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I've been looking into rollercoaster track designs and wondered how they mathematically calculate the ideal curves to fit some specific layout without creating excessive Gs or other rider discomfort or for maximising speeds.

One thing I do know is they don't use perfect arcs, due to G forces especially on the loop-the-loop which is why they take on the tear-drop shape (do normal curves also there for not take perfect curves for the same reason?).

My current guess is they must be using some bezier type spline stuff, but there are many ways to design a bezier curve between two points. So what are they actually using to find optimal curves for rider safety / comfort and speed?

Are there special types of curves they use or do they shape the curves manually in 3D software then run simulations to see if its rideable?

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  • $\begingroup$ How were the first ones done before 3D? $\endgroup$ – Solar Mike Jun 5 at 5:23
  • $\begingroup$ There is a lot of ways to design a curve between two points. But you can eliminate the choices which violate the constraints like max g, max curvature, max speed, min speed, self intersecting etc. The remaining choices will be a much smaller set. $\endgroup$ – AJN Jun 5 at 7:18
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Although, I've never had the opportunity to work in the design of a rollercoaster, I suspect the idea is that there is a general idea of the rollercoaster, and then every section is checked that it does not exceed the maximum acceleration on the participants.

The main equation that you need to consider is the equation for the centrifugal/centripetal acceleration at a specific point on the roller coaster.

$$a_c = \frac{v^2}{R}$$

where:

  • $a_c$ is the acceleration of the body (this is usually set to a g value as a maximum e.g. 3g)
  • $v$ is the velocity at point on the rollercoaster
  • $R$ is the Radius at point on the rollercoaster

So at point P, in the following image:

enter image description here

your options are the following:

  • either for a given curve, fit the largest possible circle which is tangent to the trajectory of the rollercoaster, and then estimate the maximum velocity, using $$v_{crit}= v_{max} = \sqrt{aR}$$
  • Given a velocity profile, you can adjust the radii of curvature at each point to be greater than the critical value

$$R_{crit}= R_{min} = \frac{v^2}{a}$$

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