1. Linear methods of setting out curves

(a) Linear methods of setting out curves The following methods of setting out simple circular curve are linear as measurement is done using chain/tape/distance/EDM:

1. By ordinates from the long chord
2. By successive bisection of arcs
3. By offsets from the tangents
4. By offsets from chords produced

This section introduces the topic of linear methods used to set out circular curves in surveying. These methods rely on physical measurements using tools like chains or tapes rather than angular calculations. The methods listed include:

1. Using ordinates from the long chord.
2. Successive bisections of arcs.
3. Offsets from the tangents.
4. Offsets from chords produced. Each of these methods allows surveyors to determine points along a curve accurately.

Imagine planning a road that bends smoothly. Surveyors use various methods to measure where to place the road markers. Using a long piece of string (the long chord), they compute where to place each marker based on measurements. This is similar to how a gardener lays out a curved flower bed using stakes and ropes.

1. By ordinates from the long chord: In this method, the perpendicular offsets are erected from the long chord to establish points along the curve, as shown in Figure 2.5. If T T is the length of the long chord (L), ED = O which is the offset at mid-point (E) of the long chord (the versine), and PQ = O which is the offset at distance x from E. Draw a line QQ parallel to T T which meets OD at Q, and line OQ which cuts T T at point E.

This method involves setting offsets from a straight line called the long chord to locate points on the circular curve. The offsets indicate how far each point on the curve deviates from the chord. The process uses the relationship of various points along the chord and curve to calculate distances accurately. It helps to establish several points along the curve symmetrically, making it easier to layout short curves, such as street bends.

Think of laying tiles in a curved design. The straight lines are the long chord. You place a ruler to see how far out from the ruler (long chord) each tile (offset) should be to create a perfect curve. By measuring offsets, you can get a nice smooth edge where the tiles meet.

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 - (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. (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 on the curve.

The successive bisection method, also known as the Versine method, works by continually dividing the arc into smaller sections to find points along the curve. Starting at the chord endpoints, it finds the midpoint and uses geometrical calculations based on known properties of circles to create additional points along the curve. Each step involves measuring offsets based on the bisected arcs, allowing for precision in reaching the desired curve shape.

Imagine slicing a pizza. If you want to find where to place the toppings evenly, you cut the pizza into halves, then those halves into quarters, and so on. Just like that, each bisection helps to mark exactly where the decorations should go to make the pizza (or curve) perfect.

3. By offsets from the tangents: The offsets may be either radial or perpendicular to the tangents. (a) By radial offsets In Figure 2.7, if O = PP which is the radial offset at P from O at a distance of x from T along the tangent AB, then - PP = OP – OP where OP = R^2 + x^2 and OP = R. O = R^2 + x^2 - R (Exact).

This method uses offsets measured from tangent lines that intersect the curve. The offsets can be radial (pointing directly toward the center of the curve) or perpendicular from the tangential line. Using geometrical properties of circles, surveyors can calculate how far along the tangents they need to measure to correctly represent points along the curve. It is convenient for sharp curves where ground measurement is straightforward.

Picture a bicycle that's navigating around a sharp corner. The handlebars can be thought of as the tangents, guiding where to go. By knowing how far you need to pull the handlebars outward (offsets), you can ensure that the bike stays on track along the curve.

5. By offsets from chords produced. In Figure 2.9, if AB is the first tangent, T is the first tangent point, E, F, G etc., are the successive points on the curve. Draw arc EE , so T E= T E = C which is the first chords. Similarly, EF, FG, etc., are successive chords of length C , C etc., each being equal to the full chord length.

Offsets from chords produced involve extending the lines from tangent points into the curve. As new chords are introduced, offsets from these chords are calculated to create a smooth, curved trajectory. This method tends to produce large, sweeping curves and establishes points along the arcs based on already calculated chord lengths. It facilitates laying out long curves effectively.

Think of drawing a large arc with a compass. Each time you place a new point along the drawn arc, it's like extending the chord and measuring how far outward it goes. This helps to maintain a smooth curve for whatever beautiful design you're creating.

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