I am designing a roller coaster using functions (ie. linear, cubic, logarithmic, trigonometric). At some point, one of the parts of the rollercoaster does not follow a two dimensional graph, but rather extends into three dimensional space.
The rollercoaster is the Steel Dragon 2000, and its third drop extends out into 3D space, as seen in the image below:
As seen, the image involves a turnaround at 210 feet high, followed by another turnaround at a similar height.
Is it possible to use multivariable calculus to find a function that models these two turnarounds? In addition, it is also required that the piecewise functions are differentiable at all points. How would I be able to ensure that the first two heights are differentiable when they connect to the third piece?
Alternatively, can parametric functions for spirals be useful in solving this problem? Or would Bezier curves be able to help model this part of the function?
You can certainly use Bézier curves for this if you don’t mind your rollercoaster track having discontinuities of curvature or torsion.
You just choose some points on the track, decide what direction of travel you want at each point, and then use the points and directions to construct cubic Bézier curves, one between each pair of points.
Discontinuities in curvature or torsion might be a problem, though, because (given that the car is moving at constant speed) they will lead to instantaneous changes in acceleration, and the ride will feel jerky.
I don’t know about rollercoasters, but people who design roads and railway tracks try to avoid jerkiness, so they use smoother curves called clothoids or Euler spirals.