While looking for the Original Special Instructions for a 1896 GLO Survey this morning (which I found in my desk, much to my delight), I came across the following legal description (from an old unrelated project):
“A part of the... being a strip of land 200 feet wide, to wit: 100 feet on each side of a line described as follows:”
“Beginning at a point on the west boundary of the Southwest quarter of the Northeast quarter of said Section 16, 1686.3 feet South of the North quarter corner of said section; thence South 83° 20' East 298 feet; thence on a spiral curve to the right with 11-15 foot chords, tangent to course South 80° 23' [sic] East; thence continuing on a curve to the right, tangent to course South 78° 23' East, radius 955 feet, distance 1604.9 feet; thence on spiral curve to right, with 11-15 foot chords, tangent to course South 17° 54' West; thence South 22° 51' West 740 feet, more or less to the South boundary of said Northwest quarter of the Southeast quarter of said Section 16, containing 13.6 acres.”
Although this particular deed is dated 1948 (Railroad to third party), it is essentially the same description from the original 1914 Right-of-Way deed to the Railroad from a private party (which I have [somewhere], but not in front of me just now).
I realize that the above description has some geometric problems (and typos), but combined with all of the original and subsequent deeds, the 1914 Map filed with the GLO, a County Road Plat from the 1920s, and other documents of record (and some NOT of record), I was able to reconcile the “record(s)” (and the typos & less than perfect geometry therein) with the extant physical evidence on the ground.
The point being... Sometimes Railroad Right-of-Ways ARE along the SPIRAL.
Loyal
Typos and mistakes aside, how in this modern age would you attempt to verify the area of the description in question? Would you create the centerline as described (as best as possible) and "offset" it and put a polyline around it, or what method would you use? I am just curious.
Railroad ROW Spiral
Good question Robert.
Once the CORRECT geometric parameters (Delta, Degree, Spiral definition, etc.) are determined, it is no different than any other compound curve. The “original” Centerline is solved (usually a best-fit of some sort), and parallel ROW Lines are computed.
The Searles Spiral is simply a compounding series of circular curves, so a “curve table” containing the data for the 23 curves involved will do the trick. Each of the 11 simple curves in the first spiral transition is simply repeated in reverse order on the other side of the central curve.
I try and “throw” any Total Delta variation into the Central Curve for simplicity's sake, but that depends on what the physical evidence (i.e. ROW Fences etc.) on the ground indicate too. I also try and avoid overall “scaling” whenever possible, but sometimes that IS the BEST-fit (and the right thing to do).
When entering the Searles Transition Spiral into [say] AutoCad, you must use the COMPUTED radius of each component curve if you want things to work out properly. I run them out to nine decimal places, and this seems to work quite well. In the above example, the first Searles combination would look something like this:
Searles Spiral Solution
Spiral Delta DD.MMSS/D.ec = 4.57 / 4.95
Number of Chords = 11
Chord Lengths = 15.000
c# Delta Chord Approximate Radius
Degree
1 0º 4' 30" 15.000 0º 30' 11,459.156 720 739
2 0º 9' 0" 15.000 1º 0' 5,729.579 587 554
3 0º 13' 30" 15.000 1º 30' 3,819.721 088 575
4 0º 18' 0" 15.000 2º 0' 2,864.792 248 149
5 0º 22' 30" 15.000 2º 30' 2,291.835 271 143
6 0º 27' 0" 15.000 3º 0' 1,909.864 225 850
7 0º 31' 30" 15.000 3º 30' 1,637.027 998 677
8 0º 36' 0" 15.000 4º 0' 1,432.401 032 832
9 0º 40' 30" 15.000 4º 30' 1,273.246 907 872
10 0º 45' 0" 15.000 5º 00' 1,145.923 771 533
11 0º 49' 30" 15.000 5º 30' 1,041.750 445 100
You will notice above that the solved radii are slightly different than the “true” Chord Definition radii, but that is the nature of the beast. From a layout stand point, this is a trivial variation, but AutoCad doesn't see it that way.
Other folks do it other ways, but this is how I usually handle the solution.
Once you have the above done, a polyline of the exterior will get you the area.
Sorry about miss-spelling Railroad in the post title, fat fingers!
Loyal
Railroad ROW Spiral
> Good question Robert.
>
> Once the CORRECT geometric parameters (Delta, Degree, Spiral definition, etc.) are determined, it is no different than any other compound curve. The “original” Centerline is solved (usually a best-fit of some sort), and parallel ROW Lines are computed.
>
That's where I'm having some difficulty, Loyal.
I've got the central curve figured as a 6 degree curve (5730/955 using their common short-cut method) which, given a central angle of 96d17', gives me a 100' chord-length arc of 1604.7' which checks ok with the given 1604.9' arc length (they probably got it from some table out of Searls). I've figured the 11x15' spiral segments out as most likely 11-5 minute spirals, which gives me the following chord arcs:
infinity
10313.2549
5156.6492
3437.7904
2578.3683
2062.7208
1718.9607
1473.4219
1289.2714
1146.0465
1031.4695
937.7273
Like you say, the radius' will have some "throw" in them, so the last radius sometimes exceeds the degree of curve of the central curve. The radius' I've calculated should be chord radius' using Searle's formulas.
There seems to be some inconsistency in the given bearings in and out vs the tangent bearings, so I'm having difficulty coming up with the total delta (so I can determine the spiral delta). The spiral deltas seem to be 3 degrees each by my best guess. The railroad plans should indicate the total spiral delta.
I do agree that spirals rights of way offsets were quite common on the railroads from the late 1800's and early 1900's. They were all using Searle's book, "The Railroad Spiral, The Theory of the Compound Transition Curve", by William H. Searles, 1882. The entire Utah & Salt Lake R.R. line is filled with them throughout its length acquired in 1912-1918.
JBS
I have a copy of Searles "Field Engineering, a Handbook of the Theory and Practice of Railroad Surveying, Location and Construction" which I inherited from my Father. It is the 22nd edition last copyright given is 1949. Authors listed are Searles, Ives and Philip Kissam (22nd Edition).
John
Apologies John (and others).
The Curve in question is a really BAD example simply because of all of the Booboos in the description. What's worse, is that the “ORIGINAL” document filed with the GLO in 1914, is FULL of typos, transpositions, and other ambiguities as well! I don't have the Original 1914 ROW Deeds handy (in storage down in Utah), but I was able to piece things together when I was working this project many years ago. Based on the physical evidence, AND the data contained in ALL of the documents that I had access to...
The “correct” Curve data is (stationing from original 1914 GLO filing plat):
P.S. 302+04.8
s = 4°57' 11x15'
P.C.C. 303+69.8
Delta = 96°15'
Degree = 6°
P.C.S. 319+74.0
s = 4°57' 11x15'
P.T. 321+39.0
Total Delta = 106°09'
The Central Curve length solves at 1604.194' v. 1604.2' according to the stationing. This is of course chord stationing. The arc would be 1604.9', which agrees with the length in the description. Now WHY the description uses the arc length is anybodies guess, and it is just one more of the ambiguities that tend to muddy the water on this one.
The GLO filing shows a bearing of S84°00'E going in, and S22°09'W going out, which works out to a Total Delta of 106°09'. Without the 1914 GLO Filing Map, it would have been a real stinker to figure out. In fact, all of the data on this particular curve is correct on the 1914 GLO Filing Map.
Here's some additional poop for those who might be interested:
Once you have that data, the easy way to solve for the first spiral delta is:
D1 = sD/(n*((n+1)/2))
Where:
D1 = the first spiral delta
sD = the total spiral delta
n = the number of chords (curves)
The Deltas of the several simple (chord definition) curves that form a Searles Spiral, increase from Curve 1 (D1) to Curve n (Dn, the last curve),
e.g. 7 chord Searles Spiral
D1 + D1 = D2
D2 + D1 = D3
D3 + D1 = D4
D4 + D1 = D5
D5 + D1 = D6
D6 + D1 = D7
D1+D2+D3+D4+D5+D6+D7 = sD (Spiral delta)
Hope ol' fat-fingers hasn't screwed this up...
Loyal
I think that's right now (fingers crossed)
John
Matching my numbers precisely, Loyal. (Now that I don't have to guess on the parameters) ;o)
Given: Deg Min Sec D.dddd Central Curve Data: Spiral Curve Stationing:
Degree of Curve (Dc) 6 0 0.0 6.0000 Radius (Rc) 955.366 T.S. 30204.8
Total Delta (I) 106 9 0.0 106.1500 Delta 96.250 S.C. 30369.8
Delta of Circular Curve (Ic) 96 15 0.0 96.2500 Length 1604.167(100' chords) C.S. 31974.0
Delta of Spiral Curve (Is) 4 57 0.0 4.9500 Tangent 1065.708 S.T. 32139.0
Number of spiral chords (n) 11
Length of spiral chords (ls) 15
Length of spiral (Ls) 165 (a)
First spiral arc Ds (min) 5 (b)
Where:
(a) Ls = n*ls (e) Bc= atan (Ys/Xs) (h) Yr = Ys-(Rc*sin ds)
(b) Ds = Is/(n*((n+1)/2)) (f) Lc = sqrt (Ys^2+Xs^2) (i) Xr = Xs+(Rc*cos ds)
(c) Ys = cos Bs * ls (g) Rs = ls/(sin(ds/30)*2) (j) Tt = y + x tan I/2 + Rc * (sin I/2 - Is) / (cos I/2)
(d) Xs = sin Bs * ls
ls = Position along spiral
ds = Spiral Delta (min)
Bs = Spiral Chord Brg (min)
Ys = Tangent Distance from T.S. to P.C.C. of spiral segment
Xs = Tangent Offset from tangent to P.C.C. of spiral segment
Bc = Bearing of Chord (deg) from T.S. to P.C.C. of spiral segment
Lc = Long Chord length from B.S. to point on spiral segmt
Rs = Radius of each spiral segment
Yr = Tangent Distance from T.S. to radius point of central curve
Xr = Tangent Offset from tangent to radius point of central curve
Tt = Long Tangent Total T.S. to P.I.
(c) (d) (e) (f) (g)
(ls) (ds) (Bs) (Ys) (Xs) (Bc) (Lc) (Rs)
0 0 0.0 0.000 0.000 0.000000 0.000 infinity
15 5 2.3 15.000 0.010 0.037500 15.000 11459.1690
30 9 9.0 30.000 0.049 0.093750 30.000 5729.6041
45 14 20.3 45.000 0.137 0.175000 45.000 3819.7579
60 18 36.0 59.999 0.295 0.281250 60.000 2864.8413
75 23 56.3 74.997 0.540 0.412499 74.999 2291.8966
90 27 81.0 89.993 0.893 0.568748 89.997 1909.9379
105 32 110.3 104.985 1.374 0.749994 104.994 1637.1139
120 36 144.0 119.972 2.002 0.956236 119.989 1432.4992
135 41 182.3 134.951 2.797 1.187472 134.980 1273.3574 (h) (i) (j)
150 45 225.0 149.919 3.778 1.443698 149.966 1146.0465 (Yr) (Xr) (Tt)
165 50 272.3 164.872 4.965 1.724909 164.946 1041.8854 82.4365 956.7680 1355.5745
I'd say we pretty much drilled it! Now, to offset those spirals...
JBS
John
Good deal.
Offsetting the spirals?
Heck that's the easy part considering that Searles used simple-circular curves! What is REALLY sweet, is when the offset-spiral solution FITS the 100+ year old Concrete ROW Fence posts like a glove.
The hard part is figuring out some of those ambiguous documents in the chain of title. Of course that's the reason for having ALL of the documents that you can get your hands on.
🙂
Loyal
John
I always figure if you're missing the right of way posts, it's not the RR surveyor who's in the wrong. Those guys really new how to survey. Once you get onto their alignment, it all falls right into place. Having the right numbers is just the start.
JBS
> I have a copy of Searles "Field Engineering, a Handbook of the Theory and Practice of Railroad Surveying, Location and Construction" which I inherited from my Father. It is the 22nd edition last copyright given is 1949. Authors listed are Searles, Ives and Philip Kissam (22nd Edition).
Google books has it.
John
I agree with JB about that. Have overlaid their most ancient work over new surveys and orthophotos, and I have been repeatedly shocked 100% of the time as to how accurate they were.