2.5.1. Elements of a reverse curve

Figure 2.14 shows a reverse curve made up of two different radii. In this Figure, R1 is the smaller radius (O A=O D), R2 is the larger radius (O D = O B), Δ1 is the angle subtended at the centre by the arc of smaller radius R1, Δ2 is the angle subtended at the centre by the arc of larger radius R2, V is the perpendicular distance (AJ=MN) between two straights (parallel tangents) AM and BN, h is the distance between the perpendiculars at A and B, L is the length of the line joining the tangent points A and B.

A reverse curve consists of two circular arcs which are tangent to each other and curve in opposite directions. The smaller arc has radius R1 and the larger arc has radius R2. The angles Δ1 and Δ2 correspond to the central angles of these arcs. The perpendicular distance between the two straight lines defined as V is indicative of how the arcs affect the direction of a road or railway. This setup is commonly used to smoothly transition between two straight paths that are diverging.

Imagine driving on a winding road that alternates between curves in opposite directions to avoid obstacles like hills or rivers. The reverse curves help maintain a continuous flow of travel while changing directions smoothly, much like how a snake moves.

D is the point of reverse curvature, and from it a line perpendicular to the straights AO and BO is drawn, cutting these at G and H. Draw perpendicular from O1 and O2 at line AB which will cut this line at P and Q, respectively, dividing the angle of deflection into half ∠AOP = ∠DOP, and ∠BOQ = ∠DOQ.

At the point where the two curves meet, called the point of reverse curvature D, perpendicular lines are drawn to the tangents. These lines help to determine where the curves begin and end. Points G and H represent where these perpendiculars intersect the lines AO and BO, respectively. The angle formed at point D's perpendicular is bisected, giving us a clear structure to understand how the curves relate to one another.

Consider a river that changes its course suddenly, creating a natural bend. The perpendiculars represent how one would measure the straight path of the river before and after the bend. This helps orient oneself along the floodplain.

When Δ1 = Δ2 = Δ, Perpendicular distance (V) = AG + GJ = (AG + BH) V = (O A – O G) + (O B – O H) (2.37)

When both angles subtended by the smaller and larger arcs are equal, computing the perpendicular distance V simplifies to the sum of the adjusted radius lengths. The distances AG and BH can be expressed in terms of the radii R1 and R2, yielding a formula (2.37) that computes how far apart the curves are at their closest point.

Visualize walking on a square track, executing perfect quarter-turns around each corner with equal effort. The distances you walk from corner to corner along the direct line (like the perpendiculars) help you efficiently map your route.

Total length (L) = AD + DB, L = 2R1 sin(Δ/2) + 2R2 sin(Δ/2) (2.39)

The total length L of the reverse curve can be calculated by summing the lengths of the two arcs AD and DB. Each length depends on the radius and the angle subtended at the center. The formula indicates that each segment's distance is twice the radius multiplied by the sine of half of the respective angle, creating a straightforward method to determine total curve length.

Think of a racetrack where drivers follow semi-circular paths around each turn. The total distance a driver travels while going around these curves can be determined by their path lengths being dictated by the curves’ radii and how sharply they turn.

Distance between the end points of the reverse curve measured parallel to the straights (h) = GD + DH, h = (R1 + R2) sin(Δ) (2.41)

The distance h between the endpoints of the reverse curve can be calculated by summing the distances GD and DH, which relate back to the sine of the angle Δ. This formula helps in determining how far the endpoints are when projected straight across relative to the straights of the curve. Notably, this is important for marking construction projects or roads.

Imagine drawing a straight line on a paper where the ends of curved lines meet. By figuring out how far apart those ends are using basic trigonometry, you can plan your next steps or straight connection without any misalignment, similar to ensuring a road aligns with neighboring paths.

--