158 



RAILROAD CURVES 



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 impor- 

 tance 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 perpendicu- 

 lar 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 

 EEC and EC B are 

 equal. 



3. An angle not ex- 

 ceeding 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 





