CURVE AND REDUCTION TABLES Published by Eugene Dietzgen Co. CURVE FORMULAS 1. Radius : R=\frac{50}{\sin D/2} 2. Degree of Curve: D=100 \frac{I}{L}. Also, \sin D/2=\frac{50}{R} 3. Tangent : T=R \tan \frac{1}{2} I. Also, T=\frac{T \text{ for } 1^{\circ}\text{ curve}}{D}+C. 4. Length of Curve: L=100 \frac{I}{D} 5. Long Chord : L. C.=2R \sin \frac{1}{2} I. 6. Middle Ordinate: M=R (1-\cos \frac{1}{2} I) 7. External : E=\frac{R}{\cos \frac{1}{2} I}-R. Also, E=T \tan \frac{1}{4} I. EXPLANATION AND USE OF TABLES Given P.I. Sta. 83+40.7, I=45^\circ 20' and D=6^\circ30' find: Stations-P.C.=P.I.-T. T=\frac{T \text{ for } 1^{\circ}\text{ Curve}}{D}+C. From Tables V and VI T=\frac{2392.8}{6.5}+.197=368.32=3+68.32. Sta. P. C.=83+40.7-(3+68.32)=79+72.38. P. T.=P. C.+L, and L=100 \frac{I}{D}=100 \frac{45.33}{6.5}=697.38 Therefore, P. T.=(79+72.38) +(6+97.38)=86+69.76. Offsets-Tangent offsets vary (approximately) directly with D and with the square of the distance. From Table III Tangent Offset for 100 feet=5.669 feet. Distance =80-Sta. P. C.=27.62. Hence offset=5.66 \times\left(\frac{27.62}{100}\right)^{2}=.432 ft. Also, square of any distance, divided by twice the radius equals (approximately) the distance from tangent to curve. Thus (27.62)^{2}\div(2 \times881.95)=.432 ft. Deflections-Deflection angle=\frac{1}{2} D for 100 ft., \frac{1}{4} D for 50 ft., etc. For "X" ft., Deflection Angle (in minutes) =.3 \times X \times D. For Sta. 80 of above curve Deflection Angle =.3 \times27.62 \times6.5=53.86'. Also Deflection Angle=dfI. for 1 ft. from Table III \times X=1.95 \times27.62=53.86'. For Sta. 181 Deflection Angle=53.86'+\frac{6^{\circ} 30'}{2}=4^{\circ} 8.86'. Externals-From Table V for 1^\circ curve, with central angle of 45^\circ 20', E=479.6. Therefore, for 6^\circ 30'curve, E=\frac{479.6}{6.5}+\text{Correction from Table VI}=7.378+.039=7.417. 1