## 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.

Figure 1: The Placement of a Spiral Curve

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.
• 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}

### References

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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