Tangent Arc at a 120° Path Junction
A landscape practice is laying out a new path junction on a plan at 1:200. Two straight paths meet at a vertex V with an included angle of 120°, and a circular turning area of 25 m radius must be tangent to both paths, sitting inside the angle. A bench is to stand on the locus of points equidistant from the two paths, 30 m from V, facing the turning area. The apprentice has drawn a 25 m radius arc with its centre somewhere on the angle bisector, but the arc cuts across both path edges instead of meeting them smoothly, and has marked the bench at the 30 m point. The landscape architect says the arc is misplaced and the bench would end up in the middle of the turning area.
1. Determine the correct position of the arc centre and of the two tangent points, and describe the construction to be used on the drawing.
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The centre must be 25 m from each path, so it lies on the bisector of the 120° angle. The perpendicular distance from a point on the bisector to either path is d sin(half-angle), with half-angle = 60°. Setting d sin 60° = 25 gives d = 25/0.866 = 28.9 m from V.
The apprentice's arc cuts the paths because the centre was set too close to V, so the radius overran the tangent points and the arc crossed the path edges instead of blending into them.
Construction (no calculation needed):
- Bisect the 120° angle.
- Inside the angle, draw lines parallel to each path at 25 m (125 mm at 1:200); their intersection is the centre; alternatively set off 28.9 m (144.5 mm) along the bisector.
- Drop perpendiculars from the centre onto each path. These feet are the tangent points, at 25/tan 60° = 14.4 m (72 mm) from V along each path.
- Draw the arc between the two tangent points only. At each tangent point the radius is perpendicular to the path, so the arc runs smoothly into it.
2. Test the proposed bench position and state what the acceptable alternatives are.
Read the worked answer
The bench is defined as 30 m from V on the bisector, and the arc centre is 28.9 m from V on the same bisector. The bench therefore stands 30 − 28.9 = 1.1 m from the centre of the turning area - far inside the 25 m radius. It would be sitting in the middle of the traffic area, so the position fails.
Alternatives, in order of preference:
- Keep the bench on the bisector and move it beyond the turning area: the nearest acceptable point is 28.9 + 25 = 53.9 m from V (270 mm at 1:200).
- Keep roughly the 30 m distance but leave the bisector: any point at least 25 m from the centre is acceptable, so set the bench on a line parallel to the bisector at a clear offset, or on a tangent to the arc.
- Note also that the locus of points equidistant from two intersecting lines is the pair of bisectors. Only the bisector lying inside the 120° angle is relevant here; the other bisector is also equidistant but runs outside the junction and must not be used to justify a placement.
3. The sheet is A3 (420 mm × 297 mm) and the drawing must include both paths, the arc and the setting-out lines. Check the layout at 1:200 and state what should be done.
Read the worked answer
At 1:200, 1 m on site = 5 mm on paper. The arc centre is 28.9 m = 144.5 mm from V, the tangent points are 14.4 m = 72 mm from V, and the turning radius is 25 m = 125 mm. The far edge of the turning area is therefore 28.9 + 25 = 53.9 m = 269.5 mm from V along the bisector.
With the vertex near one corner, the arc alone can be accommodated, but the two bisectors, the parallel setting-out lines at 125 mm and the 30 mm / 53.9 m bench markers push the layout past the 297 mm width and almost certainly past the 420 mm length once both paths are drawn at a sensible length.
Action: keep 1:200 if the whole junction must be shown, place V towards the lower-left and rotate the principal bisector so it runs along the long side; otherwise reduce to 1:250 (1 m = 4 mm), which brings the far edge to 215.6 mm and leaves room for the setting-out construction. Whichever is chosen, the scale must be stated in the title block, and the setting-out lines should be drawn light and left un-inked in the final presentation.