prismoidal volume.
• Mass haul diagram for balancing cut and fill.

Worked example

A curve of radius 300 m has a deflection angle of 60°.

• T = 300 tan 30° = 173.2 m; length = π × 300 × 60/180 = 314.16 m.

Practice