RAILROAD CURVES 159 



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 cen- 

 ter and also in its circumference, if the angles lie in correspond- 

 ing segments of the circle. Thus, if BG, GH, HK, and KC 

 are equal, BOG = GOH, GBH = HBK, etc. 



6. The angle FEC, palled 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 perpen- 

 dicular 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 CIR- 

 CULAR CURVE 



The degree of curvature of a curve is the central angle sub- 

 tending 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 extrem- 

 ity 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 one- 

 half the degree of curvature. The deflection angle for a sub- 

 chord that is, for a chord less than 100 ft. is equal to one- 

 half the degree of curvature multiplied by the length of the 

 subchord expressed in chords of 100 ft. The length c of a sub- 

 chord 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 



/: 

 equation just given, 



2sinD 

 12 



