curvature at a given point is directly proportional to the distance from the start of the
curve to the given point, measured along the curve.
Refer to Fig. 17. The key points are identified by the following notational system: PI =
point of intersection of main tangents; TS = point of intersection of main tangent and ap-
proach spiral (tangent-to-spiral point); SC = point of intersection of approach spiral and
circular curve (spiral-to-curve point); CS = point of intersection of circular curve and de-
parture spiral (curve-to-spiral point); ST = point of intersection of departure spiral and
main tangent (spiral-to-tangent point); PC and PT = point at which tangents to the circular
curve prolonged are parallel to the main tangents (also referred to as the offsets). Dis-
tances are designated in the following manner: Ls = length of spiral from TS to SC; L =
length of spiral from TS to given point on spiral; R 0 radius of circular curve; R = radius of
curvature at given point on spiral; Ts, = length of main tangent from TS to PI; E 3 = exter-
nal distance, i.e., distance from PI to midpoint of circular curve.
In addition, there is a long tangent (LT), short tangent (ST), and long chord (LC), as
indicated with respect to the departure spiral.
Place the origin of coordinates at the TS and the :c axis on the main tangent. Then
FIGURE 17. Notational system for transition spirals.Main tangentInfiniteradius