1.2. By successive bisections of arcs

It is also known as Versine method. In Figure 2.6, a curve T DT is to be established on the ground by this method. The steps involved are-

This section begins by introducing the 'successive bisection of arcs' method, which is also referred to as the Versine method. The purpose of this method is to accurately establish points along a circular curve on the ground. Visual aids like Figure 2.6 illustrate the method, guiding the reader through the process of setting out a curve in a practical context.

Imagine you are laying out a path in a park or garden that follows a smooth curve. Using the successive bisection of arcs method is like using a ruler to find the exact points where each segment of the path should be placed, ensuring that the curve flows nicely and appears natural.

(i) Join T T and bisect it at E. Set out the offset ED (which is equal to the versine R(1-cos(Φ/2)), thus a point D on the curve may be fixed.

The first step involves joining the endpoints of the chord (T1 and T2), then finding the midpoint E by bisecting this chord. At this midpoint, an offset called ED is set out, which represents how far off the chord the actual curve will be at that midpoint. This offset is calculated using the versine function, which relates to the radius of the curve and the angle of the curve.

Think of this process like finding the center of a circle to draw a perfect arc. The midpoint ensures that the curve starts out balanced, similar to how a tightrope walker finds their balance before stepping onto the rope.

(ii) Join T D and DT and bisect them at F and G, respectively. Then set out the offsets FH and KG at F and G, respectively, in the same manner, each equal to R(1-cos(Φ/4)). Thus, two more points H and K are fixed up on the curve.

Once the first segment is established, the next step involves connecting points T and D, and D and T2, and again finding the midpoints (F and G). Offsets are set at these points using smaller angles, which further define where the curve needs to bend. This step adds more detail to the curve, ensuring smooth transitions.

This can be likened to adding more segments to a string of beads, where each bead represents a point along the curve. By continually bisecting the segments and adjusting the offsets, you ensure that the spacing (or curvature) remains visually and structurally appealing.

(iii) Now, each of the offsets can be set out at mid points of the four chords T H, H D, D K, and K T, which is equal to R(1-cos(Φ/8)).

At this stage, more points are established on the curve by bisection of the newly created segments (from previous points established). Again, offsets are calculated to maintain the curve's smoothness and correctness. This method emphasizes how divided segments allow for precise positioning on the curve.

If you're building a smooth incline or ramp, each point you choose is crucial. It’s like checking every few feet to ensure the ramp isn't too steep or uneven – you adjust it slightly based on how it looks and feels, achieving the desired smooth transition from ground to slope.

(iv) By repeating this process, several points may be set out on the curve, as per the need.

The process of bisecting and establishing offset points is repeated as many times as needed to create a sufficiently detailed curve. The method is versatile and allows for adjustments and refinements to the curve as the construction progresses.

Imagine sculpting a figure from clay; you keep refining and adding details until the sculpture looks polished. Here, every set of points enhances the curvature of the structure, making it smoother and more aesthetically pleasing.

(v) This method is suitable where the ground distance outside the curve is not favorable for measurements by tape.

Finally, the method's applicability is noted; it is particularly useful when traditional measurement techniques cannot be employed due to uneven terrain or obstacles. This flexibility makes the method valuable in various construction and civil engineering projects.

Think of a gardener laying out a path in a garden with steep hills or obstacles; they might find it easier to 'eyeball' the curve based on previous measurements rather than relying solely on a tape measure. This method allows for practical adjustments to work where precise measurements are hindered.

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