An alternate solution to Test Yourself: Stationing a Curve from The American Surveyor / November 2015
I didn’t see the geometric solution, so I used the Pythagorean Theorem, and the Quadratic Formula to
solve the problem.
Solve via three right triangles, the Pythagorean theorem, and the Quadratic Formula:
p = 334.260
NB. Distance R-B
q = 234.567 / 2
NB. Half chord
d = distance from PI to chord
NB. Solve for this unknown
t = tangent
NB. distance A-B
r = radius
NB. Distance A-R and also (p - d)
1
Algebraic solution for distance d
𝑡 2 = 𝑞2 + 𝑑2
(1) triangle with tangent as hypotenuse
𝑟 2 = 𝑞 2 + (𝑝 − 𝑑)2
(2) triangle with radius as hypotenuse
𝑝2 = 𝑡 2 + 𝑟 2
(3) triangle with distance ‘p’ as hypotenuse
𝑃2 = 𝑞 2 + 𝑑2 + 𝑞 2 + (𝑝 − 𝑑)2
(4) Substitute (1) and (2) into (3)
𝑝2 = 2𝑞 2 + 2𝑑2 + 𝑝2 − 2𝑝𝑑
(5) Square the last expression and collect the 𝑞 2 and 𝑑2 terms
0 = 2𝑞 2 + 2𝑑2 − 2𝑝𝑑
(6) Subtract 𝑝2 from both sides
2𝑑2 − 2𝑝𝑑 + 2𝑞 2 = 0
(7) now in standard quadratic form: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0
Use Quadratic Formula to solve for d
Output from calculation script
==================================================
Solutions to Quadratic Formula: 286.197 48.063
Choose the second one.
Curve Data
t: 126.750
r: 309.296
delta: 44 34 3
arc len: 240.587
PC sta: 873.250
PT sta: 1113.837
2
J Script to do the calculations [www.jsoftware.com]
NB. module: as2015-11.ijs
NB. solve problem from American
NB. Surveyor issue 2015-11
NB. saved: 2015-12-26 14:34
o=: smoutput
NB. Return hypotenuse of y
hy=: [:%:[:+/*:
p=: 334.26
q=: -: 234.567
pi=: 1000
NB. coeffents for quadratic formula
a=: 2
b=: -2*p
c=: 2**:q
NB. apply quadratic formula
x1=:((-b)+%:(*:b)-4*a*c)%2*a
x2=:((-b)-%:(*:b)-4*a*c)%2*a
t=: hy q,x2
r=: hy q,p-x2
delta=: 2*_3&o.t%r
len=: r*delta
pc=: pi-t
pt=: pc+len
NB. Display the results
o LF,50#'='
o 'Solutions to Quadratic Formula ',10j3": x1,x2
o 'Choose the second one.',LF
o 'Curve Data',LF
o 't: ',10j3":t
o 'r: ',10j3":r
o 'delta: ',4j0 3j0 3j0":0 60 60#:delta*648000p_1
o 'arc len: ',10j3":len
o 'PC sta: ',10j3":pc
o 'PT sta: ',10j3":pt
3