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).
Tangent offset Formulas:
References
- Iowa Department of Transportation