Coordinates on the transition spiral
The setting-out of curves by traditional methods of angles and chords has been dealt with. However, these methods are frequently superseded by the use of coordinates of circular curves. The method of calculating coordinates along the centre-line of a transition curve is probably best illustrated with a worked example. Consider Worked example dealing with the traditional computation of a clothoid spiral using Highway Transition Curve Tables. As with circular curves, it is necessary to calculate the traditional setting-out data first, as an aid to calculating coordinates.
Coordinates along a transition curve
In Example the first three setting-out angles are:
If working in coordinates, the coordinates of T1, the tangent point and I, the intersection point, would be known, say.
And calculated using the coordinates of I, the WCB of T1I − 10◦ 25 35.0.
and a sub-chord length T1a = x = 5.434 m.
Using the P and R keys, or the traditional formula the coordinates of the line T1a are calculated, thus:
In triangle T1ab:
Using the sine rule, the angle at b can be calculated:
This procedure is now repeated to the end of the spiral, as follows:
In triangle T1bt1,
(1) calculate distance T1b from the coordinates of T1 and b
(2) calculate angle T1t1b by sine rule, i.e.
where bt1 is a known chord or sub-chord length and angle bT1t1 = (θ − θ2) in this instance
(3) calculate α1 = bT1t1 T1t1b
(4) WCB T1b = WCB T1l θ2
(5) WCB bt1 = WCB T1b α1 and length is known
(6) computed E, N of the line bt1 and add then algebraically to the Eb and Nb respectively, to give Et1, Nt1
Using the ‘back angle to the origin’ ( − θ) at the end of the spiral (t1), the WCB of the tangent to the circular arc can be found and the coordinates along the centre-line calculated.