Horizontal Circular Curve (Arc Definition): 18-Geom-A1 Surveying, December 2013, Q6

Question 6: Horizontal Circular Curve (Arc Definition) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Circular horizontal curve data:

Parameter	Value
Radius $R$	$900$ m
Deflection (central) angle $I$	$14^\circ45' = 14.75^\circ$
PI station	$1{+}948.800$ m
Definition	arc (metric)

Find. $L$, $T$, $E$, $M$, $LC$, and the PC and PT stations.

Figure 6 — Circular curve: back tangent PC→PI, forward tangent PI→PT, arc of radius $R$ subtending the deflection angle $I$.

Approach. Apply the standard arc-definition curve formulas in terms of $R$ and $I$, then station the PC back from the PI by $T$ and the PT forward from the PC by the arc length $L$.

Curve length (arc definition). With $I=14.75^\circ=0.257436$ rad: $$L = R\,I_{\text{rad}} = 900(0.257436) = \boxed{231.692\ \text{m}}$$
Tangent, external, middle ordinate, long chord. With $I/2 = 7^\circ22.5' = 7.375^\circ$: $$T = R\tan\tfrac{I}{2} = 900\tan 7.375^\circ = 116.490\ \text{m}$$ $$E = R\!\left(\sec\tfrac{I}{2}-1\right) = 7.508\ \text{m},\qquad M = R\!\left(1-\cos\tfrac{I}{2}\right) = 7.445\ \text{m}$$ $$LC = 2R\sin\tfrac{I}{2} = 1800\sin 7.375^\circ = 231.053\ \text{m}$$
Stationing. The PC precedes the PI by $T$; the PT follows the PC by the arc length $L$ (never by $2T$): $$\text{PC} = \text{PI}-T = 1948.800-116.490 = \boxed{1{+}832.310}$$ $$\text{PT} = \text{PC}+L = 1832.310+231.692 = \boxed{2{+}064.002}$$

Element	Value
Length of curve $L$	$231.692$ m
Tangent distance $T$	$116.490$ m
External distance $E$	$7.508$ m
Middle ordinate $M$	$7.445$ m
Long chord $LC$	$231.053$ m
PC station	$1{+}832.310$
PT station	$2{+}064.002$

Concept & Theory

Topic. Geometry and stationing of a simple horizontal circular curve under the arc definition.

Key relations. $L=RI$ (rad); $T=R\tan(I/2)$; $E=R(\sec(I/2)-1)$; $M=R(1-\cos(I/2))$; $LC=2R\sin(I/2)$; PC $=$ PI $-T$, PT $=$ PC $+L$.

Why this works. The arc definition ties length directly to the central angle in radians, which is why metric highway design prefers it over the chord (railway) definition. Because the alignment is measured as built, the PT is reached from the PC along the arc $L$, not out to the PI and back — the curve physically shortens the alignment by $2T-L$.

Common pitfall. Computing PT as PI $+T$ (wrong — it must be PC $+L$); since $L<2T$, that error throws every downstream station off by $2T-L\approx1.29$ m here.

Source. Wolf & Ghilani, Elementary Surveying §24 (horizontal curves); AASHTO A Policy on Geometric Design of Highways and Streets (design context).

Question 6 of 7: Horizontal Circular Curve (Arc Definition)

Question 6: Horizontal Circular Curve (Arc Definition) (20 marks)