Elements of a Simple Curve

Figure 2 shows the elements of a simple curve. They are described as follows, and their abbreviations are given in parentheses.

Point of Intersection (PI)

The point of intersection marks the point where the back and forward tangents intersect. The surveyor indicates it as one of the stations on the preliminary traverse.

Intersecting Angle (I)
The intersecting angle is the deflection angle at the PI. The surveyor either computes its value from the preliminary traverse station angles or measures it in the field.

Radius (R)
The radius is the radius of the circle of which the curve is an arc.

Point of Curvature (PC)
The point of curvature is the point where the circular curve begins. The back tangent is tangent to the curve at this point.

Point of Tangency (PT)
The point of tangency is the end of the curve. The forward tangent is tangent to the curve at this point.

Length of Curve (L)
The length of curve is the distance from the PC to the PT measured along the curve.

Tangent Distance (T)
The tangent distance is the distance along the tangents from the PI to the PC or PT. These distances are equal on a simple curve.

Central Angle (Δ)
The central angle is the angle formed by two radii drawn from the center of the circle (0) to the PC and PT. The central angle is equal in value to the I angle.

Long Chord (LC)
The long chord is the chord from the PC to the PT.

External Distance (E)
The external distance is the distance from the PI to the midpoint of the curve. The external distance bisects the interior angle at the PI.

Middle Ordinate (M)
The middle ordinate is the distance from the midpoint of the curve to the midpoint of the long chord. The extension of the middle ordinate bisects the central angle.

Degree of Curve (D)
The degree of curve defines the “sharpness” or “flatness” of the curve (figure 3). There are two definitions commonly in use for degree of curve, the arc definition and the chord definition.

Arc definition. The arc definition states that the degree of curve (D) is the angle formed by two radii drawn from the center of the circle (point O, figure 3) to the ends of an arc 100 feet or 30.48 meters long. In this definition, the degree of curve and radius are inversely proportional using the following formula:

As the degree of curve increases, the radius decreases. It should be noted that for a given intersecting angle or central angle, when using the arc definition, all the elements of the curve are inversely proportioned to the degree of curve. This definition is primarily used by civilian engineers in highway construction.

English system. Substituting D = 1° and length of arc = 100 feet, we obtain

Metric system. In the metric system, using a 30.48-meter length of arc and substituting D = 1°, we obtain—

Chord definition. The chord definition states that the degree of curve is the angle formed by two radii drawn from the center of the circle (point O, figure 3) to the ends of a chord 100 feet or 30.48 meters long. The radius is computed by the following formula:

The radius and the degree of curve are not inversely proportional even though, as in the arc definition, the larger the degree of curve the “sharper” the curve and the shorter the radius. The chord definition is used primarily on railroads in civilian practice and for both roads and railroads by the military.

English system. Substituting D = 1° and given Sin ½ 1 = 0.0087265355.

Metric system. Using a chord 30.48 meters long, the surveyor computes R by the formula

Chords
On curves with long radii, it is impractical to stake the curve by locating the center of the circle and swinging the arc with a tape. The surveyor lays these curves out by staking the ends of a series of chords (figure 4). Since the ends of the chords lie on the circumference of the curve, the surveyor defines the arc in the field. The length of the chords varies with the degree of curve. To reduce the discrepancy between the arc distance and chord distance, the surveyor uses the following chord lengths:

The chord lengths above are the maximum distances in which the discrepancy between the arc length and chord length will fall within the allowable error for taping, which is 0.02 foot per 100 feet on most construction surveys. Depending upon the terrain and the needs of the project foremen, the surveyor may stake out the curve with shorter or longer chords than recommended.

Deflection Angles
The deflection angles are the angles between a tangent and the ends of the chords from the PC. The surveyor uses them to locate the direction in which the chords are to be laid out. The total of the deflection angles is always equal to one half of the I angle. This total serves as a check on the computed deflection angles.