1.3. By offsets from the tangents

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 = √(R² + x²) and OP = R O = √(R² + x²) - R (Exact) (2.12)

In this section, we explore how to establish points on a circular curve using radial offsets. The aim is to determine the relationship between the radial distance (PP) used to define a point on the curve (P) from a reference point O along a tangent line (AB). To simplify calculations, we start with the Pythagorean theorem.

• We note that the distances from the center (O) to point P on the curve can be expressed in terms of the radius (R) and the offset distance (x).
• Based on this, the radial offset (PP) at point P is calculated by subtracting the radius (R) from the hypotenuse formed by the radius plus the added distance x (√(R² + x²)).
• This means that PP becomes the effective distance from point P to the center O minus the radius. By following this calculation, we can accurately set out points along the curve at specific intervals along the tangents.

Imagine you are laying out a circular flower bed in your garden. You know the radius of the circle (R), and you want to place a decorative stone at a certain distance (x) outward from the point where the circle will be anchored (point O). Using the radial offset method is akin to measuring how far out from your center point O each stone should go, ensuring they form a perfect circle around your garden.

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 P P line parallel to BT. PP = PT = x, and T P = PP = O Now T P = OT – OP Where OT = R, and OP = √(R² – x²) O = R - √(R² – x²) (Exact) (2.14)

This portion details how to calculate offsets using perpendicular measurements from the tangents to the curve.

• By constructing a vertical line from the tangent (AT) to the desired point on the curve, we can easily determine the perpendicular offset (O).
• This offset is calculated by recognizing that the total length (TP) from T (the tangent point) down to the curve consists of the radius(R) minus the vertical distance to point P (OP).
• By rearranging this relationship, the perpendicular offset can be expressed as the difference between the radius and the derived length from the right-angled triangle formed in the system.

Imagine you're marking a perpendicular line from a straight sidewalk into your lawn where you want to plant a row of bushes. By measuring straight down from the sidewalk (the tangent) to where your bushes will be planted, you're utilizing the concept of perpendicular offsets, where your sidewalk acts like the tangent and the planting line is perpendicular to it.

(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 procedure outlines the systematic approach to establish a curve from tangent references effectively.

• To begin, identify your starting points (the tangents) on the line.
• From these points, measure uniform distances along the tangent for either further calculations or offset placements.
• Using the previously discussed methods, compute the needed offsets and mark these points accurately on your planned curve.
• This symmetrical approach ensures that the second half of the curve mirrors the first, maintaining uniformity and design integrity. Following this structured method leads to a more reliable and organized layout for any circular structure.

Think of this procedure as drawing the outline for a circular track for a race. First, you establish the starting lines (tangents). Then by consistently measuring distances along those lines, you mark the spots where the inside edge of your track will go. Each point will shape the curve as you go, ensuring the opposite side mirrors the track’s outline perfectly.

--