Handbook of Civil Engineering Calculations

xc and yc = coordinates of SC; k and p = abscissa and ordinate, respectively, of PC. The
coordinates of the SC and PC are useful as parameters in the calculation of required dis-
tances. The distance p is termed the throw, or shift, of the curve; it represents the dis-
placement of the circular curve from the main tangent resulting from interposition of the
spiral.
The basic angles are A = angle between main tangents, or intersection angle; Ac = an-

gle between radii at SC and CS, or central angle of circular curve; (^6) S = angle between
radii of spiral at TS and SC, or central angle of entire spiral; Dc = degree of curve of cir-
cular curve D = degree of curve at given point on spiral; (^8) S = deflection angle of SC from
main tangent, with transit at TS; 8 = deflection angle of given point on spiral from main
tangent, with transit at TS.
Although extensive tables of spiral values have been compiled, this example is solved
without recourse to these tables in order to illuminate the relatively simple mathematical
relationships that inhere in the clothoid. Consider that a vehicle starts at the TS and trav-
erses the approach spiral at constant speed. The degree of curve, which is zero at the TS,
increases at a uniform rate to become Dc at the SC. The basic equations are
a sL^5 cD^0 L s^5 fiz\
(^6) *~ 200 ~ 2R
C
(25)

/ 0,^2 \

^-M

1

-To)

(26)

/ os o? \

*-MT-£)

(27)

k = xc-Rc sin B 3 (28)

p =yc-R 0 (I-COSO 5 ) (29)

Ls Os

*-6JTT

(30)

-(r)>«

(31)

\L's I

S=(T!''*• <

32

\ Ls ) >

Ts = (Rc + p) tan HA + k (33)

Es = (R 0 + p)(sec Y 2 A + 1) + p (34)

LT = xc-yccot6s (35)

ST=^CScOj, (36)