Circular Curve Fundamentals

Horizontal curves provide smooth transitions between tangent sections of a highway. A simple horizontal curve is a circular arc of constant radius connecting two tangents. For the PE Civil Transportation exam, horizontal alignment calculations are based on the equations found in the NCEES PE Civil Reference Handbook and design criteria in AASHTO’s Green Book (GDHS), Chapter 3.

Circular Curve Geometry and Terms

Understanding the spatial layout and definitions of horizontal curve components is critical.

graph TD
    classDef point fill:#f9f,stroke:#333,stroke-width:2px;
    classDef line fill:#fff,stroke:#333,stroke-width:1px;

    PI((PI: Point of Intersection)):::point
    PC((PC: Point of Curvature)):::point
    PT((PT: Point of Tangency)):::point
    O((O: Center of Curve)):::point

    PC -->|Tangent Length T| PI
    PI -->|Tangent Length T| PT
    PC -->|Radius R| O
    PT -->|Radius R| O
    PC -.->|Long Chord LC| PT

Definitions:

PI (Point of Intersection): The point where the back tangent and forward tangent intersect.
PC (Point of Curvature): The point where the alignment changes from a tangent to the circular curve (beginning of the curve).
PT (Point of Tangency): The point where the alignment changes from the circular curve to a tangent (end of the curve).
$\Delta$ or $I$ (Deflection Angle / Central Angle): The angle of intersection between the back and forward tangents, which is equal to the central angle subtended by the circular arc.
$R$ (Radius): The radius of the circular arc.
$T$ (Tangent Length): The distance from PC to PI, or from PI to PT.
$L$ (Length of Curve): The length of the circular arc from PC to PT.
$LC$ (Long Chord): The straight-line distance from PC to PT.
$E$ (External Distance): The distance from the PI to the midpoint of the curve along the radial line connecting the PI to the curve center.
$M$ (Middle Ordinate): The distance from the midpoint of the curve to the midpoint of the long chord.

Degree of Curve ($D$)

Degree of curve is a measure of the sharpness of a curve. There are two standard definitions:

1. Arc Definition ($D_a$)

The degree of curve is the central angle subtended by a circular arc of exactly $100\text{ ft}$. This definition is used exclusively in highway engineering.

$$D_a = \frac{360^\circ \times 100\text{ ft}}{2 \pi R} \approx \frac{5729.58}{R}$$

2. Chord Definition ($D_c$)

The degree of curve is the central angle subtended by a straight chord of exactly $100\text{ ft}$. This definition is used in railway engineering.

$$R = \frac{50}{\sin\left(\frac{D_c}{2}\right)}$$

Fundamental Circular Curve Formulas

The following geometric relationships are used to calculate the properties of a simple circular curve. Ensure your calculator is set to Degree Mode for all trigonometric functions.

Parameter	Formula	Alternative Form
Radius ($R$)	$$R = \frac{5729.58}{D_a}$$ (Arc)	$$R = \frac{50}{\sin(D_c/2)}$$ (Chord)
Tangent ($T$)	$$T = R \tan\left(\frac{\Delta}{2}\right)$$
Curve Length ($L$)	$$L = R \Delta \left(\frac{\pi}{180}\right)$$	$$L = \frac{100 \Delta}{D_a}$$
Long Chord ($LC$)	$$LC = 2 R \sin\left(\frac{\Delta}{2}\right)$$
External Distance ($E$)	$$E = R \left( \sec\left(\frac{\Delta}{2}\right) - 1 \right)$$	$$E = R \left( \frac{1}{\cos(\Delta/2)} - 1 \right)$$
Middle Ordinate ($M$)	$$M = R \left( 1 - \cos\left(\frac{\Delta}{2}\right) \right)$$	$$M = E \cos\left(\frac{\Delta}{2}\right)$$

Worked Example: Full Curve Parameter Calculation

Problem Statement

A simple horizontal circular curve is designed to connect two highway tangents. The deflection angle ($\Delta$) at the Point of Intersection (PI) is $42^\circ 30'$. The degree of curve ($D_a$, arc definition) is specified as $4.5^\circ$.

Calculate the radius ($R$) of the curve.
Find the tangent length ($T$).
Calculate the length of the curve ($L$).
Find the long chord length ($LC$).
Determine the external distance ($E$) and middle ordinate ($M$).

Solution

Calculate the Radius ($R$): Using the arc definition:
$$R = \frac{5729.58}{D_a} = \frac{5729.58}{4.5} = 1273.24\text{ ft}$$
Calculate the Tangent Length ($T$): Convert $\Delta$ to decimal degrees: $\Delta = 42^\circ + \frac{30'}{60} = 42.5^\circ$.
$$T = R \tan\left(\frac{\Delta}{2}\right) = 1273.24 \tan\left(\frac{42.5^\circ}{2}\right)$$
$$T = 1273.24 \tan(21.25^\circ) = 1273.24 \times 0.38888 = 495.14\text{ ft}$$
Calculate the Curve Length ($L$): Using the degree of curve formula:
$$L = \frac{100 \Delta}{D_a} = \frac{100 \times 42.5}{4.5} = 944.44\text{ ft}$$
Double-check with arc length formula:
$$L = R \Delta \left(\frac{\pi}{180}\right) = 1273.24 \times 42.5 \times 0.0174533 = 944.44\text{ ft}$$
(Matches!)
Calculate the Long Chord ($LC$):
$$LC = 2 R \sin\left(\frac{\Delta}{2}\right) = 2 \times 1273.24 \sin(21.25^\circ)$$
$$LC = 2546.48 \times 0.36244 = 922.94\text{ ft}$$
Calculate the External Distance ($E$) and Middle Ordinate ($M$):
- External Distance ($E$): $$E = R \left( \frac{1}{\cos(\Delta/2)} - 1 \right) = 1273.24 \left( \frac{1}{\cos(21.25^\circ)} - 1 \right)$$ $$E = 1273.24 \left( \frac{1}{0.93201} - 1 \right) = 1273.24 \times (1.07295 - 1) = 1273.24 \times 0.07295 = 92.88\text{ ft}$$
- Middle Ordinate ($M$): $$M = R \left( 1 - \cos\left(\frac{\Delta}{2}\right) \right) = 1273.24 (1 - \cos(21.25^\circ))$$ $$M = 1273.24 (1 - 0.93201) = 1273.24 \times 0.06799 = 86.57\text{ ft}$$ Check relation: $M = E \cos(\Delta/2) = 92.88 \times \cos(21.25^\circ) = 92.88 \times 0.93201 = 86.56\text{ ft}$ (Rounding agreement).

Answer

Radius ($R$): 1,273.24 ft
Tangent ($T$): 495.14 ft
Length of curve ($L$): 944.44 ft
Long Chord ($LC$): 922.94 ft
External Distance ($E$): 92.88 ft
Middle Ordinate ($M$): 86.57 ft

Crucial Exam Tips

Calculator Angle Settings: A common error is evaluating trigonometric functions in radians instead of degrees. Ensure your calculator displays “DEG”.
Arc vs. Chord Degree of Curve: If the problem does not specify, always assume highway design (arc definition: $R = 5729.58/D$). Only use chord definition if the problem specifically mentions railroads or track design.
Subdividing Deflection Angle: Make sure to divide $\Delta$ by 2 in the functions for $T$, $LC$, $E$, and $M$ ($T = R \tan(\Delta/2)$, etc.). Forgetting to divide $\Delta$ by 2 is a very common distractor choice in multiple-choice questions.