## Geometric Design - Horizontal Curve Design

Horizontal circular curves are used to join intersecting straight lines (or tangents). This section provides information relevant to simple circular curves. A simple circular curve is a constant radius arc used to join two tangents. Figure 1 shows the components of a simple circular curve.

**Figure 1: The components of a Simple Circular Curve**

### Definitions:

- PI = Point of Intersection of back tangent and forward tangent.
- PC or BC = Point of Curvature (Beginning of curve) – point of change from back tangent to circular curve.
- PT or EC = Point of Tangency (End of curve) – point of change from circular curve to forward tangent.
- LC = Long Chord – Total chord length, or long chord, from PC to PT in feet for the circular curve.
- D = Degree of curvature. The central angle which subtends a 100 foot arc. The degree of curvature is determined by the appropriate design speed: Refer to AASHTO’s A Policy on Geometric Design of Highways and Streets, 1990.
- ∆ or I = Total intersection (or delta) angle between back and forward tangents.
- T = Tangent distance in feet. The distance between the PC and PI or the PI and PT.
- L = Total length in feet of the circular curve from PC to PT measured along its arc.
- E = External distance (radial distance) in feet from PI to the mid-point of the circular curve.
- R = Radius of the circular curve measured in feet.
- θ = Deflection angle from a tangent to a point on the circular curve.
- \Delta \over 2 = Deflection angle for full circular curve measured from tangent at PC or PT.
- C = Chord length in feet, where a chord is defined as a straight line connecting any two points on a curve.
- S = Arc length in feet along a curve.
- MO or HSO = Middle ordinate. Length of the ordinate from the middle of the curve to the LC.
- RP or O = Radius point, or the center of the curve.
- POT = (any) point on the tangent
- POC = (any) point on the curve
- MPC = The midpoint of the curve (not shown on above image)

### Equations to use:

Given two variables, all others can be solved for (depending on how you rearrange the equations).

R = {{18000\over \pi}\over D} = {50\over {sin\left(D\over 2\right)}} = {5729.578\over D}

L = {R*I_{Radians}} = {100\left(\Delta\over D\right)} = {2\pi RI\over 360º}

I = \Delta = {D \times \left(L\over 100\right)}

T = R\times tan{\left(\Delta\over 2\right)}

E = {T\times tan\left(\Delta\over 4\right)} = {R\left(sec{\Delta\over 2} - 1\right)} = {R\times tan{\Delta\over 2}tan{\Delta\over 4}}

MO = {R\left(1-cos{\Delta\over 2}\right)} = {{LC\over 2}tan{\Delta\over 4}}

LC = {2R\times sin{\Delta\over 2}} = {2T\times cos{\Delta\over 2}}

D = {{18000\over \pi}\over R}

C = {2\times R\times sin\theta}

S = {{200 \over D}arcsin{C \over {2R}}}

*Tangent offset Formulas:*

x_{approximate} = {y^2\over{2R}}

x_{exact} = {R-\sqrt{R^2-Y^2}}

### References

- Iowa Department of Transportation