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A new method in road design: polynomial alignment
V. Calogero
A new method in road design: polynomial alignment The computer has gradually become an essential tool in road design, permitting the implementation of completely new methods to improve the end product and the ease of man-machine interaction. In this paper a new approach to the problem of alignment design is presented, in which the traditional straights and circles are replaced by polynomial curves (Calogero polynomials), which can only be calculated by computer. The engineer simply defines his 'ideal line' by means of required points of passage, and obtains detailed tabulations and automatic drawings of the resultant alignment. The computer follows obediently the engineers" specifications where these are 'tight', and produces as smooth, flowing and short a 'line' as possible where the engineer's specifications are 'loose'.
This section broadly outlines the problems and requirements of road alignment design and describes their traditional methods of solution. The first stage in road design occurs when it is decided for social, economic or traffic reasons to link together two areas A and B. It is then the engineer's task to select the optimum terminal positions in A and B and to specify the route between them which combines with the terrain so as to achieve the best possible balance between construction costs and user costs, which produces safe and comfortable driving conditions and which also respects any social, aesthetic and political requirements or restrictions. ~, o, 4, 7, 8, 14 18.
The engineer, with these objectives and constraints in mind, develops on a plan a number of feasible alternative 'lines' (horizontal alignments) giving regard to the associated vertical alignment. Evaluation of each line can then be accomplished (preliminary design) in some detail as a guide for the final choice. Once a route has been selected, detailed calculations and drawings must follow as a basis for contract negotiations and work in the field. The horizontal and vertical position of the centre-line must be known with great accuracy; for this reason the 'ideal line' conceived by the engineer must be put into mathematical form i.e. a sequence of mathematical curves, whether or not a computer is used. In this way the coordinates of points on the centre-line and their geometrical relationships with each other and with the terrain can be precisely determined in order to calculate quantities and data for setting out. WINTER 1969
In preliminary design the degree of accuracy required is small and the alignment may be drawn out using 'railway curves' and straights on contoured plans; this enables approximate quantities to be calculated. However, modem computer techniques (integrated system of programs, digital terrain models, etc. ~, 5, 6, g, ~2, 15, 16, 17) make the process faster and more economical if the alignments are specified, even at this early stage, by the input data of a sequence of mathematical curves. It is then left to the computer to perform terrain interpolation, cross-section design, volume calculation, etc., and to output in tabular and graphical form all the required details. In this case, of course, surveys and other preliminary operations must be computer-oriented. The problem of manually transferring into mathematical form the ideal line has been solved, all over the world, in the same way: the line is broken down into segments of straights and circular arcs. In vertical alignments, parabolas are often used in place of circles, and in horizontal alignments standard transition curves (clothoids) are usually inserted between straights and circles, to allow for easy vehicle steering. The merit of using such curves lies in the ease with which they may be handled by the engineer rather than in the dynamic or aesthetic quality of the curves themselves. Provided the standard sequence 'straight - circular arc straight' is adopted, alignments may easily be drawn using straight-edges and standard templates; the position of each curve can be easily calculated using bearings, radii, etc.; transition curves can be inserted, when necessary, using 19
standard tables; long alignments can be set out with the simple knowledge of length and defleGtion angle of each curve.
However, modem theories on optical comfort 4, 7, a 14, is together with computer techniques have tended to favour a more extensive use of non-standard radii, long transitions and compound curves, which require automatic calculation. The engineer simply tells the computer the types of curves he wants to use and a minimum number of parameters to specify them. The output usually gives the coordinates of points at regular chainage intervals along the centre-line and drawings of the alignment via a digital plotter. As a consequence, circles, straights and transitions are now rarely used immediately and directly to draw the 'ideal line'; they are only a means to specify this line to the computer. The question now arises as to whether there are more straightforward means of specifying the ideal line to the computer. An answer to this question will be proposed in the next section. Setting out methods have also been transformed as a consequence of computer techniques and more sophisticated output requirements. Coordinate system methods (national grid or other) are now generally preferred to 'deflection angle and length' methods which were used to set out one curve after the other. Intersection points (between straights) and points of tangency (between curves and straights) are set out in relation to control points (traverse stations) of known coordinates marked in the field. These are frequently the same points as those used as controls in the aerial survey or in mapping work. Coordinates, distances and angles are calculated by the computer. The curve is then set out using standard methods. In this way the accumulation of error is prevented and, therefore, the 'drift' of the alignment is controlled. Furthermore, setting out and construction work (in particular structures) may be started at points along the alignment without the need to set out long lengths of the alignment. However, three further considerations have determined a more extensive intervention of the computer in setting out work. (1) Modern alignments (non-standard curves, long transitions, etc.) require a certain amount of calculation in the field after the start and end point of each curve has been set out. (2) Intersection points may be at locations which are difficult to reach whereas ground control points used in the original survey will always be accessible and their location already marked on the ground. (3) Points at special chainages or points off the centreline (e.g. interchanges, drainage, structures, etc.) may have to be set out at the same time as or even before points on the centre-line which is not possible using traditional setting out methods. These considerations have encouraged the development of new computer programs tz, le, which calculate angles and distances necessary to set out each point individually regardless of whether it is a point on the centre-line at a regular chainage interval or not (e.g. points on cross-sections, structures, etc.). Calculations are made with reference to the control points and other setting out points in many different ways and combinations. The man in the field does not need to know what type of curve he is setting out; he has only to choose from his computer print-out 20
which of several ways (including standard methods where applicable) he prefers to use to set out each individual point. His choice may depend on the instruments available, the accuracy required, ground conditions, the points which he has already set out, etc. The impact of computers in road alignment design may be summarised in the following way: (a) Computers were first used to perform calculations for the preliminary and final design which were the same as those previously made by the engineer manually. This relieved the engineer from tedious work and enabled him to devote more time to the study of alternative solutions, to the collection of more accurate data and to the refinement of his design. Neither the pessimistic view that engineers would be out of work nor the optimistic one that they would spend more time in their gardens has come true. (b) The availability of computers has then permitted elaborations of the old methods of design and setting out, which may be achieved by complicated but automatic calculations. (c) These elaborations and extra facilities for the engineer's convenience have transformed the computer from a useful adjunct into a necessary tool. In consequence the convenience of traditional elements (i.e. circles and straights) has become irrelevant. In view of these observations, it is time to consider other mathematical types of curve for road alignments even if they require more complex computer calculations provided that the end product is improved or provided that the way in which the engineer specifies his requirements to the computer is more convenient.
A Basic Approach to Alignment Design In the previous section the design of road alignments has been analysed as a two stage process. In the first stage objectives and constraints are considered by the engineer in order to produce an ideal feasible line which is sometimes sketched on maps free hand or with the help of templates. In the second stage the engineer breaks the ideal line into straights, circles, etc., in order to obtain coordinates, angles and other necessary data, either through manual calculations or by specifying the position of the curves to a computer program. The new approach developed by the author tries to relieve the engineer completely from this second stage and to concern him only with the basic essentials of route location, leaving the mathematical difficulties to the computer. Three aspects have been considered to be basically relevant to the engineer: (i) Road standards - e.g. minimum permissible radius of curvature, maximum slope, minimum visibility distance. (2) Conditions of passage - e.g. points of passage with or without a permissible displacement, 'gates' between areas to be avoided, bearings and curvatures desirable (e.g. co-ordination of the horizontal and vertical geometry) or necessary in certain parts of the alignment. They represent economic, political or aesthetic decisions of the engineer and they can be very 'tight' in some parts of the alignment and very 'loose' in other parts, depending on land use, topography, etc. COMPUTER AIDED DESIGN
(3) Quality of the line - within the freedom left by 1 and 2 the line could still be longer or shorter, with higher or lower curvatures, with greater or smaller rate of change of curvature, etc. in a basic approach to the design of road alignments the engineer must be able to specify his line in terms of 1, 2 and 3 and leave the computer to calculate the alignment which best fits his requirements. This new procedure must be easily able to allow for refinements and modifications to be made to parts of the alignment as the engineer dictates. In theory the computer could still make use of conventional curves, though they would not be very easy to handle in an automatic computation. However, there is no reason why unconventional curves should not be used if they can be more easily computed and give better a quality of line. In an M.I.T. publication the view is expressed that: 'General opinion now favours a flowing line, curving and varied but without visual breaks or interruptions. Smooth continuity is most desired. Long easy curves are recommended, blending one into the other through transition curves of gradually changing radius. Extended straight tangents, or even any straight lines at all, are to be avoided as too monotonous. Similarly, long vertical curves are deemed best, fitted easily one to the other, without apparent straight grades b e t w e e n . . . Particularly if the road swings smoothly from point to point of a fine and rather open natural landscape, it gives the same sense of vital rhythm and movement as a skier's track', x4 These considerations point towards fully transitional alignments.
Polynomial Alignment System This system of programs generates very smooth horizontal and vertical alignments of a new type, made up of a small number of long curves, known as Calogero polynomials, the calculation of which will be described later. Along these curves the radius of curvatures and the A-value change gradually and continuously. This line can be specified by the engineer in as much detail as he likes via the input data which represents, in a simple and direct way, the requirements 1, 2 and 3 described in the previous section. One could use this same data as the pre-establisl'ed objectives of a conventional design, though this would involve more 'manual work. Similarly, the polynomial alignment can be made to follow closely any existing conventional alignment; the features of the program are, however, best exploited when some degree of freedom is left for the line to 'swing smoothly from point to point'. 14 The system consists of two programs: POLYAL, for the calculation of Calogero polynomials, and TABPLT for detailed tabulation and graphical output. Input to POLYAL. This input specifies the requirements of the alignment. The following data must always be input: (1) D E L T A : arbitrary length (e.g. 20 m for horizontal alignment and 1-5 m for vertical alignment) chosen by the engineer to represent the unit displacement from Guide Points. (2) R M I N ( + ) : value below which the radius of curvature must never fall (e.g. 600 m in the case of horizontal alignment). In the case of vertical alignment this value applies to valley curves (e.g. 10000 m). (3) R M I N ( - ) : minimum summit radius for summit curves, only for vertical alignment (e.g. - 20000 m). WINTER 1969
(4) Initial conditions: X, Y coordinates (chainage and elevation for vertical alignment), bearing (or gradient) expressed as a trigonometric tangent with respect to the coordinate axes, radius of curvature and chainage at the start point of the alignment. The following data can be specified in any number and combination: (5) Guide Point: X and Y coordinates (or chainage and elevation) of the required point of passage together with the weighting W which, together with D E L T A previously specified, determines the maximum permitted displacement from this point: disp. = DELTA/W. (6) Guide Point with a tangent (i.e. bearing or gradient expressed by their trigonometric tangent): same as 5 with a tangent value and a tangent weighting U. There is no facility to specify a permitted displacement for the tangent. (7) Fixed Guide Point: X and Y coordinates (or chainage and elevation) of a compulsory point of passage (displacement = 0). (8) Fixed Guide Point with a tangent: same as 7 but with a compulsory tangent at the point. (9) Fixed Guide Point with a tangent and radius: same as 8 but a value for the radius of curvature at this point is also assigned. Facility 9 is particularly useful when the polynomial alignment has to join up an existing line or a conventional element (straight line, circular arc, etc.). Therefore, conventional elements can be introduced within polynomial alignments by assigning coordinates, tangents and radii at their start and end point, and calculating the conventional element externally to the program. Output of POLYAL. The program tabulates the following values: X, Y (elevation for the vertical alignment), Tangent, R (radius), l/R, A-value, chainage. The spacing between tabulated points is decided by the program itself (e.g. 30 points per polynomial). The program also outputs on cards the coefficients, scales, positions, etc., of each polynomial (four cards per polynomial), to be used as input to other programs (e.g. TABPLT). Input to TABPLT. The engineer can specify: interval of chainage for tabulation (e.g. 100 feet), conversion factor between X, Y units and chainage units (e.g. if coordinates are measured in metres and chainages in feet, cony. f. = 3-28), position and scales of plotted alignment (for the vertical alignment different scales can be specified for chainage and elevation), position and scales of plotted curve representing 1/R versus chainage, width of carriageway and cross-sections to be plotted, positions and mesh of reference grid to be plotted, coded map details. The coefficients of the polynomials, punched by POLYA L for the alignment in question, must also be input to TABPLT. Output of TABPLT. The program tabulates, at the specified interval of chainage, the following values: chainage, X, Y (elevation for the vertical alignmen0, tangent, R, l/R, A-value. Guide Points and other points of interest are also tabulated, at special chainages, with the additional values: X and Y coordinates of the point, distance from the alignment. The graphical output of TABPLT consists of: drawings of the horizontal or vertical alignment with grid lines, kerb lines and cross-sections (for horizontal alignment 21
only), guide points, other points of interest, map details; curve representing 1/R versus chainage for alignment evaluation, superelevation desiga, etc. Use of the system. The system implements the basic approach to alignment design described in the previous section and can be applied to rural and urban roads and to motorways as well as slip roads and junctions. The input to P O L Y A L is coded by the engineer on the basis of his 'ideal line', which he may have already sketched free hand, and covers, in a simple form, the aspects described in 1, 2 and 3 of the previous section. In particular: (1) The only 'road standard' required by the program is R M I N ( + ) (and R M I N ( - ) for the vertical alignment). The program generally produces the highest possible A-value (i.e. lowest rate of change of curvature) for any given configuration of Guide Points. Therefore, if the A-value goes below standard at some point of the calculated alignment, the engineer can easily see which requirement is causing this and, if necessary, modify his data accordingly and re-run POLYAL. Similar measures would be taken even if the design were conventional. In vertical alignment design there is no constraint input for the gradient (maximum permitted value). This is because the gradient between Guide Points is largely determined by their relative positions, which are given by the engineer, and the displacement is generally small. The value of R M I N is usually assigned on the basis of what may be reasonably allowed over the particular stretch of alignment under consideration; this value is generally greater than the absolute minimum standard suggested by the Ministry of Transport. (2) 'Conditions of passage' are always expressed in the simple form of Guide Points. The calculated alignment will always tend to go as close as possible to Guide Points (in any case within the permitted displacement) compromising with the other objectives (quality of the line) described in 3. The permitted displacement is assumed to be the same on either side of the Guide Point. When this is not applicable at some point, the Guide Point should still be located in the most desirable position (i.e. on the 'ideal line') and the most restrictive displacement used; alternatively, the one which applies to the side on which the alignment is likely to pass. A Guide Point with the permitted displacement either side can be regarded as a 'Gate' with a preferred point of passage in the middle (Fig. 1). However, some condition of passage could be represented by a gate without a preferred point of passage, for example gaps between constructions to be avoided. There is no facility to input these gates as such, because the engineer can easily recognise (before or after the first computer run) one of the following two cases: (a) The alignment tends to miss the gate by a particular side, in order to improve the quality of the alignment. In this case a Guide Point has to be input, within the gate but as close as possible to the preferred side (Fig. 2).
22
(b) The alignment goes through the gate with no need for any Guide Point at the gate (Fig. 3). Sometimes long stretches of alignment must follow conventional curves, for example to run parallel to an existing road or railway; in this case the polynomial alignment can be made to follow the curve in question very closely by specifying a few Fixed Guide Points with tangents and radii. Alternatively the polynomial could be terminated (with a Fixed Guide Point) at the beginning of the straight or circular arc and resumed again at the end of this element. In this case the difference in the two chainages would have to be calculated externally to the program. In interchange design the alignment is required to follow long curves of tight radius, which is an 'unnatural' pattern for polynomial expressions. Nevertheless the program itself copes with this difficulty by using several short polynomial curves. This is illustrated in Fig. 4 which shows combinations of polynomials which have been calculated and plotted by the program and may be seen to closely resemble circles. However, a greater number of Fixed Guide Points and a certain amount of experience with the program is generally required for the design of interchanges. Under normal conditions, in the case of highstandard roads in developed areas, experience has shown that an average of three Guide Points per mile have been used by different engineers. (3) The 'quality of the line' is not directly specified by the engineer in the input data but is built into the program (POLYAL) through a number of computational schemes, the most effective of which being that the program calculates many alternative polynomials before selecting the ones to be used. F o r a given input ( R M I N and Guide Points) the program produces a line which, first of all, respects the permitted displacements from G.P.s, then satisfies the curvature constraint if this is geometrically (and humanly) possible, then aims for the following objectives:
x Ouide points y\ / ' Polynorninat • - ; , - \ . . . . alignment, ~ / X ,\
x
\
/ / ~ -,,
/
Fig. 1. Guide Points with permitted displacements
___~p ....
x Guide points Polynominal alignment when P is not input Potynorninat alignment
Fig. 2. Gate which requires a Guide Point. COMPUTER AIDED DESIGN
(a) Closeness to Guide Points. (b) Low curvature and rate of change of curvature. (c) Short overall length. Experience has shown that in normal cases the program reproduces the line that the engineer had in mind very closely, even if he coded the input data without previous experience with the program. Furthermore, if the engineer left some parts of the line poorly specified, the program can produce good solutions which the engineer did not have in mind in the first place. However, in some cases, the engineer may decide that the calculated line must be improved in a particular way, after examining the first output of the program. This is done by modifying the input data and re-running the program. For example, if the engineer wanted to increase the radius of a certain curve, he could proceed in a number of ways, for example: (i) Assign one or two new Guide Points on the ideal curve of greater radius. (ii) Select the coordinates and the tangent of a point on the calculated line and include it in the new input as a Fixed Guide Point with the old tangent and the desired radius. (iii) Increase the value of RMIN and re-run only the stretch of alignment to be modified. It is now clear that this approach to the problem of coding the 'ideal line' into some input data can claim simplicity and ease of use, but may require two or three computer runs before the results were fully acceptable. The philosophy behind this approach is the following: Experience shows that the user of engineering programs requires, most of the times, re-runs on the same design problem. This is because his ideas clarify or change when he examines the computer output, data errors come to light, colleagues and superiors make suggestions and comments on the results. Therefore it is most undesirable to have an extensive and difficult input layout and/or a long and complex computation if the inevitable re-runs
can easily incorporate refinements of the design. In practice only POLYAL is re-run as its output enables, in most of the cases, judgements on the acceptability of the solution. Stretches of alignment can be re-run individually (using Fixed Guide Points with tangent and radius at both ends) if the rest of the alignment must not be modified. The card output from different runs of POLYAL can be ultimately combined in one input for TABPLT, in order to obtain the final tabulation and drawings. In preliminary design the system is used as a tool to specify quickly a trial line (horizontally and vertically) and obtain automatic drawings and data for subsequent evaluation (Digital Ground Models, earthworks, etc.) as described in the introduction. In final design the alignment is 'polished up' introducing those adjustments on the horizontal and vertical location which are required by economic and aesthetic consideration, drainage requirements, horizontal-vertical coordination, details on the ground which had not been considered before, etc. This is done introducing new Guide Points, with tangents and radii where appropriate, and/or modifying the values of the old ones. In this connection the output obtained at the preliminary design stage for the line in question is very useful, as it provides accurate 'starting' values for coordinates, tangents and radii at any chainage. For detailed evaluation of the geometrical quality of the horizontal alignment it is useful to obtain the curvature curve (I/R versus chainage) via graphical output of TABPLT. Sometimes a conventional alignment and a polynomial one could almost coincide if drawn in plan at a scale of 1:2500, but their curvature curves would still show a clear difference of the type of Fig. 5. For superelevation design the engineer divides the polynomial alignment into segments along which a certain value of superelevation seems appropriate, for example normal crossfall for segments where the radius is always above 3350 m, 1 in 14 for segments where the radius is always below 600 m, etc. Changes of superelevation can
\\ x Guide points Polynominat alignment
Fig. 3. Gate w h i c h does not require a Guide Point.
f
1
f-chainage Polynominal alignment - - - - - Conventional alignment
Fig. 5. Curvature curve. WINTER 1969
-.t- Guide points
Fig. 4. 23
then be calculated with conventional methods and programs. Alternatively the engineer can.use the curvature curve. as a superelevation curve, changing the scale according to the formula: tan x -
Ve
KR
In this case he would still select the segments of alignment where crossfall must be normal, and the points of reversion, taking also into account the vertical gradient. But he would then obtain the amount of superelevation directly from the curvature curve, for those chainages where the radius falls below a certain value, e.g. 3350 m. Setting out of polynomial alignments can be done using conventional methods. However, standard tables and formulae cannot be used, and values of angles and distances are generally calculated by computer programs 12, 16 which can be 'integrated' with the polynomial system, The advantages of this approach have been discussed in the Introduction. The computer print out is used directly in the field and generally contains the type of data shown in Fig. 6, to permit the setting out of each point in several alternative ways. The choice is then made by the man in the field. The polynomial system can be linked, quite conveniently, to other systems of programs. The horizontal alignment output can be obtained on cards and used as input to other programs for ground interpolation (D.G.M.s), setting out, automatic superelevation design, drawings of perspectives, etc. Three-dimensional Guide Points can be specified by the engineer and the system used to calculate the horizontal alignment first and then the vertical one, using the original input elevations and the calculated chainages at Guide Points. Points of passage (horizontal or vertical) calculated by optimisation programs ~, 2o as a result of costs and constraints fed in by the engineer, can be directly input to the polynomial system in order to integrate fully the optimisation and alignment programs.
Discussion of advantages and disadvantages Though the polynomial method introduces a number of conceptual and practical innovations in highway engineering, its function is still that of a tool in the hands of the engineer. Polynomial and conventional alignments are difficult to compare in terms of the end-product: if one engineer was asked to design the same scheme with both methods, he would produce two lines almost identical; if a different engineer was using each method, the comparison would be basically related to the two engineers and not to the methods themselves. However, some observations can be made about the flexibility and ease of use of the two methods at the design and construction stages. Setting out work will probably be more demanding in the case of the polynomial alignment, though there is no experience to support this opinion with actual figures. Similarly, until a certain amount of experience is available, superelevation will require some extra attention at the design stage, in the case of polynomial alignment. If curvature or superelevation varied along viaducts, standardisation of the elements of the structure would be impossible. 24
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X Point to set out-on the alignment A Traverse stations (control points) • Points previously set out Fig. 6.
Input to conventional programs requires precise values of coordinates and parameters even when the physical requirements or the accuracy of the maps do not justify such a precision. On the other hand with the polynomial method input data must be provided only if and where there are practical requirements for the alignment, and 'freedom" is used by the program to improve the geometry of the line. This, however, does not apply to the design of interchanges. When necessary, the polynomial can be specified in a more precise manner than a single conventional curve. F o r example, if the positions of a circular arc and a straight line were determined in plan for some particular reason, there would only be one possible transition curve in between. In the same circumstances the polynomial could still accept further specifications and generally produce a smoother change of curvature or better avoid an obstacle on the ground. If some change of alignment was required at the setting out or construction stage, this could only be done, in the case of the polynomial, through a computer run. Alternatively, some appropriate length of alignment should have to be replaced manually with conventional elements. Refinements are easily made on a polynomial alignment because individual Guide Points can always be added or adjusted independently. Furthermore the same programs, the same basic data and, sometimes, the same results can be used in both preliminary and final design. On the other hand, conventional programs generally use different methods for preliminary and final design (e.g. no transitions in preliminary design) and alterations in one curve very often need re-adjusting the position of the other ones. Engineers can learn how to use the polynomial programs in less than an hour. Experience of Use. The polynomial system has been used experimentally on a wide range of study cases, covering rural roads, urban motorways, interchanges, etc., supplied by local authorities, consulting engineers and Ministry of Transport. In many of them a conventional alignment was already designed and used for comparison. The system has also been used for some actual route planning and for the final design of a By-Pass. To the knowledge of the author, no one has yet driven on a polynomial road.
Example This example consists of the horizontal alignment design of the Sutton-Coldfield By-Pass carried out by the Warwickshire Sub-Unit, of the Midlands R.C.U. The scheme is a trunk road improvement, five miles long, and construction is programmed to start in 1969. Input data to COMPUTER AIDED DESIGN
~ I T Y OF LONDON, INSTITgTE OF COHPUTER SC]3ENCE,
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POLYAL (This program fits CalqKero's polynomials to Guide Points) INPUT
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Fig 7 Input data to POL YAL (Sutton Coldfield By-pass) POLYNOHIkL ALIGNMENT
*
T k B P L T
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V,CALOGERO NOVEMBER 1967
POLYNONIkL OF DEGREE 7 ST4RTS ~T NEXT PT 0.000000000*00 9,25437276E-01 3 . 7 5 0 5 3 4 6 9 0 * 0 3 " 3 , 8 1 0 7 1 6 8 9 E - 0 1 9 . 2 4 5 4 5 5 0 5 | ' 0 1 2.926000000*05 4, 15302000E*05 0,00000000E-00 ¢,O0000000E*O!! O.O000flOOOB*O0 O,O000000nE*o0-9, 78052489E-01 1 . 7 9 9 0 4 4 9 9 0 - 0 0 " 1 , 5 3 1 3 ~ t $ 2 2 " 0 0 - 1 . 4 1 2 0 3 6 3 0 | * 0 0 5 . 9 4 8 3 9 7 2 4 0 * 0 0 - 6 , 35148569E*00 2.222897520*00 CNAINAGE XA 0,00 2 9 2 6 0 0 . 0 0 0 0
YA 415302.0000
TA -0,4033
R 2852.79
IlR 0.0003505
4.'VALUE
CH4INAGE XN YN 0.00 292600,0001 4~5302,0000
TN -0,4033
R 2852.79
~/R 0.0003505
4=VALUE XG YG " 2 2 1 0 . 6 9 292600.0000 415302.0000
CHAINAGE 100.00 200,00 300,00 400.00 500.00 600.00 700.00 flOG.On
1/R 0.0003562 0.0003617 0.0003669 0.0003718 0,0003764 0.0003807 0,0003846 ~-n003882
4=VALUE -|315.76 -2363.45 -2420.35 -2486.27 -|569.57 "~667.41 - 0 7 8 6 . ;~4
_,ol.80
292620,3260 292656.7780 292~85.3460 2927~4,0311 292742,8310 292771.7437 292~n.7663 °
YA 415290.7516 415279.8114 4~5269.1054 415258.8796 415248,8994 415239.2504 415229,9374 '~5220,9653
TA -0,3908 -0.3783 -0,3656 -n.3529 -0,3401 -0,3273 -0.3Sa= -"
R 2807.27 2764.92 2725.70
¢~ 29427/ . . . . 204307,4290 294337.2726 204367.0744 294396.8338 294426,5504 294456,2240 294405,6542
. . . o041 415190,8789 415197.0761 415203,4717 415~10.0617 4152~6.8420 415223,8084 415230.9567
g,1825 0.1890 0,1970 t~,2041 n,2111 n,2180 n,2248 0,2315 0,2380 ~.2444
q=~ 434t,77 4418.43 4498,15 4581.58 4669.49 4762.78 4862,46 4969,73 500~.98
O.uv~.=.; 0.0002223 0.0002183 0.n002142 0.0002100 0.0002057 0.0002012 0.0001966
CHA[NAGE XN YN 6336.42 294496,6348 415233,6045
TN 0.2468
R ~130.06
I/R O.flOOt949
AIVALUE
CHk]NAGE 6400,00 6500.00 6600,00 6700.00 6800,00
YA 415238.2824 415245.7010 415253.4477 415261.2776 415269,2654 415277,4059 415285,6933 "~294,1217 • 6850
TA 0,2507 0.2569 n.2629 9,2687 0,2744 0.2799 0,2852 0,2903 0,2952
R 5212.02 5352,17 5506.27 5677.04 5870,15 6087.21 6334 " 6"
~IR 0.000~918 0.000~668 0.0001816 0.000~761 0,0001784 N.nOg1643
A=VALUE ~524.07 2470.41 B414.33 2356.73 2296.39 2240.02 E182.22 ~125,45
414~ 414915.83~r 414905.0940 414893,9188 414082,2885 4~4870.1777 414857.5564 414844,3902 414830.6427 41482~,0000
,/ -11,3601 -0,3051 -0,4032 -0,4226 -0.4436 -0.4666 -0,4918 -0.5194 -0,5397
-2160.27 -2137,44 -2076,33 -1994,94 -1895.35 -1782.77 -1664.02 -1545.97 -1434,42 -1333.70 -1273.24
-O.uv~ -0.0004679 -0.0004016 -0.0005013 -0.0005276 -0.0005609 =0.0006010 -0.0006460 -0.0006971 -0.0007490 -0.0007854
CHA[NAGE XN YN 22267.60 299141.0006 414821.0009
TN -0.5397
R -1273.24
1/R -0.0007854
qn n
b~u~. 5500,00 5600,00 5700,00 5800,00 5900,00 6000,00 6100.00 6200,00 6300.00
X4
XA 294515,4411 204544,9847 294q74,4850 20460~,9425 204633,3576 6900.00 2 9 4 ~ 2 , 7 3 0 8 7000." "~q 7"
.uO .~400,00 21500.00 21600.00 21700.00 21800,00 ~1900.00 22000.00 22100.00 22200.00 02267.61
298069,821298490,6457 200927,3256 2989~5.8507 298984,2082 29901203821 299040.3528 299n68.0967 299095,5862 299~22.7896 299~41,0008
2689.6t
2656.63 2626.73
....
~
DN 0.0001
2756.68 2743.09 2723°47 2695,69 ~661.09 2620.36 2574.38 2540.71
XG YG 294500.0000 4 1 5 2 2 0 - 0 0 0 0
oN 14,0148
2487o82 2245.49 1075.75 956.49
072,63 014.96
776.43 760.89 760.76 AuVALUE XG YG 760.75 299141.0000 414821.0000
ON 0,0011
Fig. 8. Line printer output from TABPL T. WINTER 1969
I
RADIUS OR
.... ....
151"Iil
,
,,
COOl.IT
x . . . . . . I ~. . . . . . . . . I c~'rNAGE. . . . . . 17,9,2,6,0,0, • , , , | , , , [ 4 , 1 , 5 , 3 , 0 , 2 r , ...... [I,OtO, O," . . . . . . . GUIDE POINT CAEDS ( L a s t a . 9 . card to be followed bT a blank car'd) X X W~ZG]~
2,9,8,5,4.2..,,,
|ID~I'Z~CaTZON
414144 414114T 40 q~llll lll~l!~41111!r~ ll ll l~ 11111114 IIIIIIII~ IIIIII~II'/III~I 14"/111111~T~I~IIIIII , . I . . . . . . . | ......... I' ......... °I .........
25
,
P O L Y A L are represented in Fig. 7. Standard displacement ( D E L T A ) is 50 m, m i n i m u m r a d i u ~ o f curvature ( R M I N ) is 1000 m. There are 13 G u i d e P o i n t s with coordinates in metres, established mainly to circumvent farms and properties. Line-printer output from T A B P L T is shown in Fig. 8. Graphical output from T A B P L T is shown in Fig. 9. The line has also been redrawn manually on maps using tabulated coordinates (Fig. 10).
The Algorithm The Polynomial Alignment system is based on the calculation of the sequence of Calogero polynomials which best fits the specified Guide Points. This calculation is performed by the program POLYAL, while T A B P L T simply tabulates and plots the alignment.
Calogero Polynomials
/
Consider: (i) an ordered set of points G = {Gill = 1, 2 . . . . 1}, where Gi = (Xi, Yi, wi, ti, Ui); (ii) a length a. (iii) a radius of curvature r,,in. (iv) the trigonometric tangent to and the second derivative Y0" of the polynomial at the origin. A new set of points Q can be now defined in the following way. Let: k be an arbitrary n u m b e r of new points at equal distance on the straight line between points Gi and Gi.1; Ix = l + ( 1 - l)k; I = ( I , 2 + k, 3 + 2k . . . . . i + ( i - 1)k . . . . . ll); Q = {QjIj = 1, 2 . . . . 1~}, Qj = ( X l j , ylj, w~i, t~j, u~j); Xxj : Xi, Y l j = Yl, wlj = Wlj(j, Wi), tlj : ti, ul1 = u l j (j, u i ) , j d , i = (j + k)/(l + k), where Wlj and u~j are arbitrary functions. xli --
y l j --
(xi+1 -
xi)
k+l
/j/
//¢/
{j - [i + (i - 1)k]} + xi,
(Yi* 1 Yi) {j -- [i + (i -- 1)k]} + Yi, k+l -
//
)
-
w,j = f(j),txj=0, u~j=0, i + (i+ ik, i = 1, 2 , . . . , 1 - 1, where f is an arbitrary function.
1)k < j
< i + 1
Given i, ii, iii, iv, a Calogero polynomial is defined by: 1 y = tax + ~y0 x2 + a3x a + a4x 'l + . . . +
/
/
/
anxn,
where n, a3, a4,.., a , , have the following properties: (a) at each point Gi the following constraint (permitted displacement) is satisfied: ly(x3
-
Y~I
~/1 + y'2(xi)
d ~--; wi
(b) at each point Qj the following constraint (curvature) is satisfied :
Fig 9. (Right). TABPL T.
Graphical output from
Fig 10. (Far Right). TABPLT output redrawn manually on O.S. map. 26
COMPUTER AIDED DESIGN
delta/w=5 m _ _
1 ~5000 MN
if
Warwickshire County Council Polynomial Alignment for Sutton Coldfield By- Pass
;.17
I
I
I
O
Fair Ficw L Far~
I ~/2MI LE
W-
'w=
S~O~li G I #
"e e le i i q li Ii I
L
~
• !
• •
i
e •
II
•
li
•
•
\ T I / f M ## 1. A"
l','.',T]) -
-
")--i'l
'.F,r
/
-
\ \\
~ =
///iA\\
"7"
)~-/Peddtmore
~delta!w - 5 m
2cJ4000 MN
c" /'/ / /
•'. ... ,...,.,%,~.. _..
~,°,, __,~.~,-,.,-
f
~
- ~_
_~-
--
-
,
i
.
delta/w =
2 m
4
83000 MN
\ /I Lane
.......
~
I,
t
-
ixed Point7.~' .'. ~tart of Alignment~,,
__-~-~ ".'11
WINTER
1969
27
lrll >/r.,i.. [1 + y'=(xlj)] "~/~'. Y~(xlj) "'
where rj =
(c) aa, a4 . . . . . a. are such as to minimise the following expression: h a =
+
,E { w ~ i [ y l y j=i
-
y(xly)]" + uli[t~/ - y'(x~])]~
ag(j)[y"(xtj)]-"},
where g is an arbitrary function and ,~ is a variable. (d) A is such as to minimise n. It is obvious that, having chosen k, w~i, u~j, f and g, and given i, ii, iii, iv, there may exist one, none or an infinite number of Calogero polynomials, because A may vary within a certain range and still give the same minimum value ofn. From the alignment point of view any one of the range of polynomials can be acceptable. The arbitrary k, w~i, vjj and f, used by POLYAL, have been determined by the author through experimental work, while g is 'learned' by the program at running time. The basic definitions of the Calogero polynomials can be extended to the case where some points G~ are fixed (wt = oo and ui = oo) and the radius of curvature at the points may be fixed. The problem of calculating n, a.~, aa . . . . . a,, given i, ii, iii, iv, can be solved in many ways. The one used by P O L Y A L consists of a heuristic approach, whereby the linear system of equations: (~6t -
0, m
=
3,4,
...,n
8am
is solved for many different values of A and for increasing values of n, until a feasible solution is reached. Geometrically speaking a Calogero polynomial will always: (1) start at the origin of his system of coordinates, which is generally different from the one used originally for Guide Points; (2) assume a given bearing (to) and curvature (Yo') at the origin, to guarantee continuity with the line calculated so far; (3) keep close to the straight line (points Qi, J d ) between pairs of points G;; this tends to keep the length to a minimum; (4) keep close to the points Gi (a = min); (5) reverse the curvature as few times as possible (n = min), which also tends to keep the length to a minimum; (6) keep the curvature as low as possible (n = min and A#0).
polynomials being accepted which fit the conditions but behave erratically. Two points are worth mentioning: (I) Overlap between consecutive polynomials. In each cycle of POLYAL, the i.dh Guide Point is the end point of a particular trial segment of alignment, but is not the last point Ge used in the fitting. Some more points, whose number is calculated by the program, are always included in G to produce a convenient bearing and curvature at the i2 th point. (2) Axes for calculations. The coordinate axes of points Gi must have the origin coinciding with the start point of the new polynomial. The orientation of the axes affects considerably the quality of the fitting. Appropriate rescaling reduces the effect of rounding-off errors. Therefore POLYAL calculates each time (Block 3) the coordinate system to be used for the fitting. TABPLT reads in the cards output (Block 10) by POLYAL together with the engineer's specifications, and tabulates and plots the alignment.
Acknowledgments I would like to thank Professor R. A. Buckingham and K. Wo]fenden of the University of London Institute of Computer Science, W. Oxburgh of Planning and Transport Research and Computation Co. and Dr. J. P. Stott of the Road Research Laboratory for their co-operation and helpful discussions on this topic over the past three years. 1 am also grateful to the Ministry of Transport for financial support and to Warwickshire County Council, W. S. Atkins and Ptnrs and R. Travers Morgan and Ptnrs for their co-operation. [ am indebted to R. Timber]ake, K. Rivers, M. Lee and H. Abrahams for assistance in producing this paper.
3 I Ch°°S-erbie~t~YS~er mi,, i,~ ]
1,
I Fit Calogero Potynomiat i to G.P's from il toi=
YES
Choose i~ N0 ~1 and p°lyn°miM
OuI
The Computation The mechanism of P O L Y A L is explained by the flow chart of Fig. 11. Basically the program selects trial sets G of Guide Points (Block 3), performs polynomial fitting on them (Block 4), chooses the set and polynomial which gives the best fitting (Block 8), and proceed to the next segment of alignment (Block 12), This method prevents
28
I
from stored t r i a l oo,,nom,at, I
©
'11 G.P: guide points / L : number of G,R's read in il : index of first G,R of current polynomiat YES i;~ : index of last G.R of potynorniat being catcutated [~: chosen index of fast G.P, of best current polynomial ~ E
Fig. 11. Flow chart of POL YAL COMPUTER AIDED DESIGN
Bibliography x Manheim, M. L.: 'Highway Route Location as a Hierarchically-Structured Sequential Decision Process', M.LT. R6415 (May 1966). Roberts, P. O., and Suhrbier, J. H.: 'Highway Location Analysis', M.LT. Report n.5 (1966). 3 Leisch, J. E.: 'New Techniques in Alignment Design and Stakeout', Journal of the Highway Division, Proceedings of the American Society of Civil Engineers, Vol. 92, No. HWI (March 1966). 4 Godin, P.: 'Avants-projects d'Autoroutes et Tendances Actuelles en Mati6re de Trac6 Autoroutier', Revue Gdn~rale des Routes (January 1963). s Calogero, V.: 'The Application of Electronic Computers for the Design of Roads in France', Institute of Computer Science (July 1965). 6 Calogero, V.: 'The Application of Electronic Computers for the Design of Roads in Sweden', h~stitute of Computer Science, (July 1965). 7 Spencer, W. H.: 'The Co-ordination of Horizontal and Vertical Alignment of High-Speed Road', Institute Civ. Engineers Road Paper No. 27, (1948). s Smith, B. L., and Fogo, R. D. : 'Some Visual Aspects of Highway Design', 46th Annual Meeting, Committee on Geometric Highway Design. g Calogero, V.: 'Some details of the mechanics of handling D.T.M.'s by computer', Proceedings of P.T.R.C. Co. Ltd., Seminar (17 October 1967). ~o Calogero, V., and Tocchetti, A.: 'Un metodo veloce per il
calcolo dei volumi delle sezioni trasversali', Rivista Autostrade No. 2 (February 1965). H Calogero, V.: 'Polynomial Horizontal Alignment Program', Proceedings of P.T.R.C. Co. Ltd., Seminar (18 October 1967). 12 'Dossier Pilote TE. G167 - Trac~s [lectroniques en Geometric Imposee', Service Special des Autoroutes (1967). 1:3 Deligny, J. L.: 'Les projects d'autoroutes /t t'Sre du dessin automatique', Review G~n~rale des Routes, No. 398 (April 1965). 1.L Appleyard, D., Lynch, K., and Myer, J. R. : 'The View from the Road', M.LT. Press (1964). 15 'Programme EVASOM: Evaluation sommaire d'un avantprojet, Service Special des Autoroutes (1967). 16 'British Integrated Program System for Highway Design', Ministry of Transport, County Surveyor's Society, Association of Consulting Engineers, Part 1, May 1967, Part II, December 1968. 17 Cowling, H.: 'Highway Design', Computers in Highway Engineering - 1, The Journal of the htstitution of Highway Engineers, Vol. XV, No. 7 (July 1968). is Godin, P., Deligny, J. L., Antoniotti, P., Day, J. A., and Bernede, J. F. : 'Visual Quality Studies in Highway Design', B.C.O.M., Washington, D.C. (January 1968). 19 Calogero, V.: 'Computer Aided Highway Design', Ph.D. Thesis, Institute of Computer Science, University of London, to be submitted. 20 Calogero, V.: 'Optimisation of the Vertical Profile', Proceedings of P.T.R.C. Co. Ltd., Seminar (18 January 1968). Received December 1968
Dott. Ing. V. Calogero studied Electronic Engineering at the University of Naples (Italy) specialising in computers. In early 1964 he obtained his degree of "Dottore in Ingegneria Elettronica'. Subsequently he carried out research on the application of computers in civil engineering at the French and Swedish Ministries of Transport. At the end of 1965 he was appointed Research Fellow at the Institute of Computer Science, University of London, and registered for a Ph.D. in "Computational Methods in Operational Research' supported by the British Council and the M.O.T. The thesis, entitled 'Computer Aided Highway Design', will be submitted early in 1969. Since 1968 he has been a Lecturer at the Institute of Computer Science, University of London, with research responsibilities to the M.O.T. He is also currently involved in consulting and lecturing work for government bodies and commercial firms in England and abroad.
WINTER 1969
29
A new method in road design: polynomial alignment The computer has gradually become an essential tool in road design, permitting the implementation of completely new methods to improve the end product and the ease of man-machine interaction. In this paper a new approach to the problem of alignment design is presented, in which the traditional straights and circles are replaced by polynomial curves (Calogero polynomials), which can only be calculated by computer. The engineer simply defines his 'ideal line' by means of required points of passage, and obtains detailed tabulations and automatic drawings of the resultant alignment. The computer follows obediently the engineers" specifications where these are 'tight', and produces as smooth, flowing and short a 'line' as possible where the engineer's specifications are 'loose'.
This section broadly outlines the problems and requirements of road alignment design and describes their traditional methods of solution. The first stage in road design occurs when it is decided for social, economic or traffic reasons to link together two areas A and B. It is then the engineer's task to select the optimum terminal positions in A and B and to specify the route between them which combines with the terrain so as to achieve the best possible balance between construction costs and user costs, which produces safe and comfortable driving conditions and which also respects any social, aesthetic and political requirements or restrictions. ~, o, 4, 7, 8, 14 18.
The engineer, with these objectives and constraints in mind, develops on a plan a number of feasible alternative 'lines' (horizontal alignments) giving regard to the associated vertical alignment. Evaluation of each line can then be accomplished (preliminary design) in some detail as a guide for the final choice. Once a route has been selected, detailed calculations and drawings must follow as a basis for contract negotiations and work in the field. The horizontal and vertical position of the centre-line must be known with great accuracy; for this reason the 'ideal line' conceived by the engineer must be put into mathematical form i.e. a sequence of mathematical curves, whether or not a computer is used. In this way the coordinates of points on the centre-line and their geometrical relationships with each other and with the terrain can be precisely determined in order to calculate quantities and data for setting out. WINTER 1969
In preliminary design the degree of accuracy required is small and the alignment may be drawn out using 'railway curves' and straights on contoured plans; this enables approximate quantities to be calculated. However, modem computer techniques (integrated system of programs, digital terrain models, etc. ~, 5, 6, g, ~2, 15, 16, 17) make the process faster and more economical if the alignments are specified, even at this early stage, by the input data of a sequence of mathematical curves. It is then left to the computer to perform terrain interpolation, cross-section design, volume calculation, etc., and to output in tabular and graphical form all the required details. In this case, of course, surveys and other preliminary operations must be computer-oriented. The problem of manually transferring into mathematical form the ideal line has been solved, all over the world, in the same way: the line is broken down into segments of straights and circular arcs. In vertical alignments, parabolas are often used in place of circles, and in horizontal alignments standard transition curves (clothoids) are usually inserted between straights and circles, to allow for easy vehicle steering. The merit of using such curves lies in the ease with which they may be handled by the engineer rather than in the dynamic or aesthetic quality of the curves themselves. Provided the standard sequence 'straight - circular arc straight' is adopted, alignments may easily be drawn using straight-edges and standard templates; the position of each curve can be easily calculated using bearings, radii, etc.; transition curves can be inserted, when necessary, using 19
standard tables; long alignments can be set out with the simple knowledge of length and defleGtion angle of each curve.
However, modem theories on optical comfort 4, 7, a 14, is together with computer techniques have tended to favour a more extensive use of non-standard radii, long transitions and compound curves, which require automatic calculation. The engineer simply tells the computer the types of curves he wants to use and a minimum number of parameters to specify them. The output usually gives the coordinates of points at regular chainage intervals along the centre-line and drawings of the alignment via a digital plotter. As a consequence, circles, straights and transitions are now rarely used immediately and directly to draw the 'ideal line'; they are only a means to specify this line to the computer. The question now arises as to whether there are more straightforward means of specifying the ideal line to the computer. An answer to this question will be proposed in the next section. Setting out methods have also been transformed as a consequence of computer techniques and more sophisticated output requirements. Coordinate system methods (national grid or other) are now generally preferred to 'deflection angle and length' methods which were used to set out one curve after the other. Intersection points (between straights) and points of tangency (between curves and straights) are set out in relation to control points (traverse stations) of known coordinates marked in the field. These are frequently the same points as those used as controls in the aerial survey or in mapping work. Coordinates, distances and angles are calculated by the computer. The curve is then set out using standard methods. In this way the accumulation of error is prevented and, therefore, the 'drift' of the alignment is controlled. Furthermore, setting out and construction work (in particular structures) may be started at points along the alignment without the need to set out long lengths of the alignment. However, three further considerations have determined a more extensive intervention of the computer in setting out work. (1) Modern alignments (non-standard curves, long transitions, etc.) require a certain amount of calculation in the field after the start and end point of each curve has been set out. (2) Intersection points may be at locations which are difficult to reach whereas ground control points used in the original survey will always be accessible and their location already marked on the ground. (3) Points at special chainages or points off the centreline (e.g. interchanges, drainage, structures, etc.) may have to be set out at the same time as or even before points on the centre-line which is not possible using traditional setting out methods. These considerations have encouraged the development of new computer programs tz, le, which calculate angles and distances necessary to set out each point individually regardless of whether it is a point on the centre-line at a regular chainage interval or not (e.g. points on cross-sections, structures, etc.). Calculations are made with reference to the control points and other setting out points in many different ways and combinations. The man in the field does not need to know what type of curve he is setting out; he has only to choose from his computer print-out 20
which of several ways (including standard methods where applicable) he prefers to use to set out each individual point. His choice may depend on the instruments available, the accuracy required, ground conditions, the points which he has already set out, etc. The impact of computers in road alignment design may be summarised in the following way: (a) Computers were first used to perform calculations for the preliminary and final design which were the same as those previously made by the engineer manually. This relieved the engineer from tedious work and enabled him to devote more time to the study of alternative solutions, to the collection of more accurate data and to the refinement of his design. Neither the pessimistic view that engineers would be out of work nor the optimistic one that they would spend more time in their gardens has come true. (b) The availability of computers has then permitted elaborations of the old methods of design and setting out, which may be achieved by complicated but automatic calculations. (c) These elaborations and extra facilities for the engineer's convenience have transformed the computer from a useful adjunct into a necessary tool. In consequence the convenience of traditional elements (i.e. circles and straights) has become irrelevant. In view of these observations, it is time to consider other mathematical types of curve for road alignments even if they require more complex computer calculations provided that the end product is improved or provided that the way in which the engineer specifies his requirements to the computer is more convenient.
A Basic Approach to Alignment Design In the previous section the design of road alignments has been analysed as a two stage process. In the first stage objectives and constraints are considered by the engineer in order to produce an ideal feasible line which is sometimes sketched on maps free hand or with the help of templates. In the second stage the engineer breaks the ideal line into straights, circles, etc., in order to obtain coordinates, angles and other necessary data, either through manual calculations or by specifying the position of the curves to a computer program. The new approach developed by the author tries to relieve the engineer completely from this second stage and to concern him only with the basic essentials of route location, leaving the mathematical difficulties to the computer. Three aspects have been considered to be basically relevant to the engineer: (i) Road standards - e.g. minimum permissible radius of curvature, maximum slope, minimum visibility distance. (2) Conditions of passage - e.g. points of passage with or without a permissible displacement, 'gates' between areas to be avoided, bearings and curvatures desirable (e.g. co-ordination of the horizontal and vertical geometry) or necessary in certain parts of the alignment. They represent economic, political or aesthetic decisions of the engineer and they can be very 'tight' in some parts of the alignment and very 'loose' in other parts, depending on land use, topography, etc. COMPUTER AIDED DESIGN
(3) Quality of the line - within the freedom left by 1 and 2 the line could still be longer or shorter, with higher or lower curvatures, with greater or smaller rate of change of curvature, etc. in a basic approach to the design of road alignments the engineer must be able to specify his line in terms of 1, 2 and 3 and leave the computer to calculate the alignment which best fits his requirements. This new procedure must be easily able to allow for refinements and modifications to be made to parts of the alignment as the engineer dictates. In theory the computer could still make use of conventional curves, though they would not be very easy to handle in an automatic computation. However, there is no reason why unconventional curves should not be used if they can be more easily computed and give better a quality of line. In an M.I.T. publication the view is expressed that: 'General opinion now favours a flowing line, curving and varied but without visual breaks or interruptions. Smooth continuity is most desired. Long easy curves are recommended, blending one into the other through transition curves of gradually changing radius. Extended straight tangents, or even any straight lines at all, are to be avoided as too monotonous. Similarly, long vertical curves are deemed best, fitted easily one to the other, without apparent straight grades b e t w e e n . . . Particularly if the road swings smoothly from point to point of a fine and rather open natural landscape, it gives the same sense of vital rhythm and movement as a skier's track', x4 These considerations point towards fully transitional alignments.
Polynomial Alignment System This system of programs generates very smooth horizontal and vertical alignments of a new type, made up of a small number of long curves, known as Calogero polynomials, the calculation of which will be described later. Along these curves the radius of curvatures and the A-value change gradually and continuously. This line can be specified by the engineer in as much detail as he likes via the input data which represents, in a simple and direct way, the requirements 1, 2 and 3 described in the previous section. One could use this same data as the pre-establisl'ed objectives of a conventional design, though this would involve more 'manual work. Similarly, the polynomial alignment can be made to follow closely any existing conventional alignment; the features of the program are, however, best exploited when some degree of freedom is left for the line to 'swing smoothly from point to point'. 14 The system consists of two programs: POLYAL, for the calculation of Calogero polynomials, and TABPLT for detailed tabulation and graphical output. Input to POLYAL. This input specifies the requirements of the alignment. The following data must always be input: (1) D E L T A : arbitrary length (e.g. 20 m for horizontal alignment and 1-5 m for vertical alignment) chosen by the engineer to represent the unit displacement from Guide Points. (2) R M I N ( + ) : value below which the radius of curvature must never fall (e.g. 600 m in the case of horizontal alignment). In the case of vertical alignment this value applies to valley curves (e.g. 10000 m). (3) R M I N ( - ) : minimum summit radius for summit curves, only for vertical alignment (e.g. - 20000 m). WINTER 1969
(4) Initial conditions: X, Y coordinates (chainage and elevation for vertical alignment), bearing (or gradient) expressed as a trigonometric tangent with respect to the coordinate axes, radius of curvature and chainage at the start point of the alignment. The following data can be specified in any number and combination: (5) Guide Point: X and Y coordinates (or chainage and elevation) of the required point of passage together with the weighting W which, together with D E L T A previously specified, determines the maximum permitted displacement from this point: disp. = DELTA/W. (6) Guide Point with a tangent (i.e. bearing or gradient expressed by their trigonometric tangent): same as 5 with a tangent value and a tangent weighting U. There is no facility to specify a permitted displacement for the tangent. (7) Fixed Guide Point: X and Y coordinates (or chainage and elevation) of a compulsory point of passage (displacement = 0). (8) Fixed Guide Point with a tangent: same as 7 but with a compulsory tangent at the point. (9) Fixed Guide Point with a tangent and radius: same as 8 but a value for the radius of curvature at this point is also assigned. Facility 9 is particularly useful when the polynomial alignment has to join up an existing line or a conventional element (straight line, circular arc, etc.). Therefore, conventional elements can be introduced within polynomial alignments by assigning coordinates, tangents and radii at their start and end point, and calculating the conventional element externally to the program. Output of POLYAL. The program tabulates the following values: X, Y (elevation for the vertical alignment), Tangent, R (radius), l/R, A-value, chainage. The spacing between tabulated points is decided by the program itself (e.g. 30 points per polynomial). The program also outputs on cards the coefficients, scales, positions, etc., of each polynomial (four cards per polynomial), to be used as input to other programs (e.g. TABPLT). Input to TABPLT. The engineer can specify: interval of chainage for tabulation (e.g. 100 feet), conversion factor between X, Y units and chainage units (e.g. if coordinates are measured in metres and chainages in feet, cony. f. = 3-28), position and scales of plotted alignment (for the vertical alignment different scales can be specified for chainage and elevation), position and scales of plotted curve representing 1/R versus chainage, width of carriageway and cross-sections to be plotted, positions and mesh of reference grid to be plotted, coded map details. The coefficients of the polynomials, punched by POLYA L for the alignment in question, must also be input to TABPLT. Output of TABPLT. The program tabulates, at the specified interval of chainage, the following values: chainage, X, Y (elevation for the vertical alignmen0, tangent, R, l/R, A-value. Guide Points and other points of interest are also tabulated, at special chainages, with the additional values: X and Y coordinates of the point, distance from the alignment. The graphical output of TABPLT consists of: drawings of the horizontal or vertical alignment with grid lines, kerb lines and cross-sections (for horizontal alignment 21
only), guide points, other points of interest, map details; curve representing 1/R versus chainage for alignment evaluation, superelevation desiga, etc. Use of the system. The system implements the basic approach to alignment design described in the previous section and can be applied to rural and urban roads and to motorways as well as slip roads and junctions. The input to P O L Y A L is coded by the engineer on the basis of his 'ideal line', which he may have already sketched free hand, and covers, in a simple form, the aspects described in 1, 2 and 3 of the previous section. In particular: (1) The only 'road standard' required by the program is R M I N ( + ) (and R M I N ( - ) for the vertical alignment). The program generally produces the highest possible A-value (i.e. lowest rate of change of curvature) for any given configuration of Guide Points. Therefore, if the A-value goes below standard at some point of the calculated alignment, the engineer can easily see which requirement is causing this and, if necessary, modify his data accordingly and re-run POLYAL. Similar measures would be taken even if the design were conventional. In vertical alignment design there is no constraint input for the gradient (maximum permitted value). This is because the gradient between Guide Points is largely determined by their relative positions, which are given by the engineer, and the displacement is generally small. The value of R M I N is usually assigned on the basis of what may be reasonably allowed over the particular stretch of alignment under consideration; this value is generally greater than the absolute minimum standard suggested by the Ministry of Transport. (2) 'Conditions of passage' are always expressed in the simple form of Guide Points. The calculated alignment will always tend to go as close as possible to Guide Points (in any case within the permitted displacement) compromising with the other objectives (quality of the line) described in 3. The permitted displacement is assumed to be the same on either side of the Guide Point. When this is not applicable at some point, the Guide Point should still be located in the most desirable position (i.e. on the 'ideal line') and the most restrictive displacement used; alternatively, the one which applies to the side on which the alignment is likely to pass. A Guide Point with the permitted displacement either side can be regarded as a 'Gate' with a preferred point of passage in the middle (Fig. 1). However, some condition of passage could be represented by a gate without a preferred point of passage, for example gaps between constructions to be avoided. There is no facility to input these gates as such, because the engineer can easily recognise (before or after the first computer run) one of the following two cases: (a) The alignment tends to miss the gate by a particular side, in order to improve the quality of the alignment. In this case a Guide Point has to be input, within the gate but as close as possible to the preferred side (Fig. 2).
22
(b) The alignment goes through the gate with no need for any Guide Point at the gate (Fig. 3). Sometimes long stretches of alignment must follow conventional curves, for example to run parallel to an existing road or railway; in this case the polynomial alignment can be made to follow the curve in question very closely by specifying a few Fixed Guide Points with tangents and radii. Alternatively the polynomial could be terminated (with a Fixed Guide Point) at the beginning of the straight or circular arc and resumed again at the end of this element. In this case the difference in the two chainages would have to be calculated externally to the program. In interchange design the alignment is required to follow long curves of tight radius, which is an 'unnatural' pattern for polynomial expressions. Nevertheless the program itself copes with this difficulty by using several short polynomial curves. This is illustrated in Fig. 4 which shows combinations of polynomials which have been calculated and plotted by the program and may be seen to closely resemble circles. However, a greater number of Fixed Guide Points and a certain amount of experience with the program is generally required for the design of interchanges. Under normal conditions, in the case of highstandard roads in developed areas, experience has shown that an average of three Guide Points per mile have been used by different engineers. (3) The 'quality of the line' is not directly specified by the engineer in the input data but is built into the program (POLYAL) through a number of computational schemes, the most effective of which being that the program calculates many alternative polynomials before selecting the ones to be used. F o r a given input ( R M I N and Guide Points) the program produces a line which, first of all, respects the permitted displacements from G.P.s, then satisfies the curvature constraint if this is geometrically (and humanly) possible, then aims for the following objectives:
x Ouide points y\ / ' Polynorninat • - ; , - \ . . . . alignment, ~ / X ,\
x
\
/ / ~ -,,
/
Fig. 1. Guide Points with permitted displacements
___~p ....
x Guide points Polynominal alignment when P is not input Potynorninat alignment
Fig. 2. Gate which requires a Guide Point. COMPUTER AIDED DESIGN
(a) Closeness to Guide Points. (b) Low curvature and rate of change of curvature. (c) Short overall length. Experience has shown that in normal cases the program reproduces the line that the engineer had in mind very closely, even if he coded the input data without previous experience with the program. Furthermore, if the engineer left some parts of the line poorly specified, the program can produce good solutions which the engineer did not have in mind in the first place. However, in some cases, the engineer may decide that the calculated line must be improved in a particular way, after examining the first output of the program. This is done by modifying the input data and re-running the program. For example, if the engineer wanted to increase the radius of a certain curve, he could proceed in a number of ways, for example: (i) Assign one or two new Guide Points on the ideal curve of greater radius. (ii) Select the coordinates and the tangent of a point on the calculated line and include it in the new input as a Fixed Guide Point with the old tangent and the desired radius. (iii) Increase the value of RMIN and re-run only the stretch of alignment to be modified. It is now clear that this approach to the problem of coding the 'ideal line' into some input data can claim simplicity and ease of use, but may require two or three computer runs before the results were fully acceptable. The philosophy behind this approach is the following: Experience shows that the user of engineering programs requires, most of the times, re-runs on the same design problem. This is because his ideas clarify or change when he examines the computer output, data errors come to light, colleagues and superiors make suggestions and comments on the results. Therefore it is most undesirable to have an extensive and difficult input layout and/or a long and complex computation if the inevitable re-runs
can easily incorporate refinements of the design. In practice only POLYAL is re-run as its output enables, in most of the cases, judgements on the acceptability of the solution. Stretches of alignment can be re-run individually (using Fixed Guide Points with tangent and radius at both ends) if the rest of the alignment must not be modified. The card output from different runs of POLYAL can be ultimately combined in one input for TABPLT, in order to obtain the final tabulation and drawings. In preliminary design the system is used as a tool to specify quickly a trial line (horizontally and vertically) and obtain automatic drawings and data for subsequent evaluation (Digital Ground Models, earthworks, etc.) as described in the introduction. In final design the alignment is 'polished up' introducing those adjustments on the horizontal and vertical location which are required by economic and aesthetic consideration, drainage requirements, horizontal-vertical coordination, details on the ground which had not been considered before, etc. This is done introducing new Guide Points, with tangents and radii where appropriate, and/or modifying the values of the old ones. In this connection the output obtained at the preliminary design stage for the line in question is very useful, as it provides accurate 'starting' values for coordinates, tangents and radii at any chainage. For detailed evaluation of the geometrical quality of the horizontal alignment it is useful to obtain the curvature curve (I/R versus chainage) via graphical output of TABPLT. Sometimes a conventional alignment and a polynomial one could almost coincide if drawn in plan at a scale of 1:2500, but their curvature curves would still show a clear difference of the type of Fig. 5. For superelevation design the engineer divides the polynomial alignment into segments along which a certain value of superelevation seems appropriate, for example normal crossfall for segments where the radius is always above 3350 m, 1 in 14 for segments where the radius is always below 600 m, etc. Changes of superelevation can
\\ x Guide points Polynominat alignment
Fig. 3. Gate w h i c h does not require a Guide Point.
f
1
f-chainage Polynominal alignment - - - - - Conventional alignment
Fig. 5. Curvature curve. WINTER 1969
-.t- Guide points
Fig. 4. 23
then be calculated with conventional methods and programs. Alternatively the engineer can.use the curvature curve. as a superelevation curve, changing the scale according to the formula: tan x -
Ve
KR
In this case he would still select the segments of alignment where crossfall must be normal, and the points of reversion, taking also into account the vertical gradient. But he would then obtain the amount of superelevation directly from the curvature curve, for those chainages where the radius falls below a certain value, e.g. 3350 m. Setting out of polynomial alignments can be done using conventional methods. However, standard tables and formulae cannot be used, and values of angles and distances are generally calculated by computer programs 12, 16 which can be 'integrated' with the polynomial system, The advantages of this approach have been discussed in the Introduction. The computer print out is used directly in the field and generally contains the type of data shown in Fig. 6, to permit the setting out of each point in several alternative ways. The choice is then made by the man in the field. The polynomial system can be linked, quite conveniently, to other systems of programs. The horizontal alignment output can be obtained on cards and used as input to other programs for ground interpolation (D.G.M.s), setting out, automatic superelevation design, drawings of perspectives, etc. Three-dimensional Guide Points can be specified by the engineer and the system used to calculate the horizontal alignment first and then the vertical one, using the original input elevations and the calculated chainages at Guide Points. Points of passage (horizontal or vertical) calculated by optimisation programs ~, 2o as a result of costs and constraints fed in by the engineer, can be directly input to the polynomial system in order to integrate fully the optimisation and alignment programs.
Discussion of advantages and disadvantages Though the polynomial method introduces a number of conceptual and practical innovations in highway engineering, its function is still that of a tool in the hands of the engineer. Polynomial and conventional alignments are difficult to compare in terms of the end-product: if one engineer was asked to design the same scheme with both methods, he would produce two lines almost identical; if a different engineer was using each method, the comparison would be basically related to the two engineers and not to the methods themselves. However, some observations can be made about the flexibility and ease of use of the two methods at the design and construction stages. Setting out work will probably be more demanding in the case of the polynomial alignment, though there is no experience to support this opinion with actual figures. Similarly, until a certain amount of experience is available, superelevation will require some extra attention at the design stage, in the case of polynomial alignment. If curvature or superelevation varied along viaducts, standardisation of the elements of the structure would be impossible. 24
Ir
~'T
~t~j......~-
X Point to set out-on the alignment A Traverse stations (control points) • Points previously set out Fig. 6.
Input to conventional programs requires precise values of coordinates and parameters even when the physical requirements or the accuracy of the maps do not justify such a precision. On the other hand with the polynomial method input data must be provided only if and where there are practical requirements for the alignment, and 'freedom" is used by the program to improve the geometry of the line. This, however, does not apply to the design of interchanges. When necessary, the polynomial can be specified in a more precise manner than a single conventional curve. F o r example, if the positions of a circular arc and a straight line were determined in plan for some particular reason, there would only be one possible transition curve in between. In the same circumstances the polynomial could still accept further specifications and generally produce a smoother change of curvature or better avoid an obstacle on the ground. If some change of alignment was required at the setting out or construction stage, this could only be done, in the case of the polynomial, through a computer run. Alternatively, some appropriate length of alignment should have to be replaced manually with conventional elements. Refinements are easily made on a polynomial alignment because individual Guide Points can always be added or adjusted independently. Furthermore the same programs, the same basic data and, sometimes, the same results can be used in both preliminary and final design. On the other hand, conventional programs generally use different methods for preliminary and final design (e.g. no transitions in preliminary design) and alterations in one curve very often need re-adjusting the position of the other ones. Engineers can learn how to use the polynomial programs in less than an hour. Experience of Use. The polynomial system has been used experimentally on a wide range of study cases, covering rural roads, urban motorways, interchanges, etc., supplied by local authorities, consulting engineers and Ministry of Transport. In many of them a conventional alignment was already designed and used for comparison. The system has also been used for some actual route planning and for the final design of a By-Pass. To the knowledge of the author, no one has yet driven on a polynomial road.
Example This example consists of the horizontal alignment design of the Sutton-Coldfield By-Pass carried out by the Warwickshire Sub-Unit, of the Midlands R.C.U. The scheme is a trunk road improvement, five miles long, and construction is programmed to start in 1969. Input data to COMPUTER AIDED DESIGN
~ I T Y OF LONDON, INSTITgTE OF COHPUTER SC]3ENCE,
~Am]~
POLYNOMIAL A L I G n :
POLYAL (This program fits CalqKero's polynomials to Guide Points) INPUT
DATA
Y
~uo== V. C a l o ~ r o D,ATZ= 1Btk N o v 'BB
~zm'os
~
Y
~o..o,l
BHER~
A l l nu=bers mue.._~contain a decimal p o i n t . CONSTRAINTS
I
OZZ,~
CARD
•
I
I~IZ.(+)
[m,EN(-) ( o p t ¢ o . ~ l . )
i
I 314 111 ? I I I I I I I I I I I 4 1 1 1 1 1 ? I I U ~ N I I I I I ~ I I I I I I I ' ~ I I I I ~ 4 ',' ,',' ,' ,' ,' ,' ,',1011 , . . . I ..... , . . . . . . . I , , . . . . . . . . . 15,o,. ....... ,,. I n,o,o,o,., , ....... I, , .........
INITIAL
I
CONDITIONS
~9,2;6,0',01.,,, ;,' 2,9,3,1,0,0,., , , I . 2,9,3.6,4,6..,,, I , 2,9,4,5,0,0,.,,, ,.,, 2,9,5,2,0,0,..., ~ , 2,9,5,9,0,0,.,., = , 2,9,6,2,0,0..,,, =,,. 2,9,6,5.00..,, ~,,, 2,9,7,2,3,2..',, , ~9.7,7.7.8.,,,, , 2,9.8,2,9{0.,,, |,
',','
, 4, 1,5,3,0,21., , , 4,1,5,1,5,5=., , , 4,1,5.1,1,5|., 4,1,5,2
~,0|',
......
LI.'| .... , 12,5=', il,Ol.,
...... ..... .....
,. 4,1,5,~,~8=., , , 4 I 5639C, 4. 1 . 5 , 6 , 7 , 6 r , 4 , 1 , 5 , 6 , 8 , 2 | .. . . . . . , , 4,1,5,5,0,0C, , , 4,1,5,2,6,5~ • . . . . . , , 4, 1 , 5 , 0 , 8 , 5 | . , I, , , 4, 1 , 5 . 0 , 2 , 2 | , , , , = , , , 4, 1,4,8,2, I ~.
2,9,9,1.4,1,.,
l'le~
C01.~tlE~
,.
I.
. . . . . . .
,
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,,,
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.
l.Or,
, ,
~,51",,, ~,51",,,
I
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,
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5~i',
,
11 17 1l 1l ~i It II Ii Ii Ii Ii
~ ' ,
[ ~ANG~IT | , F,0,',4,0,5,3, I .....
i ,
....
,,,
,,,,,
....
I
, ,
!
I,.,, I II
llll
,,,,,,,l
,,,, III
I I I I I I I I i
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,,,I
i I
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(~t.)
i , o , , , , ,
,,I ,,I III
,,
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,,I IlII ,,,I,
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......
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i |, I,
, , , ,
. . . . . . . .
I
,
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- p , . ,5,3,9,7,
,
,.
, I ,
,
,
,
,
,
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,
-,I,2,7,3,.,~
.',,'.'.',,V,,'.',,'.'.'.'.'....'.'.
~V.'='.',,'.'.'~
.'.~'.'#.',,'##.
Fig 7 Input data to POL YAL (Sutton Coldfield By-pass) POLYNOHIkL ALIGNMENT
*
T k B P L T
*
V,CALOGERO NOVEMBER 1967
POLYNONIkL OF DEGREE 7 ST4RTS ~T NEXT PT 0.000000000*00 9,25437276E-01 3 . 7 5 0 5 3 4 6 9 0 * 0 3 " 3 , 8 1 0 7 1 6 8 9 E - 0 1 9 . 2 4 5 4 5 5 0 5 | ' 0 1 2.926000000*05 4, 15302000E*05 0,00000000E-00 ¢,O0000000E*O!! O.O000flOOOB*O0 O,O000000nE*o0-9, 78052489E-01 1 . 7 9 9 0 4 4 9 9 0 - 0 0 " 1 , 5 3 1 3 ~ t $ 2 2 " 0 0 - 1 . 4 1 2 0 3 6 3 0 | * 0 0 5 . 9 4 8 3 9 7 2 4 0 * 0 0 - 6 , 35148569E*00 2.222897520*00 CNAINAGE XA 0,00 2 9 2 6 0 0 . 0 0 0 0
YA 415302.0000
TA -0,4033
R 2852.79
IlR 0.0003505
4.'VALUE
CH4INAGE XN YN 0.00 292600,0001 4~5302,0000
TN -0,4033
R 2852.79
~/R 0.0003505
4=VALUE XG YG " 2 2 1 0 . 6 9 292600.0000 415302.0000
CHAINAGE 100.00 200,00 300,00 400.00 500.00 600.00 700.00 flOG.On
1/R 0.0003562 0.0003617 0.0003669 0.0003718 0,0003764 0.0003807 0,0003846 ~-n003882
4=VALUE -|315.76 -2363.45 -2420.35 -2486.27 -|569.57 "~667.41 - 0 7 8 6 . ;~4
_,ol.80
292620,3260 292656.7780 292~85.3460 2927~4,0311 292742,8310 292771.7437 292~n.7663 °
YA 415290.7516 415279.8114 4~5269.1054 415258.8796 415248,8994 415239.2504 415229,9374 '~5220,9653
TA -0,3908 -0.3783 -0,3656 -n.3529 -0,3401 -0,3273 -0.3Sa= -"
R 2807.27 2764.92 2725.70
¢~ 29427/ . . . . 204307,4290 294337.2726 204367.0744 294396.8338 294426,5504 294456,2240 294405,6542
. . . o041 415190,8789 415197.0761 415203,4717 415~10.0617 4152~6.8420 415223,8084 415230.9567
g,1825 0.1890 0,1970 t~,2041 n,2111 n,2180 n,2248 0,2315 0,2380 ~.2444
q=~ 434t,77 4418.43 4498,15 4581.58 4669.49 4762.78 4862,46 4969,73 500~.98
O.uv~.=.; 0.0002223 0.0002183 0.n002142 0.0002100 0.0002057 0.0002012 0.0001966
CHA[NAGE XN YN 6336.42 294496,6348 415233,6045
TN 0.2468
R ~130.06
I/R O.flOOt949
AIVALUE
CHk]NAGE 6400,00 6500.00 6600,00 6700.00 6800,00
YA 415238.2824 415245.7010 415253.4477 415261.2776 415269,2654 415277,4059 415285,6933 "~294,1217 • 6850
TA 0,2507 0.2569 n.2629 9,2687 0,2744 0.2799 0,2852 0,2903 0,2952
R 5212.02 5352,17 5506.27 5677.04 5870,15 6087.21 6334 " 6"
~IR 0.000~918 0.000~668 0.0001816 0.000~761 0,0001784 N.nOg1643
A=VALUE ~524.07 2470.41 B414.33 2356.73 2296.39 2240.02 E182.22 ~125,45
414~ 414915.83~r 414905.0940 414893,9188 414082,2885 4~4870.1777 414857.5564 414844,3902 414830.6427 41482~,0000
,/ -11,3601 -0,3051 -0,4032 -0,4226 -0.4436 -0.4666 -0,4918 -0.5194 -0,5397
-2160.27 -2137,44 -2076,33 -1994,94 -1895.35 -1782.77 -1664.02 -1545.97 -1434,42 -1333.70 -1273.24
-O.uv~ -0.0004679 -0.0004016 -0.0005013 -0.0005276 -0.0005609 =0.0006010 -0.0006460 -0.0006971 -0.0007490 -0.0007854
CHA[NAGE XN YN 22267.60 299141.0006 414821.0009
TN -0.5397
R -1273.24
1/R -0.0007854
qn n
b~u~. 5500,00 5600,00 5700,00 5800,00 5900,00 6000,00 6100.00 6200,00 6300.00
X4
XA 294515,4411 204544,9847 294q74,4850 20460~,9425 204633,3576 6900.00 2 9 4 ~ 2 , 7 3 0 8 7000." "~q 7"
.uO .~400,00 21500.00 21600.00 21700.00 21800,00 ~1900.00 22000.00 22100.00 22200.00 02267.61
298069,821298490,6457 200927,3256 2989~5.8507 298984,2082 29901203821 299040.3528 299n68.0967 299095,5862 299~22.7896 299~41,0008
2689.6t
2656.63 2626.73
....
~
DN 0.0001
2756.68 2743.09 2723°47 2695,69 ~661.09 2620.36 2574.38 2540.71
XG YG 294500.0000 4 1 5 2 2 0 - 0 0 0 0
oN 14,0148
2487o82 2245.49 1075.75 956.49
072,63 014.96
776.43 760.89 760.76 AuVALUE XG YG 760.75 299141.0000 414821.0000
ON 0,0011
Fig. 8. Line printer output from TABPL T. WINTER 1969
I
RADIUS OR
.... ....
151"Iil
,
,,
COOl.IT
x . . . . . . I ~. . . . . . . . . I c~'rNAGE. . . . . . 17,9,2,6,0,0, • , , , | , , , [ 4 , 1 , 5 , 3 , 0 , 2 r , ...... [I,OtO, O," . . . . . . . GUIDE POINT CAEDS ( L a s t a . 9 . card to be followed bT a blank car'd) X X W~ZG]~
2,9,8,5,4.2..,,,
|ID~I'Z~CaTZON
414144 414114T 40 q~llll lll~l!~41111!r~ ll ll l~ 11111114 IIIIIIII~ IIIIII~II'/III~I 14"/111111~T~I~IIIIII , . I . . . . . . . | ......... I' ......... °I .........
25
,
P O L Y A L are represented in Fig. 7. Standard displacement ( D E L T A ) is 50 m, m i n i m u m r a d i u ~ o f curvature ( R M I N ) is 1000 m. There are 13 G u i d e P o i n t s with coordinates in metres, established mainly to circumvent farms and properties. Line-printer output from T A B P L T is shown in Fig. 8. Graphical output from T A B P L T is shown in Fig. 9. The line has also been redrawn manually on maps using tabulated coordinates (Fig. 10).
The Algorithm The Polynomial Alignment system is based on the calculation of the sequence of Calogero polynomials which best fits the specified Guide Points. This calculation is performed by the program POLYAL, while T A B P L T simply tabulates and plots the alignment.
Calogero Polynomials
/
Consider: (i) an ordered set of points G = {Gill = 1, 2 . . . . 1}, where Gi = (Xi, Yi, wi, ti, Ui); (ii) a length a. (iii) a radius of curvature r,,in. (iv) the trigonometric tangent to and the second derivative Y0" of the polynomial at the origin. A new set of points Q can be now defined in the following way. Let: k be an arbitrary n u m b e r of new points at equal distance on the straight line between points Gi and Gi.1; Ix = l + ( 1 - l)k; I = ( I , 2 + k, 3 + 2k . . . . . i + ( i - 1)k . . . . . ll); Q = {QjIj = 1, 2 . . . . 1~}, Qj = ( X l j , ylj, w~i, t~j, u~j); Xxj : Xi, Y l j = Yl, wlj = Wlj(j, Wi), tlj : ti, ul1 = u l j (j, u i ) , j d , i = (j + k)/(l + k), where Wlj and u~j are arbitrary functions. xli --
y l j --
(xi+1 -
xi)
k+l
/j/
//¢/
{j - [i + (i - 1)k]} + xi,
(Yi* 1 Yi) {j -- [i + (i -- 1)k]} + Yi, k+l -
//
)
-
w,j = f(j),txj=0, u~j=0, i + (i+ ik, i = 1, 2 , . . . , 1 - 1, where f is an arbitrary function.
1)k < j
< i + 1
Given i, ii, iii, iv, a Calogero polynomial is defined by: 1 y = tax + ~y0 x2 + a3x a + a4x 'l + . . . +
/
/
/
anxn,
where n, a3, a4,.., a , , have the following properties: (a) at each point Gi the following constraint (permitted displacement) is satisfied: ly(x3
-
Y~I
~/1 + y'2(xi)
d ~--; wi
(b) at each point Qj the following constraint (curvature) is satisfied :
Fig 9. (Right). TABPL T.
Graphical output from
Fig 10. (Far Right). TABPLT output redrawn manually on O.S. map. 26
COMPUTER AIDED DESIGN
delta/w=5 m _ _
1 ~5000 MN
if
Warwickshire County Council Polynomial Alignment for Sutton Coldfield By- Pass
;.17
I
I
I
O
Fair Ficw L Far~
I ~/2MI LE
W-
'w=
S~O~li G I #
"e e le i i q li Ii I
L
~
• !
• •
i
e •
II
•
li
•
•
\ T I / f M ## 1. A"
l','.',T]) -
-
")--i'l
'.F,r
/
-
\ \\
~ =
///iA\\
"7"
)~-/Peddtmore
~delta!w - 5 m
2cJ4000 MN
c" /'/ / /
•'. ... ,...,.,%,~.. _..
~,°,, __,~.~,-,.,-
f
~
- ~_
_~-
--
-
,
i
.
delta/w =
2 m
4
83000 MN
\ /I Lane
.......
~
I,
t
-
ixed Point7.~' .'. ~tart of Alignment~,,
__-~-~ ".'11
WINTER
1969
27
lrll >/r.,i.. [1 + y'=(xlj)] "~/~'. Y~(xlj) "'
where rj =
(c) aa, a4 . . . . . a. are such as to minimise the following expression: h a =
+
,E { w ~ i [ y l y j=i
-
y(xly)]" + uli[t~/ - y'(x~])]~
ag(j)[y"(xtj)]-"},
where g is an arbitrary function and ,~ is a variable. (d) A is such as to minimise n. It is obvious that, having chosen k, w~i, u~j, f and g, and given i, ii, iii, iv, there may exist one, none or an infinite number of Calogero polynomials, because A may vary within a certain range and still give the same minimum value ofn. From the alignment point of view any one of the range of polynomials can be acceptable. The arbitrary k, w~i, vjj and f, used by POLYAL, have been determined by the author through experimental work, while g is 'learned' by the program at running time. The basic definitions of the Calogero polynomials can be extended to the case where some points G~ are fixed (wt = oo and ui = oo) and the radius of curvature at the points may be fixed. The problem of calculating n, a.~, aa . . . . . a,, given i, ii, iii, iv, can be solved in many ways. The one used by P O L Y A L consists of a heuristic approach, whereby the linear system of equations: (~6t -
0, m
=
3,4,
...,n
8am
is solved for many different values of A and for increasing values of n, until a feasible solution is reached. Geometrically speaking a Calogero polynomial will always: (1) start at the origin of his system of coordinates, which is generally different from the one used originally for Guide Points; (2) assume a given bearing (to) and curvature (Yo') at the origin, to guarantee continuity with the line calculated so far; (3) keep close to the straight line (points Qi, J d ) between pairs of points G;; this tends to keep the length to a minimum; (4) keep close to the points Gi (a = min); (5) reverse the curvature as few times as possible (n = min), which also tends to keep the length to a minimum; (6) keep the curvature as low as possible (n = min and A#0).
polynomials being accepted which fit the conditions but behave erratically. Two points are worth mentioning: (I) Overlap between consecutive polynomials. In each cycle of POLYAL, the i.dh Guide Point is the end point of a particular trial segment of alignment, but is not the last point Ge used in the fitting. Some more points, whose number is calculated by the program, are always included in G to produce a convenient bearing and curvature at the i2 th point. (2) Axes for calculations. The coordinate axes of points Gi must have the origin coinciding with the start point of the new polynomial. The orientation of the axes affects considerably the quality of the fitting. Appropriate rescaling reduces the effect of rounding-off errors. Therefore POLYAL calculates each time (Block 3) the coordinate system to be used for the fitting. TABPLT reads in the cards output (Block 10) by POLYAL together with the engineer's specifications, and tabulates and plots the alignment.
Acknowledgments I would like to thank Professor R. A. Buckingham and K. Wo]fenden of the University of London Institute of Computer Science, W. Oxburgh of Planning and Transport Research and Computation Co. and Dr. J. P. Stott of the Road Research Laboratory for their co-operation and helpful discussions on this topic over the past three years. 1 am also grateful to the Ministry of Transport for financial support and to Warwickshire County Council, W. S. Atkins and Ptnrs and R. Travers Morgan and Ptnrs for their co-operation. [ am indebted to R. Timber]ake, K. Rivers, M. Lee and H. Abrahams for assistance in producing this paper.
3 I Ch°°S-erbie~t~YS~er mi,, i,~ ]
1,
I Fit Calogero Potynomiat i to G.P's from il toi=
YES
Choose i~ N0 ~1 and p°lyn°miM
OuI
The Computation The mechanism of P O L Y A L is explained by the flow chart of Fig. 11. Basically the program selects trial sets G of Guide Points (Block 3), performs polynomial fitting on them (Block 4), chooses the set and polynomial which gives the best fitting (Block 8), and proceed to the next segment of alignment (Block 12), This method prevents
28
I
from stored t r i a l oo,,nom,at, I
©
'11 G.P: guide points / L : number of G,R's read in il : index of first G,R of current polynomiat YES i;~ : index of last G.R of potynorniat being catcutated [~: chosen index of fast G.P, of best current polynomial ~ E
Fig. 11. Flow chart of POL YAL COMPUTER AIDED DESIGN
Bibliography x Manheim, M. L.: 'Highway Route Location as a Hierarchically-Structured Sequential Decision Process', M.LT. R6415 (May 1966). Roberts, P. O., and Suhrbier, J. H.: 'Highway Location Analysis', M.LT. Report n.5 (1966). 3 Leisch, J. E.: 'New Techniques in Alignment Design and Stakeout', Journal of the Highway Division, Proceedings of the American Society of Civil Engineers, Vol. 92, No. HWI (March 1966). 4 Godin, P.: 'Avants-projects d'Autoroutes et Tendances Actuelles en Mati6re de Trac6 Autoroutier', Revue Gdn~rale des Routes (January 1963). s Calogero, V.: 'The Application of Electronic Computers for the Design of Roads in France', Institute of Computer Science (July 1965). 6 Calogero, V.: 'The Application of Electronic Computers for the Design of Roads in Sweden', h~stitute of Computer Science, (July 1965). 7 Spencer, W. H.: 'The Co-ordination of Horizontal and Vertical Alignment of High-Speed Road', Institute Civ. Engineers Road Paper No. 27, (1948). s Smith, B. L., and Fogo, R. D. : 'Some Visual Aspects of Highway Design', 46th Annual Meeting, Committee on Geometric Highway Design. g Calogero, V.: 'Some details of the mechanics of handling D.T.M.'s by computer', Proceedings of P.T.R.C. Co. Ltd., Seminar (17 October 1967). ~o Calogero, V., and Tocchetti, A.: 'Un metodo veloce per il
calcolo dei volumi delle sezioni trasversali', Rivista Autostrade No. 2 (February 1965). H Calogero, V.: 'Polynomial Horizontal Alignment Program', Proceedings of P.T.R.C. Co. Ltd., Seminar (18 October 1967). 12 'Dossier Pilote TE. G167 - Trac~s [lectroniques en Geometric Imposee', Service Special des Autoroutes (1967). 1:3 Deligny, J. L.: 'Les projects d'autoroutes /t t'Sre du dessin automatique', Review G~n~rale des Routes, No. 398 (April 1965). 1.L Appleyard, D., Lynch, K., and Myer, J. R. : 'The View from the Road', M.LT. Press (1964). 15 'Programme EVASOM: Evaluation sommaire d'un avantprojet, Service Special des Autoroutes (1967). 16 'British Integrated Program System for Highway Design', Ministry of Transport, County Surveyor's Society, Association of Consulting Engineers, Part 1, May 1967, Part II, December 1968. 17 Cowling, H.: 'Highway Design', Computers in Highway Engineering - 1, The Journal of the htstitution of Highway Engineers, Vol. XV, No. 7 (July 1968). is Godin, P., Deligny, J. L., Antoniotti, P., Day, J. A., and Bernede, J. F. : 'Visual Quality Studies in Highway Design', B.C.O.M., Washington, D.C. (January 1968). 19 Calogero, V.: 'Computer Aided Highway Design', Ph.D. Thesis, Institute of Computer Science, University of London, to be submitted. 20 Calogero, V.: 'Optimisation of the Vertical Profile', Proceedings of P.T.R.C. Co. Ltd., Seminar (18 January 1968). Received December 1968
Dott. Ing. V. Calogero studied Electronic Engineering at the University of Naples (Italy) specialising in computers. In early 1964 he obtained his degree of "Dottore in Ingegneria Elettronica'. Subsequently he carried out research on the application of computers in civil engineering at the French and Swedish Ministries of Transport. At the end of 1965 he was appointed Research Fellow at the Institute of Computer Science, University of London, and registered for a Ph.D. in "Computational Methods in Operational Research' supported by the British Council and the M.O.T. The thesis, entitled 'Computer Aided Highway Design', will be submitted early in 1969. Since 1968 he has been a Lecturer at the Institute of Computer Science, University of London, with research responsibilities to the M.O.T. He is also currently involved in consulting and lecturing work for government bodies and commercial firms in England and abroad.
WINTER 1969
29