# Roller Coaster Topology

The motivation for this exercise was an attempt to classify roller coaster layouts in some systematic way. Knoebel's Phoenix and the Roller Coaster at Lagoon Park can both be described as double out-and-back coasters, for example, but their layouts are quite distinct. The system described here appears to work better than several more rigorously topological approaches.

At the outset, I should mention that this system is useful only for a subset of roller coasters. Coasters which have two ends, and in which the turns at each end can be described as being about a common axis or pole, can be classified using this system. Fortunately, this includes a large number of coasters, including most compact coasters. However, the system does not work for those double out-and-back coasters such as the Grizzly at PKD (below) in which the two turns at one end are not around a common pole.

Obviously, this system will not work for free-form twisters such as the Wildcat at Hersheypark. Although it can be applied to some wild mouse coasters, it does not appear to be particularly useful for this purpose.

## The System (Basic Description)

We consider the layout (plan) of the coaster, with poles at each end. The actual order of crossing at the intersections is ignored. Three simple examples are shown below:

We will represent the turns at the ends by R (right turn) or L (left turn). The runs between the poles which cross from one side to the other are represented by X, and those that do not cross by O. The first layout above can be described as OROR or OLOL (depending on the direction of travel). The second layout can be represented as XLXR or XRXL. The third layout is equivalent to the first. The runs between the poles begin and end on the same side, and the turns are taken in the same direction as in the first layout.

We note that the descriptions for turns and the legs between the turns are redundant. RO must always be followed by R, and RX must always be followed by L. For this reason, I will use an abbreviated form which describes only the legs, omitting the directions of the turns. Using this more compact notation, the first and third layouts above would be described as OO, while the layout in the middle is XX.

Using this representation, mirror image layouts are not distinguished from each other. OO and XX are the only possibilities for simple layouts with two legs. The number of crossed legs X must be even, and the number of uncrossed legs O must also be even. (Zero is, of course, even.)

For single out-and-back coasters, this is fairly uninteresting, since the layouts are equivalent either to an oval or to a simple figure eight. If we consider double out and back layouts (those having four legs), we find the eight possibilities listed below:

OOOO
OOXX    OXXO    XXOO    XOOX
OXOX    XOXO
XXXX

Patterns listed on each line are permutations of each other, but we find it useful to distinguish these by beginning the description with the leg containing the lift, and by proceeding in the direction of travel. The station is often in the same leg as the lift (the first), but if there is a turn between the station and the lift (as in many compact coasters), the station will be found in the last leg.

Here are examples of wooden coasters exhibiting three of these patterns:

OOOO       Roller Coaster, Lagoon Park (double out and back, all left turns)

The turn between the lift and the station does not precisely coincide with the second turnaround, but it is not much of a stretch to describe this as if it did.

XXOO        Sea Dragon, Wyandot Lake Park (and many other junior wooden coasters)

XXXX       Knoebel's Phoenix, ignoring the tunnel between the station and the lift

One feature of this approach is that it is indifferent to whether multiple turns about one pole are stacked, or whether an upper turn is inside or outside a lower turn.

## A More Complete System

This last example above brings up the problem of additional elements such as helices. In this case, it is simplest to consider the pre-lift tunnel as being an extra turn around one of the poles. If we represent this by / the Phoenix can be described as XXXX/ with a helix after the return leg but before the lift. A helix with an additional turn can be described as //, and so forth.

## Application to Compact Coasters

This formulation is particularly adaptable to many compact coasters because they typically occupy a rectangular space, and the prototypical layout (with stacked layers superimposed) is approximately as shown below:

For three-layer (six-leg) coasters, the number of possibilities (ignoring helices and other features) jumps to 32. Here are some examples among production-model compact coasters:

OOOXX/O       Pinfari RC40
O//OOXXO      Zierer Flitzer
OOXO/X/O      Schwarzkopf Wildcat
OXO/X/OO      Pinfari Z47 & Z64 Zyklons (also the recently lost Gebirgsbahn at Phantasialand)
OXX//O/OO     standard (47 m) SDC Galaxi
XOX/O/OO      small (40 m) SDC Galaxi
XXO/OOO       Z40 Zyklon
XXOX/X/O      RC48/RC50/RC60
OXXXXO        side-friction figure 8 (e.g. Leap the Dips)

If we consider looping coasters, we can represent the segment containing the loop as O* or X*:

OO*XOXO      Schwarzkopf Looping Star
OXO*/XOO       Pinfari ZL42
OX*X//O/OO    Roller Coast Loop, Funtown Pier
OXXO*/XX      Pinfari ZL50
OXX*O/X*/X    Pinfari RC70 (double loop in 3rd segment and corkscrew in 5th)

Note that the Funtown Pier coaster is identical in layout to a standard Galaxi with a loop inserted in the leg containing the first drop.

## More Complications

One difficulty with this approach is how to handle coasters with irregular features that do not quite fit the model. For example, how should we describe a mid-structure turn:  as being a short end or part of a large helix?

As drawn on the left, this could be considered a large helix and notated as OO/.
As drawn on the right, it could be notated instead as OOOO, similar to the Lagoon coaster.
An alternative is to use lower case to indicate the ambiguous status of this turn, and to describe this layout as OOoo. This approach seems preferable because it allows handling the following cases:

That on the left can be described as OOxx, while that on the right is OXxo.
Using this notation, we might describe the Coney Island Cyclone as XXOOoo.

The current method is somewhat more complicated when applied to coasters with helical lifts. In this case, the convention of beginning the description with the leg containing the lift does not work. My preference is to describe these layouts with helical lift as the first element. The Jet Star 2 is then ////OXO/X/OO, and the City Jet is ///XXXX/OO.

The SDC/Zamperla Windstorms can be described as XOXO*/O/O where * represents not a loop, but the steeply-inclined helical turn in the middle of the fourth leg. However, the current method of description does not work for the Pinfari Xpress, illustrated schematically below:

At the end of the leg following the lift, the track rises into an Immelman-like element which is clearly neither a right turn nor a left turn.