# Spiral Curve

Spirals are used to overcome the abrupt change in curvature and superelevation that occurs between tangent and circular curve. The spiral curve is used to gradually change the curvature and superelevation of the road, thus called **transition curve**.

## Elements of Spiral Curve

*TS*= Tangent to spiral*SC*= Spiral to curve*CS*= Curve to spiral*ST*= Spiral to tangent*LT*= Long tangent*ST*= Short tangent*R*= Radius of simple curve*T*= Spiral tangent distance_{s}*T*= Circular curve tangent_{c}*L*= Length of spiral from*TS*to any point along the spiral*L*= Length of spiral_{s}*PI*= Point of intersection*I*= Angle of intersection*I*= Angle of intersection of the simple curve_{c}*p*= Length of throw or the distance from tangent that the circular curve has been offset*X*= Offset distance (right angle distance) from tangent to any point on the spiral*X*= Offset distance (right angle distance) from tangent to_{c}*SC**Y*= Distance along tangent to any point on the spiral*Y*= Distance along tangent from_{c}*TS*to point at right angle to*SC**E*= External distance of the simple curve_{s}*θ*= Spiral angle from tangent to any point on the spiral*θ*= Spiral angle from tangent to_{s}*SC**i*= Deflection angle from*TS*to any point on the spiral, it is proportional to the square of its distance*i*= Deflection angle from_{s}*TS*to*SC**D*= Degree of spiral curve at any point*D*= Degree of simple curve_{c}

## Formulas for Spiral Curves

**Distance along tangent to any point on the spiral:**

$Y = L - \dfrac{L^5}{40R^2{L_s}^2}$

At *L* = *L _{s}*,

*Y*=

*Y*, thus,

_{c}$Y_c = L_s - \dfrac{{L_s}^3}{40R^2}$

**Offset distance from tangent to any point on the spiral:**

$X = \dfrac{L^3}{6RL_s}$

At *L* = *L _{s}*,

*X*=

*X*, thus,

_{c}$X_c = \dfrac{{L_s}^2}{6R}$

**Length of throw:**

$p = \frac{1}{4}X_c = \dfrac{{L_s}^2}{24R}$

**Spiral angle from tangent to any point on the spiral (in radian):**

$\theta = \dfrac{L^2}{2RL_s}$

At *L = L _{s}*,

*θ*=

*θ*, thus,

_{s}$\theta_s = \dfrac{L_s}{2R}$

**Deflection angle from TS to any point on the spiral:**

$i = \frac{1}{3}\theta = \dfrac{L^2}{6RL_s}$

At *L = L _{s}*,

*i = i*, thus,

_{s}$i = \frac{1}{3}\theta_s = \dfrac{L_s}{6R}$

This angle is proportional to the square of its distance

$\dfrac{i}{i_s} = \dfrac{L^2}{{L_s}^2}$

**Tangent distance:**

$T_s = \dfrac{L_s}{2} + (R + P)\tan \dfrac{I}{2}$

**Angle of intersection of simple curve:**

$I_c = I - 2\theta_s$

**External distance:**

$E_s = \dfrac{R + P}{\cos \dfrac{I}{2}} - R$

**Degree of spiral curve:**

$\dfrac{D}{D_C} = \dfrac{L}{L_s}$