Minute of symmetry

Shortly before every full hour there is a train crossing (meeting) on ​​many railway lines in Central Europe with a regular timetable

The symmetry minute is a characteristic figure in clock schedules of public transportation . At this point in time, a journey in a clocked timetable meets its counterpart on the same line . Transfer times between lines with identical symmetry minutes are always the same in both directions.

introduction

Train crossing in Süderbrarup on the Kiel – Flensburg line at minute 28/29 (hourly)

If the same number of trains (or other vehicles) run in both directions in a clocked timetable , with the same travel and transfer times and with the same travel order, this timetable is completely symmetrical. The trains going there and back always meet at half the pitch at constant points on the given route.

At each symmetry time , the timetable of the direction and the opposite direction is "mirrored". At the end of the line, the middle of the turning time coincides with the minute of symmetry. The time interval between two successively occurring symmetry times (a sufficiently long distance) is equal to half the cycle time. With an hourly service, there is an intersection with trains on the same line every 30 minutes. With a pitch of two hours, there is a symmetry time every hour.

The following table is used for easy reading of the departure time in the opposite direction at hourly intervals. This is based on the symmetry minute, which is by far the most common in Central Europe. For shorter periods, the further departure times can be calculated based on this. The last line shows the bars at the respective meeting times.

Arrival one minute	58½	00	03	06	09	12	13½	15th	18th	21st	24	27	28½	↰
Departure return minute	58½	57	54	51	48	45	43½	42	39	36	33	30th	28½	↲
Encounter at ... min. Cycle	120			15th			30th			15th			60

Since 15-minute intervals are seldom found, the yellow-highlighted encounters do not appear often, but for example on the local railway Vienna – Baden , on the Trogenerbahn and on the regional tram from Liberec to Jablonec nad Nisou . The Rhein-Main S-Bahn also has a basic cycle of 15 minutes.

Network every hour with zero symmetry

In principle, an intersection can be defined at any point in time. However, in order to be able to create a symmetrical connection relationship at the transition between two networks or lines, it is advisable to define uniform symmetry minutes. In coordination with the long-distance traffic cycle systems of the ÖBB and SBB , the Research Association for Roads and Transport for Germany recommends minute 58, so that with a four-minute transfer time there is a departure at minute 0. In the meantime, most railways in Central Europe and some other transport companies have established the symmetry minute 58½, with which the departure minute 0 results with a three-minute stopping time. In accordance with the symmetry, this results in additional symmetry minutes for shorter cycles, which are shifted by half the cycle time. So at hourly the minute 28½, with 30-minute clocks the minutes 58½, 13½, 28½ and 43½ etc.

Basics

With an n -minute cycle, symmetry nodes occur every hour over a sufficiently long route , these are the times at which two journeys on the same line meet. In the case of hourly journeys there are, for example, two symmetry minutes per hour, each occurring at an alternating location, so that the encounters at one location logically correspond to the rhythm. In general, this results in a minute of symmetry on a line ${\displaystyle {\tfrac {120}{n}}}$ ${\displaystyle t_{\mathrm {sym} }}$

${\displaystyle t_{\mathrm {sym} }={\frac {t_{\mathrm {an} }+t_{\mathrm {ab} }}{2}}}$

with the arrival time in the outward and the departure time of the return journey at any stop (or the corresponding transit times at any location). With an hourly cycle there is another symmetry minute offset by 30 minutes, with an even denser cycle there are further symmetry minutes that are each spaced apart. ${\displaystyle t_{\mathrm {an} }}$ ${\displaystyle t_{\mathrm {ab} }}$ ${\displaystyle {\tfrac {n}{2}}}$

A symmetry node does not necessarily have to be in a train station where travelers get on or off, but can also be between such stations. On single-track routes, however, there must be an alternative, which restricts the scope for design.

Knowing the minute of symmetry, if the minute of arrival is known, a passenger can immediately approximate the minute of departure in the opposite direction in a regular clock schedule .

The symmetry minute is also important for transfer connections: When traveling from place A on a certain route to place B on another route and back, the transfer time in the opposite direction is only the same if both routes have the same symmetry time. With different times of symmetry, the difference in transfer times from direction and opposite direction is always twice the difference in times of symmetry.

The waiting time of the means of transport at the crossing points can be different. When different types of train cross, for example, a regional train waits until the faster train, which is not available in the opposite direction at the time because of a different cycle, has passed. The speed can also be different in different directions (see example). Both result in deviations from the symmetrical course.

Every 2 hours

In Mannheim z. For example, at the half-hour clock junction , ICE lines 11 and 12, which run every two hours, are swapped with lines 42 and 43 every hour, in such a way that trains on the same line always meet just before the full hour on their respective route. Trains from different ICE lines always meet in Mannheim.

With a 2-hour cycle, there are departure times with different numbers of hours in each direction. When arriving in one direction at the even hour (e.g. 14:40), the departure in the opposite direction always takes place at the odd hour (e.g. 15:17) and vice versa (in the examples, the symmetry minute 58½ was again assumed) . A clock shift of one direction of travel by one hour results in departure hours for both directions at the same even or odd hour. Exceptions in this case are departures at the (approx.) Full hour.

In some regions, for example Brandenburg , this time shift is used more often at 2-hour intervals. Examples are also the EVB on the Bremerhaven – Buxtehude railway line and the Rügensche Bäderbahn (with a symmetry minute that differs greatly from the usual one). If this does not happen network-wide, this leads to different connections in the direction and opposite direction. So on Sundays in Bremerhaven you have a direct connection from Bremervörde to Cuxhaven, but you have to wait over an hour in the opposite direction.

The formation of circulation groups can offer operational advantages, since depending on the travel time on the routes and the frequency, the vehicles have long waiting times at the ends. If several lines end there that travel in an "alternating" cycle, the arriving vehicles of one can continue on the other; you save vehicles without worsening the offer.

Symmetry defect

The deviation of the minute of intersection of a single line to the minute of symmetry of the network is called a symmetry defect. This is calculated from the formula

${\displaystyle 2\cdot s-2\cdot d\leq T_{i}+T_{j}-z_{a}\cdot t_{T}\leq 2\cdot s+2\cdot d}$

With

${\displaystyle d\to min}$

${\displaystyle z_{a}\in \mathbb {Z} }$

${\displaystyle d_{a}}$ : = Symmetry defect

${\displaystyle s}$ : = Minute of symmetry

${\displaystyle T_{i}}$ : = Arrival execution

${\displaystyle T_{j}}$ : = Departure in the opposite direction

${\displaystyle t_{T}}$ : = Cycle time

example

A line runs in a network with a symmetry minute 58½ every 60 minutes. One direction arrives at minute 55, the opposite direction leaves at minute 58. The formula is accordingly:

${\displaystyle 2\cdot 58,5-2\cdot d_{a}\leq 55+58-z_{a}\cdot 60\leq 2\cdot 58,5+2\cdot d_{a}}$

${\displaystyle 117-2\cdot d_{a}\leq 113-z_{a}\cdot 60\leq 117+2\cdot d_{a}}$

with results . ${\displaystyle z_{a}=0}$ ${\displaystyle d_{a}=2}$

The intersection of the line at minute 56.5 is 2 minutes away from the symmetry minute of the network.

Practical examples

Switzerland

The original plan for regular traffic in Switzerland from 1972 provided for a symmetry minute 10.

Contrary to the previous agreement with the neighboring railways DB and ÖBB, the symmetry minute 0 is often used in Switzerland today. In the 2014 timetable year, for example, the long-distance lines from Basel SBB train station, from Zurich HB train station to the west, south and south-east, and from Bern train station to north and south all lines almost without exception operate with exactly this symmetry time. The lines that are not exactly at this symmetry time move in the range of ± 0.5 minutes to this.

Netherlands, Belgium, Czech Republic

The Nederlandse Spoorwegen had a regular timetable long before December 10, 2006 - the beginnings go back to 1934. Under the name Spoorslag '70 , a clock timetable was introduced in 1970/71 with a symmetry time at minutes 16 and 46, which differed greatly from current practice on other European railways.

With the 2007 annual timetable, the Nederlandse Spoorwegen introduced a symmetry time to the minute 00/30. This regulation also exists in the Belgian Knooppuntdienstregeling (intersection timetable).

The Czech Railways operate on routes with a regular schedule, e.g. B. Prague – Cheb , usually the agreed minute of symmetry in German-speaking countries. In contrast, the symmetry time in some other European countries is exactly on the hour.

Take Mettmann, for example

S8 (red) and S28 (black) of the VRR between Mettmann and Düsseldorf. Right further to Mönchengladbach (S8) and Kaarst (S28), left to Wuppertal (S8)

Take a look at the illustrated timetable for the KBS 450.28 railway line between Mettmann-Stadtwald and Düsseldorf-Bilk, on which the S28 runs from Mettmann to Kaarst. From Düsseldorf-Gerresheim, the S8 from Wuppertal to Mönchengladbach also runs on this route under KBS 450.8. The plan shows the following:

• Both lines run every 20 '.
• The S8 meets itself at 10, 30 and 50 in Düsseldorf Hbf; further, not shown encounters must take place at 0, 20 and 40 in order to arrive at six symmetry nodes. These encounters can also be guessed from the diagram. An obvious symmetry minute is therefore 0.
• The S28 meets itself at 10, 30 and 50 in Bilk. The meetings between Gerresheim and Flingern do not take place as expected at 0, 20 and 40, but rather ½ ′ later. The reason is as follows: A punctual meeting would take place according to the plan in Gerresheim. There, however, the route for the S28 can only be used on a single track. This also affects the encounter between Neanderthal and Erkrath, which also does not take place on time.
• In Bilk, where both lines have the same symmetry minute 0, the transfer times are the same for both directions: 6 ′ for Mettmann – Mönchengladbach and 14 ′ for Wuppertal – Kaarst. In Gerresheim, where the symmetry minutes no longer match, the transfer times for different directions are no longer the same, e.g. 18 ′ from Mettmann to Wuppertal, but only 1 ′ from Wuppertal to Mettmann.

Connections using the example of Bielefeld

NWB 75577	KBS 402	NWB 75564	NWB 75425	KBS 403	NWB 75424
9:55	Bielefeld Central Station	9:51	9:39	Bielefeld Central Station	10:16
10:04	Brackwede	9:46	9:44	Brackwede	10:11
10:06	Copper heather	9:42	9:48	Senne	10:04
10:13	source	9:40	9:51	Diaper bleach	10:03
10:16	Steinhagen	9:36	9:55	Sennestadt	10:00
Hourly, course book excerpt 2012

Graphic timetable of trains NWB NWB 75577 and 75564 (red) and NWB NWB 75425 and 75424 (Blue)

Take a look at the single-track branch lines Paderborn – Bielefeld and Osnabrück – Bielefeld . At Bielefeld -Brackwede train station, the connections cannot be reached due to the short distance to the crossing point at Sennestadt. There are waiting times of just under an hour. This is exactly what can be read from the pictorial timetable on the right:

• The two lines have different minutes of symmetry. For the line running via Sennestadt the symmetry minute is 57½, for the line running via Steinhagen it is 53. This means that the transfer times in the different directions are not the same; This can be seen clearly in the pictorial timetable.
• Both pairs of trains intersect in the diagram just above Brackwede. Passengers transferring there from NWB 75564 Sennestadt (first pair of lines) and from NWB 75424 to Steinhagen (second pair of lines) have to wait almost an hour for the next train, as NWB 75425 and NWB 75577 have just left. The connection from Bielefeld would be good; Unfortunately, both trains start there, so that no passengers from Bielefeld will change in Brackwede.

Long waiting times at a transfer hub can result in connection losses at subsequent hub stations. To improve connection times, the points of symmetry of individual lines can be shifted, but this is usually not possible with single-track lines. Another option is to take into account the flow of passengers at peak times. By shifting the clocks in morning and afternoon traffic, connection times for the more heavily used direction can be improved at the expense of the weaker traffic flow, but this requires a deviation from fixed symmetry minutes.

Different systems

The “jumping” minute of symmetry

On single-track routes, trains often arrive at the crossing station a few minutes later. It can also happen that a route section is traveled faster in one direction than in the other. As a result, only one part of the route has the usual symmetry time, while the other part has one that is shifted by several minutes.

Major deviations

There are major deviations on individual railway lines, for example the Heidekrautbahn near Berlin, the Oberwesterwaldbahn from Limburg to Au (Sieg), the Rhönbahn between Fulda and Gersfeld or the Bavarian Oberlandbahn .

The S1 line of the Salzburg S-Bahn has a symmetry minute of 52½. The consequence of this is that the transfer times at Salzburg main station to other lines are around 13 minutes longer in one direction than in the opposite direction. The arrival time of the line is 45 minutes (or 15 minutes) and thus corresponds to the usual arrival times at junction stations, the departure time is the same as the full hour aimed at with a symmetry minute 58½ (this, however, creates tight connections to some IC / ÖEC / local transport lines). Overall, this larger deviation from the usual symmetry time has only minor effects - for example, the S3 is missed in one direction.

In the entire Hamburg rapid transit network (S-Bahn, U-Bahn and partly AKN railway) the symmetry minute is, with few exceptions, a few seconds after minute 31.

With the Regio-S-Bahn Bremen / Lower Saxony , line RS2 deviates from the usual symmetry minute, especially north of Bremen Hbf: Trains in the north reach Bremen Hbf at: 27 and continue at: 40, trains in the south leave the main station at: 30 and leave it at: 33. The symmetry minute of the southern section is thus 00 and 30, while it is 04 and 34 for the northern section (taking the other stations into account). The RS1 drives in and out of the route section also used by the RS2, for example in both directions 6 minutes before the RS2 in the main station, so that a change from the RS1 to the RS2 is possible regardless of the direction, but not on the way back.

Minor deviations: Munich S-Bahn

On the main line of the Munich S-Bahn , the symmetry time is two minutes later than usual, i.e. half a minute after : 00 . The following simplified representation is a "snapshot" of such a point in time, due to the 20-minute cycle, it is also valid for minutes 20.5 and 40.5.

S3 - (S2) - S4 - S1 - S8 - (S3) - S2 - S6 - S7 - (S8) (in brackets amplifier trains)

From left to right, i.e. from west to east: the trains of the S3 meet at the Donnersbergerbrücke, at the Hackerbrücke at the same time (during rush hour) the intersection of the compressor trains of the S2, the trains of the S4 meet between the main station and Karlsplatz. At the same time, the S1 trains meet at Marienplatz and the S8 at Isartor.

This applies accordingly to the other five routes, each offset by ten minutes.

Urban transport

Arrival one way	Departure in the opposite direction
Min: sec
0:00	0:00
0:20	9:40
0:40	9:20
1:00	9:00
1:20	8:40
1:40	8:20
2:00	8:00
...	...

Inner-city public transport networks also often have a uniform symmetry time. This is especially true for night networks and modern city ​​bus systems in smaller cities.

All underground lines in Munich (apart from a few exceptions during rush hour traffic, as well as the U4 and U6 in general) and most tram lines have a uniform minute of symmetry. With the timetable change on December 15, 2013, this was changed to zero symmetry. This results in encounters at the minute 0:00 (20-minute intervals) or 5:00 (10-minute intervals), which can be observed in the simultaneous departure from certain stops in both directions: for example Heimeranplatz and Max-Weber -Space on the U5, Columbusplatz and Hauptbahnhof on the U1 (except in the morning rush hour), Fraunhoferstraße on the U2, Implerstraße on the U3, Scheidplatz simultaneously on the U2 and U3. Analogous to the hourly cycle, the table opposite results for the 10-minute cycle in Munich. The sum is 0 or 10 in each row; Rounding off the displayed timetables can - only seemingly - result in a total of 9.

Other cities such as Braunschweig , Chemnitz , Halle (Saale) , Hanover , Linz (Danube), on the other hand, are oriented towards rail traffic in this regard and consequently have the symmetry time at 3:30 and 8:30, although there are not so differences here due to the denser cycle very important.

On inner-city tram and bus networks, for reasons of optimization, there are more and more circulation networks , even with more than two lines, e.g. B. in Kassel and Nuremberg . This is facilitated by the technology of the matrix display of the line and destination, since the previously necessary time-consuming changing of train route signs is no longer necessary.

Remarks

1. ↑ For the U- and S-Bahn, the largest cycle time is usually 20 minutes and the symmetry minute shortly after minute x1. The 60-minute cycle on the S3 is followed by a symmetry minute shortly after minute 01 or just after minute 31. The AKN runs on its rapid transit lines A1 and A2 on Sundays only every 40 minutes, here the symmetry minutes are shortly after minutes 11 , 31 and 51. Only the symmetry minute 31 is compatible with all of the lines mentioned.

References

1. ↑ ^{a b c} Samuel Stähli: "On the development of the clock timetable in Switzerland", in "Tasks of the railway in a future transport system", University of Natural Resources and Life Sciences, Vienna , 1988.
2. ^ Felix Thoma: The integral cycle timetable , 2013. Here: page 7.
3. ↑ Departure board "Inzersdorf Personenbahnhof" of the Wiener Lokalbahnen .
4. ↑ Departure board “Spisertor” of the Trogenerbahn .
5. ↑ [1] Departure board Lekarna to Jablonec and Liberec.
6. ↑ ^{a b} Research Society for Roads and Transport (ed.): Leaflet on the integral cycle timetable . Definition, boundary conditions, possible uses and limits of use in long-distance, regional and local transport. Cologne 2001, p. 21, 32 .
7. ↑ Michael Kümmling, Peter Großmann, Karl Nachtigall, Jens Opitz, Reyk Weiß: "The State-of-the-art Realization of Automatic Railway Timetable Computation", in Proceedings of the 3rd International Conference on Models and Technologies for Intelligent Transport Systems 2013, Dresden
8. ↑ J.-P. Berthouzoz, H. Meiner, S. Stähli: “Timed timetable Switzerland - a new passenger train concept”, 1972, Swiss Federal Railways, Bern.
9. ↑ SMA: Netzgrafik 2014-1 , accessed on May 22, 2014.
10. ↑ Openstreetmap , accessed on October 3, 2012.
11. ^ Online course book , accessed on October 2, 2012.
12. ^ Online course book , accessed on October 2, 2012
13. ^ Online course book , accessed on October 2, 2012.
14. ^ Online course book , accessed on October 2, 2012.

Web links

• Student thesis by Jan Zbikowski on the creation of timetables with explanations on the symmetry minute (PDF file; 446 kB)

Examples: Departure boards with departures according to the symmetry minute shown at the beginning

• Current departure board of the Frauenfeld-Wil-Bahn for Matzingen
• Crossing on the Pinzgau Railway to the minute: 58
• ICE departure in Hanau in both directions at the minute: 29
• Express train in Janoshalma, departure in both directions just before the hour