R PA P P A 777 MATHEMATICS ?,!5 PAPER LOCATION OF A RAILROAD J, C. L. FISH MATHEMATICS OF THE PAPER LOCATION OF A RAILROAD J. C. L. FISH Assoc. M. Am. 8oc. C. E. Associate Professor of Civil Engineering Leland Stanford, Jr., University, Palo Alto, Cal. NEW YOKK M. C. OLAEK 13-21 PARK Row 1905 Copyright, 1905, by M. C. Clark The object of this article is to present the mathematical steps involved in preparing, from the paper location of a railroad, a set of mathe- matically consistent alinement notes by which to run the corresponding field location; and to sug- gest an orderly arrangement for the necessary computations. Mr. W. B. Storey, Jr., in discussing a paper by Michael L. Lynch on Railroad Location, says:* On the Southern Pacific System the location ia trans- ferred from the paper to the ground, not by scaling, but by calculation. Each tangent is fixed, and the connect- ing curves are all calculated in the office by carrying the line from one fixed tangent around through the prelim- inary to the next tangent. These calculated notes are then given to a machine known as the locating party and put on, the ground by it. The writer does not remember having seen else- where any reference to the calculation of field notes; but in various articles on railroad location he has noted direct or implied reference to scal- ing. To make clear what is meant by scaling and by calculating, and to show the advantage of calcu- lating over scaling, let us find, by the two methods, the station and "plus" via the prelimi- nary and via the location for the check point A, Fig. 1, the first crossing of the location and pre- liminary w;hich have a common starting point, Pi. BY SCALING. The angle at Pi, between pre- liminary and location, is scaled in the course of preparing the notes for field location. Simply for the purpose of getting a check on the location at Trans. Am. Soc. C. E., Vol. XXXI., p. 92. (Reprinted from Engineering News, March 16, 1905.) 381115 5000 FT. 4000 (1500,4000) R 1000 Pape (3000, 4200) fg <fr2f.' /r P 7 (4000, 4-500) 4000 FT. FIG. 1. METHOD OF PLOTTING PAPER LOCATION OF A RAILWAY LINE. the point A we scale the distance PiA on the loca- tion and the distance PaA on the preliminary, thus obtaining the station and plus for A via each of the two lines. We now have of the triangle PiPaA the given side PiPa (taken from the field notes of preliminary), the given angle PiPaA, the two scaled sides PiA and PaA, and one scaled angle, APiPa. All the scaled quantities are af- fected 'by the errors of scaling, which are large compared with field errors, and it is evident that our values for the five parts of the triangle are mathematically inconsistent, since any three parts (which include one side) of a triangle determine the other parts. The result is that when the field party has located to A, and found they do not check on the preliminary within several feet, they have no means of telling how much of the error is due to surveying and how much to scaling. In this case the surveying checks the scaling, but there practically is no check on the field work. BY CALCULATION. iBy this method we scale only so many parts of the triangle PiPaA as will, together with the given part or parts, determine the size and shape of the triangle. Side PiPa and angle PiPaA are known from the preliminary notes. If, then, we scale the angle PaPiA we shall have values for three parts of the triangle, from which we can compute the sides PiA and PaA. These five parts of the triangle 'two given, one scaled and two computed 'are mathematically consistent; and when the location party arrives at A and find they do not exactly check on the pre- liminary, they know that the error is all charge- able to field work (assuming that no errors have been made in computing). In this case the field work is really checked. Of course the error may be in running the preliminary or the location, or more probably in both. The check is on the sur- veying on both lines between Pi and A. If the error is not within the limit of error permitted, the location is re-run (using the same notes), and if this is found to be practically correct, the next step is to re-run the preliminary from Pi to A. If the error is not discovered here it must be found in the office work computations or copying of notes. It cannot be due to scaling. 5 MATHEMATICS OF PAPER LOCATION: USING RECTANGULAR CO-ORDINATES. The chief points of the preliminary, surveyed by any method or combination of methods, have been plotted by rectangular co-ordinates, and our map shows points Pi, P 2 , P 3 , P4, Ps, Pe and PT. The co- ordinates of each point are written by the point, the value of the abscissa first. The engineer has drawn in pencil on the map the location Pi, PCi, PTi, PCs, PT 2 , PCs. PCi and PTi are respectively the beginning and end of a 6-degree curve; and PCz and PTa are the beginning and end respec- tively of an 8-degree curve. PCs is the beginning of a curve, and is considered the end of this loca- tion. It will be noticed that each curve is num- bered, and that each letter standing for a curve element is given for a subscript the number of its tmrve; e. g., the central angle of curve No. 2 is AS. All dimensions which appear on the map at this time are put in parentheses on our figure in order to distinguish them from the quantities computed in the work of getting out the location notes, which are written without parentheses. We have nothing to do here with the question of the proper place for the location line on the map: that is an engineering problem, while our work is to take the location line as we find it, and get out the notes for transferring it to the ground. Hav- ing no need of the topography in what follows, It has been omitted from our map. The distances have been computed here to the nearest foot only, although it is customary to consider tenths of a foot. Of course, in any given location the limit of error in field work should control the precision of computation. To save space there have been omitted from Fig. 1 the station numbering of the preliminary and of the location. 1. PRODUCE THE TANGENTS to intersect at Vi and V 2 . 2. SCALE THE CO-ORDINATES of the initial point, the final point, and of all the points of tan- gent intersection. In our case, the initial point of the location is at the origin of co-ordinates, and coincides with the initial point of the preliminary. The co-ordinates of Vi and V 2 and PCs are written by the points on the map as they are scaled off, x first and y second. 6 No further scaling is done, except that every computed distance and angle should be roughly checked by scaling. We are now ready to take up the computing. 3. COMPUTE THE BEARINGS AND LENGTHS OF TANGENTS PRODUCED TO INTERSECTION. The plus direction of the y-axis is taken as north. The tangent of the bearing of PiVi is tan *S PV = X Y ,-XP, 3,zso~o (Throughout, the subscript of an x or a y of any point is the symbol which designates that point; and the subscript of is composed of the two symbols which designate the terminal points of a line.) log 3,250 3.51188 log 3,250=3.51188 log 1,610 = 3.20683 log sin 63 39' = 9.95236 log tan 63 39' = 0.30505 log 3,627 = 3.65952 From the logarithmic computation we find the bearing and length of PiVi to be N 63* 39' E, 3,627 ft. The tangent of the bearing of ViVa is , 3,640-3,250 2,030 and log 2,100 = 3.32222 n log 2,100 = 3.32222 log 2,080 = 3.30750 log sin 46^ 58' - 9.85609 45 58' = 0.01472 n 2,921 = 3.46553 That is, the bearing and length of ViVi are N 45 58' W, 2,921 ft. (The negative value of the tangent shows that the bearing is either S E or N W, but the map indicates that the bearing is NW.) The tangent of the bearing of VaPCs is and log 2,380 = 3.37658 logr 2,380 3.37658 log 480 = 2.68124 log sin 78 36' = 9.99135 log tan 78 36' = 0.69534 log 2,428=3.38523 giving- for VaPCa the bearing and length of N 78 36' E, 2,428 ft. We now scale the map for a rough check on these computed bearings and distances. 4. COMPUTE CENTRAL, ANGLES OF CURVES. It is evident from the map that the central angle for curve No. 1 is Ai = ^ P]VI + ^v!V 2 = 63 39 ' + 45 58/ = 109 37/ ; and the central angle for curve No. 2 is Aa = ^Vj + ^V 2 PC 3 = 45 58 ' + 78 36/ = 124 34/ - We roughly check these values by scaling the map with the protractor. 5. COMPUTE TANGENT - DISTANCES OP CURVES. Using a "table of tangent-distances for a 1. curve" we find the tangent-distance for curve No. 1 is Ti = 1,355; and for curve No. 2, T 2 = 1,364. We obtain a rough check on these by scaling. If in any case the value of A is beyond the limits of the table, of course the tangent-distance must be computed from the formula: T = R tan A/2. 6. COMPUTE CURVE LENGTHS. Length of curve is equal to one hundred times the ratio of central angle to degree of curve. Length of curve No. 1 is Lj 100 Ai / I>i = 100 (109 370/6 = 100 (109.62) / 6 = 1,827. 8 Length of curve No. 2 is L s 100 Aa/D a 100 (124 34') / 8 = 1OO (124.57) / 8 1,557. We now step off the curve lengths on the map to obtain a rough check on the computed values. 7. COMPUTE LENGTHS OF CENTER-LINE TANGENTS. The length of the first tangent, as the map shows, is PiPCi = PiVi Ti = 3,627 1,355 = 2,272. The second tangent is PTi PC 2 = ViVs (Ti + T 2 ) = 2,921 (1,355 + 1,364) = 202; And the third tangent is PTa PCs = Va PCs T 2 = 2,428 1,364 = 1,064. The rough check by scaling is now employed. 8. COMPUTE STATION AND PLUS FOR CURVE AND TANGENT POINTS. The station and plus for Pi is + 00. The station and plus for PCi Is PiPCi = (22 + 72). The station and plus for PTi is (22 + 72) + La. = (22 + 72) + 1,827 = (40 + 99). The station and plus for PCs is (40 + 99) + PTi PCs = (40 + 99) + 202 = (43 + 01). The station and plus for PTa is (43 + 01) + L = (43 + 01) + 1,557 = (58 + 58). The station and plus for PCs is (58 + 58) + PT PCs = (58 + 58) + 1,064 = (69 + 22). Now we prick off the stations on the map, thus checking roughly these values. 9. COMPUTE ELEMENTS FOR CHECK POINTS. On our map the location crosses the preliminary at the point A, which we use as a check point. There is no corresponding point at the other end of the line, and, in order to obtain a check, we draw the line PCsP? and compute its bearing and length, to be run in the field as an auxiliary line to check on point PT. CHECK POINT A. We first write the equa- tions for lines, P 2 Ps and PiVi. The general equa- tion for a straight line is y = ax + b, where a = (y n y^ / (x n x k ) and b = y k ax k . (The subscripts k and n refer to the initial and final points respectively of any line.) 9 For line P g P 3 a = (y 3 y 2 ) / (x 3 x 2 ) (1,700 600) (2,600 1,600) = 1,100/1,000 = 1.1; and b = y, ax 2 = 600 1.1 (1,600) = 1,160. Equation of line P a P 8 is therefore, y=l.l x + (1,160) or y - 1.1 x 1,160. For line PiVi i, 6W0 _ 04.954 3,250-0 and making the equation of PiVj y = 0.4954 x. We now compute the co-ordinates of the point A of intersection of PaP2 and PiVi. If the equation of the first line be written y = ax + b and the equation of the second be written y = a' x + b' then the co-ordinates of this common point are : and a check is had in the equation y ~a!x> +V Substituting numerical values of a, b, a', b', for rou point A, we get : (- J.160J - 951 We scale the map to get a rough check on the computations. It remains to find for A the station and plus via each line. The distance log 319 = 2.50379 log sin 42 16' = 9.82775 log 474.8 = 2.67604 The logarithmic computation makes P 2 A 474.3. 10 The station and plus of A on the preliminary Is PI P 2 -f ?2 A. = 1,709 + 474.3 - 21 + 83.3. The distance as computed by logarithms here : log 1,919 = 3.28299 log sin 63 39' = 9.95236 log 2,141 = 3.330Q3 The station and plus for A on the location is, then, 21 + 41. Scale the map to roughly check these values. It is evident that station 21 + 41 on the location survey should coincide with station 21 + 83.3 on the preliminary. In cases in practice a check point may be con- veniently obtained by producing a location tan- gent to intersect the preliminary, and making the computation in the foregoing manner. CHECK LINE PCaF?. When a crossing of the location with the preliminary is not near at hand for a. desired check on the field work, a check line, or tie line, is drawn between a chosen point of the location and a chosen point of the pre- liminary; and the bearing and length of the check line are computed. As an example, draw the check line PCsP?, and find the bearing by the equation: Vr, -ypc 3 4,500-4,120 and find the length log 470 2.67210 log 470 2.67210 log 380 = 2.57978 log sin 51 03' = 9.89081 log tan 51o 03' = 0.09232 log 604.4 = 2.78129 We find the bearing and length of PCaP? to be N 51 03' E, 604.4 ft. The location having been carried in the field to the point PCs, the transit- man deflects to the left at this point the angle ^PCs - ^PCsPr = 78 36 ' ~ 51 3 ' ---- 27 33/ ' and the chainmen lay off 604.4 on this course, and 11 should by so doing- arrive precisely at PT on the preliminary. 10. COMPUTE CURVE DEFLECTIONS. PCi is 22 + 72 making- the first sub-chord on the 6-degree curve, 28 ft. The corresponding deflection is .28 x 3 = .28 x 180' == 50', i. e., the tangent deflection for station 23 is 50 minutes. The deflection for sub-chord at .PCa is .99 x 240' = 238' = 3 58'. The deflection for sub-chord at PTi is .99 x 180' = 178' = 2 58'. The deflection for sub-chord at PT 8 is .58 x 24tf = 139' = 2 19'. ALGORITHMS. To facilitate explanation the foregoing computations have been put down in a Red- Fig. 2. Design for Rulings on Computing Paper. rambling manner which makes the computed quantities hard to find when wanted. For a lon- ger location than this, economy of time and effort requires that the routine computations be syste- matically arranged, that like operations as well as like quantities may be brought together. By this means entering the data and making the computa- tions in a short time become largely mechanical processes, and at the end the computed quantities stand in tabular order and may be quickly found when wanted. For the computations of this ar- tide the following computation forms, or algo- rithms, are suggested. While these may not pre- sent the best arrangement, they will at least show the advantage of order over disorder. The reader may be interested in the fact that the writer's stu- dents buy for their computing, letter-size sheets of paper ruled on one side with the special design shown in Fig. 2. By the use of this paper any algorithm may be followed without drawing lines, and columns of digits are kept vertical automat- ically. Algorithm 1 is for computing the bearing and length of lines which are terminated by points of known co-ordinates. Column A contains the num- bers of the horizontal lines of the algorithm. Col- umn B contains the symbols. Xk, yk designate the co-ordinates of the initial point of any line; x n , y n designate the co-ordinates of the final point of the line; )3kn is the bearing, and dk n is the length of the line. The lower half of this algorithm has to do with the check equation: hypot. = (base 2 + altitude 2 ) 1 / 2 . Notice that the values on line 21 check those on line 12. Of course this check will not detect errors made in entering values of co-ordinates on lines 1, 2, 3, 4. When a table of squares is at hand it may well be used in place of logarithms for the computation of checks. Columns C, D, E, F give for the line Pi Vi, Vi Va, Va PCs, and PCsP? respectively, the numerical quantities corresponding to the symbols of Column B. It is suggested that the order of steps in computing be: (1) Enter the co-ordinates for all the lines concerned; (2) Set down on lines 5 and 6 the differences; (3) Enter all the logs on line 7; (4) Enter all the logs on line 8, and so on; every operation in Column C being immediately repeated for the succeeding columns. Algorithm 2: Column A contains the numbers which have been used to designate the curves. In B will be found the degree of each curve. Each degree of curve is repeated in parenthesis, with the minutes expressed in decimal of a degree. In C are entered the central angles, each of which has been computed from two values of taken from Algorithm 1. For example: C2 (i. e., the 13 ICC*'* CO COO _, ^ibw^wS ^ ^ CO T-J iM iH CO kO l> ko kOko koc* 2s: OoOo rHOkOr-iCO.SlOCS CO CO<N-* CD rHTjt >> H-^OS-^iMCD'H k(5 iHkOiM CO COC^ <D l^ 05 0500 CO'M'O'OS'CO' t-coio l O O C<l t- ^ OOOOOO L- ^^^ O7 kO rH kO r-l 00 CO kC CD Cq (M - CD CO O O CS ' iMCD<NCDOOOO O COkCCO ^5 t-DCDC<lkOo* CO r-l CO rH rH CD ifl C^ 05 CO CO CO CO kO 05 iH 05 05 CO rHO O kOkO Q C* rH O kO CO r-l kO eoco o' oico *co t-co s ss , be M 6C tfi 6C 00 O 00 II .t : : : :3 rH <N CO "* kO CD t- 00 05 OrH(M 14 -9p paoqoqng PITB "BIS snid pire " iu -i . . . , o .to : : : M 01 <S1 . . Z jo urns jstp -Sum I S H -UOT!J09e.T9!lUI g 05 p9onpojd ^ c^ 4 (ITTQ^TTTJI CD iS i i| : 3" 9AJUO :g:: s ::: ifi : '% : ' : : <] r 15 quantity in Column C and on line (2) is 109 37', which is the sum of C13 (Alg. 1) and D13 (Alg. 1). C3 and C6 give the central angle with minutes re- duced to decimal of a degree. D contains the tangent distance for each curve. E contains the curve lengths. In P are entered the lengths of tangents produced to intersection points, copied from line 12 of Alg. 1. G contains the amounts to be subtracted from quantities in F to obtain the quantities in H. For example: G4 is the sum of D2 and D5, and H4 is F4 less G4. I contains the station and plus for each point of curve; and J contains the station and plus for each point of ALGORITHM III. Line Equations. Formulas : a = (y n = y k x n Line P 2 P 3 Line PiVj 1600 600 5 2600 3250 4 y n 1700 1610 5 y n y k 1100 1610 6 x n x k 1000 3250 7 log(y n -y k ) 3.04139 3.20683 8 log(x n x k ) 3.00000 3.51188 9 log a 0.04139 9.69495 10 a 1.1 0.4954 11 logx k 3.20412 12 logax k 3.24551 13 ax k 1760 14 b 1160 15 (Eq.)y = ax + b y = l.lx 1160 y = 0.4954x + ABC ALGORITHM IV. Co-ordinates of Intersection Point. Formulas : x = (b' b) / (a a') ; y = ax + b ; check, y = a'x + b'. Check Point A. 1..., (Eq.iy = ax + b y=l.lx 116O 2 (Eq.)'y = a'x + b' y= 0.4954 + 3 a 1.1 4 a' 0.4954 5 b 1160 6 b' 7 b' b 1160 8 B a' 0.6046 9 log(b b) 3.06446 10 log (a a' 9.78147 11 log x 3.28299 12 log a 0.04139 13 log a' 9.69495 14 log ax 3.32438 15 log a'x 2.97794 16 ax 2110. 17... a'x 950.5 18 x 1919. 19 y(=ax + b) 950. 20 y(=a'x + b') 950.5 1G tangent. 12 is rewritten from HI; J2 is 12 plus E2; 15 is J2 plus H4; J5 is 15 plus E5, and so on. K and L contain the subchord deflections. Algorithms 3 and 4 will need no explanation. 11. ALINEMENT NOTES FOR LOCATION. We are now ready to enter the results of our com- putations in the field book in the usual form of alinement notes. Location Alinement Notes. Computed Sta. Curve Point. Deflections. Bearings. 69 + 22 PC 8 Check : Deflect 27 33' (bearing N 51 03' E) to left and run 604/4 to P 7 which .... is Sta. 84 + 26 on preliminary. 60 59 N 78 36' E 58 + 58 PT 2 62 17' N 16 20' E 58 59 58' 57 55 58' 45 7 58' 44 3 58' 43 + 01 PC 2 8 R 00' N 45 58' W 43 42 41 N 46 58' W 40 + 99 PT! 54 48.5' N 8 51' E 40 51 50' 39 48 50' 25 6 50' 24 3 50' 23 50' 22 + 72 PCi 6 L 00' N 63 39 E 22 21 + 41 Check : This point is 21 + 83.3 on preliminary. 21 "2" 1 This is Sta. of preliminary. For first location course deflect 5 48' to left of first preliminary course . THE MATHEMATICS OF PAGPER LOCATION : USING POLAR CO-ORDINATES.-^To compute the location field notes without employing rect- angular co-ordinates, proceed thus: 1. The location having been laid down on the map as shown, scale PiVi, ViV2, "WFCa, and with the protractor scale the bearings of these three lines. 2. OomDUte the central angles A i and . 8. Compute curve lengths La. and L 2 . 4. Compute tangent distances Ti and Tz. 5. Compute lengths of tangents PiPCi, PTi PCs and PTaPCs. 17 6. Write station and plus for Pi, PCi, PTi, PC, PT a , and PC 8 . 7. Find preliminary station and plus, and loca- tion station and plus, for check point A, thus: (a) Compute angles of triangle PilPsA; (b) Prom the known side and angles of this triangle compute the sides PiA and P2A; (c) The preliminary sta- tion and plus for A is PiP 2 + P^A; and the loca* tion station and plus for A is PiA. 8. Having drawn on the map the check line PCsP?, compute the length and bearing 'of this line. This is done by treating PCaP? as the "miss- ing side" of the closed figure APaP^PriPftPTPCs- VaViA. 9.' Write the alinement notes in the field book. The work of finding the "missing side" in Step 8, just above, involves practically all the operations required to compute the co-ordinates of the chief points of our map from the field notes of the pre- liminary survey. Computing from the field notes the rectangular co-ordinates of the chief points of a survey is analogous to computing from the level notes the elevations of the stations on a profile, and has similar advantages. 18 O S^\ f\ IS C on any subject in which you may be interested sup- plied promptly on receipt of price. Send for catalogue. M. C. CLARK, PUBLISHER AND BOOKSELLER, 13-21 PARK Row, NEW YORK. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. 17 LD 21-100TO-12, f 43 (8796s) Gaylord Bros Maker. Syracuse. N. Y. PAT. JAN. 21. IMS ; 381115 ,;;:::A LIBRARY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. JUN 10 193! ) MJP, 2 19 40