My Roller Coaster Assignment
Entry posted by JoshC.
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Whilst this entry may seem technically coaster related (and, in a way, is), the core idea about the blog isn't, so I'm giving this blog a bit more love...
For my university course, one module I've had to take is Geometry and Motion, with the key element of the module being to learn about curves. Each week, we're given assignments (which count towards our final grade) which consists of multiple questions. In last week's assignment, the final question was this:
Simply put, we were asked to sketch out a line representing a roller coaster, and give equations for the various segments. This had to be shown in 3D on the paper; basically show where the 'track' goes behind / in front of itself. There was no need for it to make mechanical sense / being realistic in any other way, as the point of the question was to have the roller coaster lines as an equation in terms of time (so that at any given time, it would show the 'coordinates' of where the track would be).
Of course, this doesn't have to be wholly difficult. In theory, the question can be answered by giving the equation of a vertical loop (due to the need for the 'fun feature') and then a circle to join itself back to the beginning (which, in terms of what we have been taught, is straightforward). This wasn't the general case, with many that I spoke to going for the idea of a lift and drop, with a spiral or two, vertical loop and some general straight lines to join it all together. Others went for a much more basic layout, but instead focused on a variety of different 3D sketches.
"But where's the fun in that?" I thought to myself, when I was thinking about the assignment. Being interested in coasters as I am, to be able to do something like this should be great fun. Only having a week to do this, along with the other questions on the assignment and general life, I didn't spend that a lengthy period of time in roller coaster terms (as in, you can spend multiple hours in creating coasters in RCT or No Limits), but I probably spent about 8-9 hours during the week on the question. In terms of how long I would spend on other questions (or even whole assignments!) that is extremely large.
Like with any 'creative' project I would do, I jotted down ideas quickly as they came into my head. Rough layouts, fun features, anything where I thought 'Oh, that would be great', I would jot it down as quickly as possible, maybe even draw quick sketches of what I had in mind. When it came to actually doing it though, it seemed as though I had been hugely over-ambitious, with a lot of ideas I had thought out just being too difficult to achieve in the set time period. Many fun elements I had considered, such as a variety of different elements, 'beyond-vertical' drops and such were more complicated to create equations for than originally anticipated. One thing which was feasible, however, were interlocking loops, inspired by the roller coaster 'Loch Ness Monster' at Busch Gardens Williamsburg.
However, I decided to put my own little twist on it, and instead combine a vertical loop with a helix (which, in this case, was basically a horizontal loop, to have different 'fun elements'). I don't know if a roller coaster(s) in the world features an element similar to that, but I wouldn't be surprised either way.
So then, I had my fun element, which was pretty much the key feature to the ride and the design. Combine that with a lift hill (pretty standard), a vertical drop (because that's so much easier than anything else!), airtime hill / 'bunny hop' and a couple of other bits and bobs to connect it all together and I had my coaster...visualised in my head. A few sketches later and I worked out how exactly it would look, and when the track went in front / behind itself. Now came the fun part - actually working out the equations of the coaster.
Following the hint from the question, I split the 'track' into several different sections, thirteen in fact, to make it easier to create equations. Owing to the need to plot it onto a '3D graph', I started the track at the origin, and went from there. The beginning was relatively straightforward, basically being combined of multiple straight lines / easy curves, and due to the easy numbers, it wasn't difficult to 'stick' the curves together so that they were continuous. The first real challenge came in the creation of the 'loop de loop', as it needed to be of suitable height and width to allow for the helix to easily go through it later on. After lots of thought, and playing about with the Maths program Matlab, I eventually got there.
A screenshot of the 'vertical loop' produced using Matlab, including the equations of the curve and the 3D graph produced.
One thing which may be worth noting here is how the loop is circular. Whenever you look at roller coasters, these loops are by no means circular - all such loop de loops are actually clothoid loops; basically meaning the radius increases as you get further to the top of the loop. Whilst I originally tried to recreate this, it was a bit too difficult to achieve in the time period (the usual reason..)
After a bit more playing about with equations, it got to the stage of the helix. This was much more challenging than expected, due to the need for the track to be carefully positioned before it started to ensure the actual equation for the helix, which was already somewhat scary-looking, was not even more complicated. Generally, it took a lot of hard work and thought procedure, and when I ended up creating it, I realised that the helix was going upwards, not downwards, making it the most mechanically inaccurate section in my opinion. But, it was there, and the mechanical sense was not a necessity, so I decided to leave it. A long straight section follows, which in real terms would probably consist of a twist or something, but minute details such as that could be left out for the sake of the question, as it would have added extra complexity to the equations. Basically, what I just said is that it's a straight line, but pretend it isn't.. 
This was then pretty much the end of the circuit. The numbers were getting more and more complicated, so a quick and simple finish was required really. Basically I wanted to scrap the evil-looking numbers and get it over with. Had I taken more care with the numbers earlier on and thought about the equations' knock on effects to later ones, I could have added in a spiral or something as a 'big finale', but alas, an easy finish it ended up being. So, a few straight lines to act as drops and curves, and we were done.
So then, that's pretty much the roller coaster I designed. This isn't actually the final design, as the submitted one had markings for each section, to correspond to the given equation, and the section between the turnaround on the far right hand side and the helix should only be diagonal line (I drew it incorrectly in this sketch..). Hopefully once I get my assignment back sometime in the next week, I can scan up to finished sketch and the equations for anyone interested.
As you have probably got from this, I had a real feel of designing something which was as mechanically accurate as possible, in the sense that if you were to build this, the train would make its way around the track. I think that the final design would be possible (though maybe with a bit of help from a quick launch mid-way through), so that's a positive. In second year, there is 'Second Year Essay' that needs to be done, which basically is a project around a section of Maths that is of interest to the person. If it is allowed, I would most certainly be interested in extending this idea further, to simply just use geometry and equations to design, or recreate, a roller coaster, which is mechanically accurate, safe to ride, and has equations / expressions to find things such as the speed, force, acceleration and so forth at any given point during the roller coaster.
Any questions or comments would be greatly appreciated! 
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