1.3.2. By offsets perpendicular to the tangents

In Figure 2.8, O = PP which is the perpendicular offset at P at a distance of x from T along the tangent AB. Draw PP line parallel to BT. PP = PT = x, and TP = PP = O. Now TP = OT – OP, where OT = R, and OP = R² – x².

This chunk introduces the concept of setting out curves by taking perpendicular offsets from the tangents. When we have a point P along a tangent, a perpendicular line is drawn to the circle at this point to establish the offset. Essentially, this helps determine the distance from the tangent to the actual curve.

1. The point P is where the curve meets the tangent.
2. The distance OP represents a radius adjusted by the perpendicular distance from the tangent to the circle's edge.
3. The total offset O can then be computed using the relationship between the circle's radius and the lengths derived from the offsets.

Imagine if you're standing on a straight road (the tangent) and need to measure how far a tree (the curve) is from that road. By walking directly away from the road (the perpendicular direction from the tangent), you can measure the tree’s distance accurately rather than trying to measure at an angle, which can be tricky.

So O = R – R² – x² (exact) (2.14). Approximately, the formula may be obtained similarly as equation 2.13; O = (approximate) (2.15).

In this section, we look at the formulas used to calculate offsets. The exact formula provides a precise calculation for the offset from the tangent based on the radius and the distance x:

1. The exact formula reflects the geometric properties of the circle, ensuring accurate calculations based on the fundamental definitions of radius and distance.
2. The approximate formula is useful in practical scenarios where quick estimations are necessary and higher precision isn't critical. This makes it more convenient in everyday applications.

Think of using a ruler to measure a piece of paper. If you want a precise length, you'd measure carefully (exact method). However, if you're just estimating how long it is for a quick decision (approximate method), you might not worry if it's exactly right, as long as you're close enough.

(i) Locate the tangent points T and T. (ii) Measure equal distances, say 15 or 30 m along the tangent from T. (iii) Set out the offsets calculated by any of the above methods at each distance (say x), thus obtaining the required points on the curve. (iv) Continue the process until the apex of the curve is reached. (v) Set out the other half of the curve from the second tangent; being symmetrical in nature.

This chunk details the step-by-step method for setting out the curve using the perpendicular offsets:

1. Locate Tangent Points: Identify the points where the tangents meet the curve.
2. Distance Measurement: Measure predetermined intervals along the tangent line, which serve as reference points for offsets.
3. Calculate Offsets: Calculate the offsets at each of these measured points using the formulas established earlier.
4. Construct the Curve: Mark the curve points based on these offsets. Once you reach the curve's peak, reflect this process for the other side, ensuring both halves mirror each other to maintain symmetry.

Imagine baking a cake. First, you determine how wide the cake will be (locating tangent points), then measure equal rounds of batter to fill the pan (measuring distances), pour in the batter with some spread from the center (set offsets), and finally you ensure both sides are evenly done as the batter rises in the oven (ensuring symmetry).

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