curves may be included what are known as easement curves, transition curves, and spiral curves, now used very generally on the more important railroads.

A reverse curve is a continuous curve composed of the arcs of two circles of the same or different radii, the centers of which lie on opposite sides of the curve, as in Fig. 3. The two arcs composing the curve meet at a common point or point of reversal M, at which point they are tangent to a common line perpendicular to the line joining their centers. Reverse curves are becoming less common on railroads of standard gauge.

GEOMETRY OF CIRCULAR CURVES

The following principles of geometry. are of special importance as relating to curves:

1. A tangent to a circle is perpendicular to the radius at its tangent point. Thus, in Fig. 4, AF is perpendicular to BO

at its tangent point B. and ED is perpendicular to CO at C.

2. Two tangents to a circle from any point without the circle are equal in length, and make equal angles with the chord joining their points of tangency. Thus BE and CE are equal, and the angles EBC and ECB are equal.

3. An angle not exceeding 90° formed by

a chord and the tangent at one of its extremities is equal to one-half the central angle subtended by the chord. Thus, the angle EBC = ECB one-half BOC.

4. An angle not exceeding 90° having its vertex in the circumference of a circle and subtended by a chord of the circle, is equal to one-half the central angle subtended by the chord. Thus, the angle GBH, whose vertex B is in the

circumference, is subtended by the chord GH and is equal to one-half the central angle GOH, subtended by the same chord GH.

5. Equal chords of a circle subtend equal angles at its center and also in its circumference, if the angles lie in corresponding segments of the circle. Thus, if BG, GH, HK, and KC are equal, BOG=GOH, GBH=HBK, etc.

6. The angle FEC, called the angle of intersection, of two tangents of a circle is equal to the central angle subtended by the chord joining the two points of tangency. Thus, the angle CEF = BOC.

7. A radius that bisects any chord of a circle is perpendicular to the chord.

8. A chord subtending an arc of 1° in a circle having a radius = 100 ft. is very closely equal to 1.745 ft.

ELEMENTS AND METHODS OF LAYING OUT A CIRCULAR CURVE

The degree of curvature of a curve is the central angle subtending a chord of 100'. Thus, if, in Fig. 4, the chord BG is 100 ft. long and the angle BOG is 1°, the curve is called a one-degree curve; but if, with the same length of chord, the angle BOG is 4°, the curve is called a four-degree curve.

The deflection angle of a chord is the angle formed between any chord of a curve and a tangent to the curve at one extremity of the chord. It is equal to one-half the central angle subtended by the chord. The deflection angle for a chord of 100 ft. is called the regular deflection angle, and is equal to onehalf the degree of curvature. The deflection angle for a subchord-that is, for a chord less than 100 ft.-is equal to onehalf the degree of curvature multiplied by the length of the subchord expressed in chords of 100 ft. The length c of a subchord or of any chord is given by the equation

c = 2 R sin D,

in which R is the radius and D the deflection angle of that chord.

Relation Between Radius and Deflection Angle.-From the

If D100 is the deflection angle for a chord of 100 ft., then

50

R=

sin D100

For a 1° curve, D100=30′ and R=5,730, nearly. For curves

less than 10°, the radius may be taken as

the degree of curvature. The accompanying table gives the length of the radius, in feet, for degrees of curvature ranging by intervals of 5′ and 10' from 0' to 20°.

Tangent Distance.-The point where a curve begins is called the point of curve, and is designated by the letters P. C.; and the point where the curve terminates is called the point of tangency, and is designated by the letters P. T. The point of intersection of the tangents is called the point of intersection; it is designated by the letters P. I.