Geometric Design - Spiral Curve Design

Spiral curves (aka transition or easement curves) are generally used to provide a gradual transition in curvature from a straight section of road to a curved section (or from tangents to circular curves). Figure 1 shows the placement of spiral curves in relation to circular curves. Figure 2 shows the components of a spiral curve. Spiral curves are necessary on high-speed roads from the standpoint of comfortable operation and gradually bringing about the full superelevation? of the curves. A spiral should be utilized with a circular curve with a superelevation? of 3% or greater. Spiral curves were developed for locomotives and are still used widely today in the industry.

Spiral Curve Diagram
Figure 1: The Placement of a Spiral Curve
Spiral Curve Diagram
Figure 2: Components of a Spiral Curve[2]

Definitions [1]:

  • SCS PI = Point of intersection of main tangents.
  • TS = Point of change from tangent to spiral curve.
  • SC = Point of change from spiral curve to circular curve.
  • CS = Point of change from circular curve to spiral curve.
  • ST = Point of change from spiral curve to tangent.
  • LC = Long chord.
  • LT = Long tangent.
  • ST = Short tangent.
  • PC = Point of curvature for the adjoining circular curve.
  • PT = Point of tangency for the adjoining circular curve.
  • Ts = Tangent distance from TS to SCS PI or ST to SCS PI.
  • Es = External distance from the SCS PI to the center of the circular curve.
  • Rc = Radius of the adjoining circular curve.
  • Dc = Degree of curve of the adjoining circular curve, based on a 100 foot arc (English units only).
  • D = Degree of curve of the spiral at any point, based on a 100 foot arc (English units only).
  • l = Spiral arc from the TS to any point on the spiral (l = ls at the SC).
  • ls = Total length of spiral curve from TS to SC.
  • L = Length of the adjoining circular curve.
  • θs = Central (or spiral) angle of arc ls.
  • ∆ = Total central angle of the circular curve from TS to ST.
  • c = Central angle of circular curve of length L extending from SC to CS.
  • p = Offset from the initial tangent.
  • k = Abscissa of the distance between the shifted PC and TS.
  • Yc = Tangent offset at the SC.
  • Xc = Tangent distance at the SC.
  • x and y = coordinates of any point on the spiral from the TS.

Equations to use:

D_c = {{18000\over \pi}\Bigg/ R_c}
D_c = {200 \times {\theta_s\over l_s}}
l_s = {200 \times {\theta_s\over D_c}}
\theta_s = {{l_s \times D_c}\over 200}
\Delta = {{180\times L}\over {\pi \times R_c}}
\theta_s\hbox{ (decimal degrees) }= {180\over \pi}\times \theta_s\hbox{ (radians)}
X_c = {\left(l_s\over 100\right)\times \left(100-0.0030462\theta^{\raise1pt 2}_s \right)}
Y_c = {\left(l_s\over 100\right)\times \left(0.58178\theta_s-0.000012659\theta^{\raise1pt 3}_s \right)}
p = {Y_c - R_c\times \left(1.0-\cos \theta_s \right)}
A = {{20000\times \theta_s}\over l^{\raise1pt 2}_s}
k = {{l_s\over 2}-0.000127A^2 \times \left(l_s\over 100\right)^5}
T_s = {\left(R_c + p\right)\times tan {\Delta\over 2} + k}
E_s = {\left(R_c+p\right)\times arcsec {\Delta\over 2} + p}
LT = {X_c - \left(Y_c\times \cot \theta_s\right)}
ST = {Y_c\over {\sin \theta_s}}
LC = {l_s-0.00034A^2\times \left({l_s\over 100}\right)^5}
\Delta_c = {\Delta-2\times \theta_s}



  1. Iowa Department of Transportation
  2. Hickerson, T.F., Route Location and Design. 2nd ed. (New York: McGraw-Hill, Inc., 1964), pg 168.



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