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If I remember my uni engineering/calculus maths class correctly, the third derivative of position is used in planning these sort of curves.
The first derivative of postion (with respect to time) is velocity. The second derivative is acceleration (ie rate of change of velocity). And the third derivative is jerk (rate of change of acceleration).
And 'jerk' has to be kept below a certain threshold for humans to find movement comfortable.
A very similar thing is done in the creation of reflective surfaces on car bodies (typically in CAD software).
They call these constraints by G and a number.
G1 would be a positional constraint: the two surfaces meet each other at the same point
G2 tangential: same as G1, but the surfaces are tangential
G3: same as G2, but the curvature (radius^-1) of the surfaces is the same at the point where the two meet. This essentially means the curvature combs of the surfaces shall meet at the same position (G1)
G4: same as G3, only now the meeting curvature combs have to be tangential as well
G5: same as G4, only now the curvature combs of the curvature combs have to meet at the same position
A simple example of this is the squircle. This page [1] has a couple of nice images that are easy to understand.
https://99percentinvisible.org/article/circling-square-desig...
Minor point (well in this case at least) but you have an off by one error. Your G1 is G0.
Here is how it is defined in terms of basis vectors. https://people.eecs.berkeley.edu/~jfc/cs184f98/lec19/lec19.h...
> G2 tangential: same as G1, but the surfaces are tangential
This makes me think "tangential to what?".
Do you mean that, along the seam between G1 and G2, the tangent plane to G1 at a given point is equal to the tangent plane to G2 at the same point?
Also seen in the planning of curves in roads (where jerk corresponds to the rate at which a steering wheel must be turned) and railways.
And this is also why the passengers jerk of a vehicle jerk backwards after it comes to a complete stop. Their muscles statically counter the relative forwards acceleration of their torsos during braking and require time to react to the acceleration suddenly going away. This effect can be prevented by gradually letting off the brake before reapplying it fully upon stopping, but few drivers and rapid transit systems seem to be aware.
> but few drivers or rapid transit systems seem to be aware
I find that amazing. What the heck are drivers ed instructors doing? It's not just hard on the passengers, it's hard on the machinery.
It's the same with the clutch. I've driven with enough people who fancy themselves as great shifters, but they jerk the hell out of the clutch every time, never attempting to match the shaft speed with the engine speed. If I comment on it, they always deny doing that :-/
If I'm on my game, I can shift smoother than an automatic. The bonus is the clutch will last a very long time.
This effect can be prevented by gradually letting off the brake before reapplying it fully upon stopping, but few drivers and rapid transit systems are aware.
This is surprising to read. Everyone whose car I've ridden in knows to do that, and it's only in extremely urgent and unexpected stops where it's neglected. Also, when fully stopped, only minimal pressure should be necessary to keep the car still.
It's a little more complicated than that in a passenger car. The deceleration compresses the front springs. When the car comes to a stop, the springs decompress and the front of the car pops up and the body of the car moves slightly backwards even though the wheels are now stationary.
>Also seen in the planning of curves in roads (where jerk corresponds to the rate at which a steering wheel must be turned) and railways.
Just for the record, the transition curve is usually (not always but very often ) a clothoid (or Euler's spiral or Cornu spiral)
This was particularly noticeable to me riding San Francisco light rail to work. When trains run underground they start and stop under computer control. Nice, smooth acceleration and deceleration.
On the surface (or when the computer control system was borked) the starts and stops were a lot less pleasant.
This is fun to try in cars. The drivetrain can have a bit of twist under deceleration, and you can feel it spring back after the wheels stop. For best comfort you need to gradually reduce the braking force not just for human comfort but also to relieve that twist.
I have a t-shirt which has "don't be a" and the equation for the third derivative
And 'jerk' has to be kept below a certain threshold for humans to find movement comfortable.
It's not strictly a matter of threshold -- people might tolerate a higher jerk if it's for a much shorter duration, for example. In practice it doesn't much matter which metric you minimize; you'll end up with similar results. The simplest option is to minimize the mean absolute jerk, which has the side benefit of utterly confusing any non-physics-literate people listening in. (You want to do what to whom?)
Jounce, crackle and pop for the 4th, 5th and sixth derivatives
This video has an excellent visual demo of that concept: https://youtu.be/aVwxzDHniEw?t=451.
Skateboarders in 2008 don't get this. https://www.youtube.com/watch?v=TkeCZfG_KaI
This seems to be the source material:
https://www.datagenetics.com/blog/march42014/index.html
Vox re-heating a gizmodo article[0] which re-publishes (with permission) the one above.
[0]: https://gizmodo.com/why-roller-coaster-loops-are-never-circu...
And in fact Data Genetics is s great blog with loads of interesting content on maths, statistics and simulation.
One that I've used as inspiration for a programming class I was teaching is his analysis of Snakes and Ladders: https://datagenetics.com/blog/november12011/index.html
Note that the narrow-loop shape in modern roller-coasters means that the tightest curve is at the top, where the G-forces are partially countered by gravity. That's an 1-G that they can subtract, and it means the radius can be significantly larger on entry to the loop, resulting in smaller G-forces.
This free 1g of turning force is apparently known among pilots as “God’s G”. It needn’t be produced as lift while you’re inverted at the top of a loop, so you get a good turn rate despite losing speed and hence lift as you get up there.
Not just gravity, but the train also loses speed at the top, so the loop needs to tighten if you want to maintain the same force (you might not want this).
Also by the time you get to the apex it’s going slower than when it entered the loop.
Certain modern(-ish) roller coasters do have more circular loops than others. Specifically Schwarzkopf[1] coasters are famous for having more circular loops (and the more intense positive Gs that come with it). Anyone in the Bay Area might remember Zonga[2] at Six Flags Discovery Kingdom which featured the more circular loops.
Also maybe of interest is Blue Flash [3], a backyard roller coaster that has a loop that reminds me of old school circular loops.
[2] https://en.wikipedia.org/wiki/Tsunami_(roller_coaster)
[3] https://www.atlasobscura.com/places/blue-flash-backyard-roll...
Honorable mention for the ill-advised looping slide at action park[0]
> More people paid to watch others ride these early coasters rather than ride themselves
I had no idea roller coasters have been around this long. The photos are laugh-out-loud terrifying. I was shocked that anyone would pay to ride them until I read the quote above. Now it makes sense. I’d pay to watch that too!
One of the most mind-blowing images on Wikipedia is this one of the first ever Ferris Wheel from the Chicago World's Fair in 1893:
https://upload.wikimedia.org/wikipedia/commons/6/68/Chicago-...
Note how large the cars relative to the tiny people standing in them. This thing was unimaginably massive. It's easy to think of 1893 as being before the modern technological era, but we were more modern than most people like to think.
Having first read your comment, the photo wasn't as bad as I expected. The supports don't look too unreasonable for a single wooden car with two passengers. Modern trains are an order of magnitude heavier--~20 people and a gross weight of ~20 tonnes.
A video of the coaster in the photo exists (the playback framerate seems somewhat too fast): https://commons.wikimedia.org/wiki/File:Flip_Flap_Railway_ea...
Modern tubular steel rails also allow for rickety-looking yet safe single-car coasters: https://commons.wikimedia.org/wiki/File:Rat%C3%B3n_Vacil%C3%...
In that era, there weren't even seatbelts on cars.
We used to have a much different view on systematic rare danger.
Thus is similar to corners in roads not working well when they're circular arcs. Going from straight to circular arc means your steering wheel needs to jump instantly from one fixed position to another. Using something like a linear change in curvature (i.e. a clothoid curve) is much smoother. This kind of thinking applies to other areas too: https://en.wikipedia.org/wiki/Euler_spiral
for exactly the same reasons the corners of apple products aren't just an arc. They change the tangent smoothly over time.
Curiously Dubins paths do have instantaneous steering changes. https://en.wikipedia.org/wiki/Dubins_path
I think never having considered this type of thing is why I could never make a sweet roller coaster tycoon coaster that didn't make everyone throw up.
A "G force spikes" is called a "jerk" in physics; as acceleration is the rate of change of speed, jerk is the rate of change of acceleration. The jerk is huge in transitioning from a straight track to a circular one immediately, since the acceleration goes from zero to nonzero instantly.
If I recall correctly the list of derivatives and derivatives of derivatives goes like this: position, velocity, acceleration, jerk, snap, crackle, pop.
A similar principle works in UI design and industrial design. Rounded rectangles often just slap circle arcs on the corners—but that looks unnatural because you don't get such a shape if you bend a straight rod. In this approach the turning radius changes from infinity to a constant at one point, but the proper way is to change it gradually. Bezier curves accomplish that, I think.
Once you see this, you begin to notice it everywhere, just like with kerning. Apple used both ways on iPhones for rounding the phone's corners, iirc. Also I've been told that the principle applies to road turns: you don't want people to have to suddenly turn into the curve. (In related news, I wish a month of bad hiccups on Herman Tilke.)
I once implemented rounded rectangles by drawing circles with regards to arbitrary p-norms [1,2] – I did not think about it at the time, but the derivatives are probably quite nice. (It did have the advantage of using the same code for rounded rectangles and for circles.)
[1] https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm
[2] https://en.wikipedia.org/wiki/Lp_space#/media/File:Vector-p-...
Apple went from using roundrects to squircles a decade ago:
https://99percentinvisible.org/article/circling-square-desig...
I really want to like Vox's video format but just can't. Which is a shame because they have lots of topics that are quite interesting.
I can't pin down which part that I hate most, the "stylish" presentation with random SFX and virtual pen circling around, or the fact they always insert some super low quality video conference footage instead of just letting the narrator paraphrasing (I get it they're the domain experts, but still..).
I couldn't watch it, I guess it felt messy and there was too much stuff going on, so it was hard to focus.
The book Curves for the Mathematically Curious ( https://press.princeton.edu/books/hardcover/9780691180052/cu... ) has a whole chapter about the shape used instead of a circle, complete with equations and derivation. Highly recommended
Am I high or is the picture of the flip flap railroad loop decidedly not circular?
You’re high. But also the photo was taken at an angle to the track, so it’s not gonna trace out a circle on your screen.
I remember the Corkscrew at Knotts as a kid, no longer exists. The Revolution at Magic Mountain arrived in the late 70s with the new parabolic shape and still running. (Southern California)
It does actually. It was moved to Silverwood in Idaho and has operated there very since.
> G-force
I looked up this term to be sure and I'm convinced it's as meaningless as I thought it was and is a strange way of saying “force” or in this case a centrifugal force.
G-force is just measuring acceleration in units of G. It's a commonly used measurement because you've got an intuitive feeling for what one G feels like.
That would be a hard pass from me
14 G's! That would knock you out. There's no way it was that high. I looked it up and it says they tested it on chimpanzees. I'm sure the monkeys were absolutely thrilled.
Could be 14G for a very short period, fist when entering the loop and again when leaving it.