Roller Coasters and Space Curves

Montu, the Egyptian-themed roller coaster at Busch Gardens, Tampa Bay

I always loved roller coasters as a kid. Hell, I love roller coasters now, though I don’t get to ride them very often now that I’m not in Tampa anymore. Busch Gardens is one of my favorite amusement parks. Back when I was in engineering school, one semester, two other friends and myself bought season passes to the park. We would go there during the weekday, when the place was damn near empty, in between classes. It was a real blast.

When we would ride one of the coasters, I was always mesmerized by the mathematics and engineering that went into the design and construction of something like Montu or Kumba. Lately, I’ve been re-familiarizing myself with some of the vector calculus I learned as an undergrad as a way to stay sharp. The roller coaster is probably the most concrete application of a space curve and vector calculus I can think of outside of the Space Program.

The Mathematics of Space Curves

As I discussed in a previous post, it is really difficult to describe things in the physical world with a natural language. It simply isn’t possible to describe how a particular space curve is supposed to look using English, Chinese, or Spanish to convey your idea. The only way to to do it is with mathematics, specifically with some type of curve parameterization, e.g.:

$\bold{r}(t)=[x(t),y(t),z(t)]^T$

Where $t$ is the curve parameter, and x, y, and z are functions describing the curve in 3-space.

By plugging in various values of the parameter $t$, a curve in 3d-space is traced out. This precisely describes the shape of the curve we want. In practice however, it would be much too difficult to try and come up with closed-form functions to describe a shape as complicated as a roller coaster. While I don’t know for sure, I would guess that actual design software uses piecewise cubic splines to interpolate smoothly between points, thus providing a piecewise-smooth numerical function for the roller coaster space curve.

TNB Frames

Instead of writing a large treatise on TNB frames, I direct the reader here to some useful notes on the subject. Briefly, a TNB (tangent, normal, binormal) frame is a very useful local coordinate frame at a specific point on our space curve. From the TNB frame description, we can calculate very important quantities of the curve, such as a maximum curvature and maximum torsion. It also allows us to decompose the acceleration vector, $\bold{a}$, into two natural components.

Safety Constraints on Curve Shape

Obviously, safety is absolute when designing a roller coaster. To the lay person, this usually means just making sure the coaster stays on the tracks – but that betrays much mathematical subtlety behind the shape of the space curve.

The human body is a fragile thing, and the design of a roller coaster requires that the human body never experiences too much force in a given direction too suddenly. Twist too suddenly, and someone snaps their neck. Too high of G-forces, and someone passes out.

But how does one describe “safe” shapes for the curve?

When designing the space curve for a roller coaster, there are presumably constraints on the maximum curvature and maximum torsion of the curve:

$\frac{d\bold{T}}{ds}\cdot\bold{N}\leq\kappa_{max}$

$-\frac{d\bold{B}}{ds}\cdot\bold{N}\leq\tau_{max}$

To prevent high G-forces from injuring people,  the T and N components of the acceleration vector must never exceed certain values (acceleration is always orthogonal to the binormal vector):

$\bold{a} = a_T\bold{T}+a_N\bold{N}$

$a_T \leq a_T^{max}$

$a_N \leq a_N^{max}$

$a_T = \frac{d^2s}{dt^2}$

$a_N = \kappa(\frac{ds}{dt})^2$

And probably also a constraint on total acceleration:

$|\bold{a}|\leq a_{max}$