[hyperpallium]

\tangent another solution is camber or cant, where the outer rail is higher, banking into the curve, as an aeroplane turns.

A problem is the beginning of a curve. If the straight simply becomes an arc of a circle, the lateral centrifugal force (see xkcd) goes instantaneously from 0 to maximum. One might think an Euler spiral, whose rate of turn increases linearly, would fix this; but now, although the lateral force increases linearly, the onset of change in force is instantaneous, and is felt as a jolt by passengers. Apparently it is possible to create turns giving a smooth ride, but requires sophisticated dynamic models that are presently beyond me.

I started making a game with very smoothly transitioning banked turns, but it's a lot more complex than I thought! (Other games use tilting-like mechanisms, such as suspension, and the natural averaging over the four contact points of a car, but that's not applicable to what I want to do). If anyone has some pointers, I'd love to hear them!

P.S. Re trains: Of course, another problem is that banked turns are "static", the same for all trains regardless of velocity; whereas the "dynamic" tilting can adjust to current velocity, on-the-fly (in addition to smoothing out transitions). The rail camber/cant can remain suited to the typical velocity of non-tilting trains using the same track.

[nightcracker]

> the onset of change in force is instantaneous, and is felt as a jolt by passengers

An instantaneous change of force (and thus acceleration) create a jolt, I understand that. But taking the derivative again - 'the onset of change in force' - I don't see how that is felt as a jolt.

[hyperpallium]

I thought so too; but that's what I read, as what happens when Euler curves were used in practice (which they were). Unfortunately, I didn't get to the point of trying it out for myself in my game.

[BTW: Like a bob-sled track, I did find that cross-sections that were sine waves or quadratic did feel strikingly smooth - those being smooth, with smooth deviratives, smooth derivatives of derivatives, and so on.]

If you imagine an x-y graph of force over time for a track that went like (start in a straight, an Euler curve to begin the turn, some time in an circular arc, and a reverse Euler to straighten it, end in a straight), the graph would be (level, slope up, level, slope down, level):

    _
  _/ \_

Apparently, humans are sensitive to the change between levels and linear slopes. Lines are uncommon in nature [though gravity is constant acceleration...?]. Perhaps related to inverse kinematics in limb movement? "Jolt" may be the wrong description.