Maybe a super ellipse ? https://procato.com/superellipse/ ( try with same height and width and default n or at least >2 ), i have no idea how one would measure the precise distance then x). I've seen those used in video games that mimick real life architecture because of their "visual appeal".
The new curves pieces have ratio of 5:1 then 4:3 and then 3:4 and 1:5 for to make 1/4 arc circle i counted from the FFF. It think it makes is so that control point are placed on integer value instead of half tile as if a 12:5 curve was used to introduce 2:1 which is what you say ?
and it seem unnecessary given than the exact lengh were given by boskid Yep, as if there are some particular adjustments / programming tricks to makes things works like removing 1/256 or rounding to the 1/256 to match the position system, it may not be accessible for players to see, but boskid gave us the new values for curves
:AFAIK numbers about "20" and "quarter circle" were told by JG which may give you some rough estimates from his playtesting experience. Exact mechanic for rail support distance is slightly more numbers involved but relatively simple:
- elevated-straight-rail has a length of 2 (equals to 2.00 required support distance)
- elevated-half-diagonal-rail has a length of 4.472135955 (equals to 4.48 required support distance)
- elevated-curved-a-rail has a length of 5.132284556 (equals to 5.14 required support distance)
- elevated-curved-b-rail has a length of 5.077891568 (equals to 5.08 required support distance)
- rail-ramp provides support distance of 9. Since rail-ramp is a rail itself, this value is measured from the upper rail connection point. Visually you may see the rail support baked into rail ramp graphics, its center is 2 tiles away from the elevated connection point, so that means when counting support distance from visible baked in support center, it would be 11 tiles support distance (2 is immediately consumed by the rail ramp, only 9 distance is remaining)
- rail-support provides support distance of 11 measuring from the rail support center.
Thank you for clarifying the wrong information i found on reddit and posted there.
Haha, yeah. All depends on your purposes. You could probably make it horribly discontinuous and disassemble trains visually.
I’d personally try cubic splines. Even approximations of much longer arcs are virtually indistinguishable. They’re cheap to evaluate, flexible (to match control points, for instance) and amenable to linear transformation. Speed preservation does suffer more for S-curves, although a single-segment ramp can remain within about ±1%. Not that real trains are so unfazed by inclines. The harder part is analyzing or reversing it (to snap trains on placement or whatever) but the game doesn’t need to concern itself with that in a way that isn’t easy to evaluate numerically when it does come up.boskid wrote: ↑Sun Oct 15, 2023 8:35 pm In 2.0 the curved rails will be defined by a single continuous function where for curved-b-rail the constant velocity assumption is slightly violated to hit the grid point at the 45deg end while the ramp has shape that is defined by 3 segments as there is also a sloped part where the horizontal travel has to slow down slightly due to height change which i did in order to keep a relatively constant travel speed in a 3D space: this is why the ramp length is more than 16.
It already isn’t feasible to compute collisions in curves analytically, and Factorio doesn’t do that. Collisions are a post-hoc check. If you ram trains into each other at high speeds, you’ll find they don’t get as close as they do at low speeds. The train is stopped where it was a tick before the collision. You can then start moving again to snuggle up. Any position function can work, and any sensible position function will have sensible outcomes.
No i can't ....quale wrote: ↑Mon Oct 16, 2023 11:18 am You can construct splines such that their parameterization is almost equal to distance. I actually did that for all curves, using a single cubic curve between the endpoints, preserving C^1 smoothness (both position and velocity are continuous within and between all curves and straights). You can do slightly better than this if you minimize extremes at the cost of the average.
I'm beginning to think it was more of a rethorical "you can" akin to "it is mathematically possible".quale wrote: ↑Mon Oct 16, 2023 11:18 am The endpoints of the cardinal-adjacent curve form a Pythagorean triple. If you allow the endpoint direction to be 5:12 instead of 1:2, you can keep it within ±0.004%, showing how good these are at approximating actual circular arcs.
See my plots. f is the curve itself. g is the local speed multiplier. a and d are endpoints. b and c are directions toward the curve at those endpoints. l is the tunable presumed length of the curve. Adjust it to make g average 1. t is the parameter in [0, 1]. lt is the approximate distance along the curve.
wait what ? My years of playing roller-coaster tells me that vertical G's and horizontal G's are totally different, curve need to be smooth only on the horizontal plane, otherwisequale wrote: ↑Mon Oct 16, 2023 11:18 am The ramp S-curve assumes a rise of 4.5 tiles which should be enough to compensate for Factorio’s weird perspective. Noticeably worse than the rest as a single segment, but I still doubt anyone would notice. Compare to the visual stretching induced by that same weird perspective, or the fact that a train maintains an exact length measured down a virtual line in the middle, disregarding more realistic bogie and coupling physics. But of course you can split the curve and do much better.
Certainly not 1.1. Maybe 2.0. I was going from first principles anyway, designing new pieces that fit the control points but don’t have to match an existing curve exactly. I figured the speed constraint is more important than a near-perfect static arc given that it’s used to position trains. The difference is very minor for arcs of ≤45° anyway, and then the control points themselves impose imperfect arcs. The power of splines is that you can match anything that’s smooth very closely. Just use shorter segments for a drastic improvement, by all measures.mmmPI wrote: ↑Sat Oct 21, 2023 2:29 am I can see how the parameter l can be tweaked and the curve changes shape so that its lengh matches the change in l in direction and magnitude at the cost of more squareness for the 1/4 circle but still try to stay similar to in game curves rails shape. It looks cool but it is not how it's done in factorio AFAIK now
The same curiosity that led to this post made me learn about it. I don’t know how often this is used. It’s just math you can apply to anything. I know the SCS trucking games define roads using splines of a similar form. AI traffic is on rails, following the parameterization directly. Theoretical speed changes are dwarfed by the lateral offset. X-Plane started using fully general cubic Bézier curves to define roads in version 10. Its road traffic is all weird because world generation forwent a speed constraint. Thankfully, it’s only decor in a flight sim.
Factorio trains seem to take ramps effortlessly. All about playability. If you did take such considerations into account, don’t forget that the optimal curve depends on overall traversal speed and train length. Carriages can’t follow their own ballistic trajectory, being forced to travel at the same speed as the rest. A longer train needs a smaller top–bottom difference because any one carriage doesn’t change speed as much on the ramp itself. A faster train needs a smaller top–bottom difference because the speed change is again smaller. Imagine a vehicle nearly coming to a stop on top. The ideal curve would be gentle at the bottom and very sharp at the top. That would be a launch ramp to a faster vehicle. If you’re designing horizontal curves, superelevation likewise depends on speed. The uniform shape I went with is good at the limit of high speed and/or a long train, without being horrible in other cases. And again, train momenta don’t seem to be affected in Factorio, so that symmetry is actually optimal.mmmPI wrote: ↑Sat Oct 21, 2023 2:29 amwait what ? My years of playing roller-coaster tells me that vertical G's and horizontal G's are totally different, curve need to be smooth only on the horizontal plane, otherwisequale wrote: ↑Mon Oct 16, 2023 11:18 am The ramp S-curve assumes a rise of 4.5 tiles which should be enough to compensate for Factorio’s weird perspective. Noticeably worse than the rest as a single segment, but I still doubt anyone would notice. Compare to the visual stretching induced by that same weird perspective, or the fact that a train maintains an exact length measured down a virtual line in the middle, disregarding more realistic bogie and coupling physics. But of course you can split the curve and do much better.the passengerscargo gets smashed outward due to centrifugal forces. Whereas for the ramp S curve, trains using their max power will inevitably be slower when going uphill so the bottom part of the ramp need to be smoother than the upper part ! But that's not very scientific way of drawing curves.
You sound like you have a better understanding of splines than me who only ever heard of them as a math tool like a hammer and chisel before. Then, your plot and formulas felt like seeing a whole piece of functionnal furniture and then watching the video felt like seeing all the different way to stick pieces of wood together. That doesn't make me a carpenter i'm still unable to piece that recently acquired knowledge together in a functionnal way but that definitly sparked my curiosity and i realized now i've been using splines without knowing i was using splines when doing curvy path with moving control point in image editing software. I feel they are everywhere now that i can recognize some in games i seen them heavily used in City Skyline where you can make "freeform" road, and there are tiling appearing on each side, pictures or videos from the game are mostly showing players draw those.quale wrote: ↑Tue Oct 24, 2023 5:43 pm Certainly not 1.1. Maybe 2.0. I was going from first principles anyway, designing new pieces that fit the control points but don’t have to match an existing curve exactly. I figured the speed constraint is more important than a near-perfect static arc given that it’s used to position trains. The difference is very minor for arcs of ≤45° anyway, and then the control points themselves impose imperfect arcs. The power of splines is that you can match anything that’s smooth very closely. Just use shorter segments for a drastic improvement, by all measures.
The same curiosity that led to this post made me learn about it. I don’t know how often this is used. It’s just math you can apply to anything. I know the SCS trucking games define roads using splines of a similar form. AI traffic is on rails, following the parameterization directly. Theoretical speed changes are dwarfed by the lateral offset. X-Plane started using fully general cubic Bézier curves to define roads in version 10. Its road traffic is all weird because world generation forwent a speed constraint. Thankfully, it’s only decor in a flight sim.
Factorio trains seem to take ramps effortlessly. All about playability. If you did take such considerations into account, don’t forget that the optimal curve depends on overall traversal speed and train length. Carriages can’t follow their own ballistic trajectory, being forced to travel at the same speed as the rest. A longer train needs a smaller top–bottom difference because any one carriage doesn’t change speed as much on the ramp itself. A faster train needs a smaller top–bottom difference because the speed change is again smaller. Imagine a vehicle nearly coming to a stop on top. The ideal curve would be gentle at the bottom and very sharp at the top. That would be a launch ramp to a faster vehicle. If you’re designing horizontal curves, superelevation likewise depends on speed. The uniform shape I went with is good at the limit of high speed and/or a long train, without being horrible in other cases. And again, train momenta don’t seem to be affected in Factorio, so that symmetry is actually optimal.
Funny you should mention RCT. That isn’t actually super realistic itself, which isn’t to say you don’t go through the same kind of design process. Accelerations are proportional to speed in the game, as opposed to speed squared. I suppose that makes the game more tolerant of extremes, which can be helpful even if your designs are exquisite, given the small assortment of fixed track pieces. I don’t know if that was deliberate or just a lucky outcome. It also doesn’t even apply gravity consistently; there are some arrangements (I think involving a half corkscrew and a half loop) that add net energy to the train. I learned about the latter part through Marcel Vos.mmmPI wrote: ↑Wed Oct 25, 2023 9:46 am Vertically that makes people's ass fly off their seat which is fun or would ruin a freight train infrastructure as you explain, the downward-to-horizontal curve having the potential for positive vertical G's, where people get the feeling of weighting a ton, being slammed into their siege stronger when the train goes fast/when the curve is changing angle fast. This is simulated on openRCT2, which is also an isometric tiled based game. This is a very significant part of the gameplay, re-shaping part of the deisgn to change how forces are felt given speed.
There is no closed form for length. What matters for Factorio, though, is that you can set a length and it will behave as such. That length is defining. The position function is basically just flair. It will work fine as long as that function isn’t gravely off, such as by causing adjacent carriages to collide. Signal-less operation is likewise defined by those lengths that have been set. Long-term integration error is tiny in binary64 and affects different trains similarly. The visual requirements are much more stringent than that, yet are also easily met.mmmPI wrote: ↑Wed Oct 25, 2023 9:46 am I know there exist some "serious" roller coaster simulation out there not marketed as game, if at all it'd be like architect software-lite compared to the sims , where physics is the focus, i can't tell if they use splines or if at some points physics and material resistance and train lengh and weight and all the other considerations you are mentionning are forcing the use of other formulas to account for the real world physic in drawing the curves. [Quote cut to save space. You can read it right above.]
Ironically, one of the places you won't find splines is real-world roads and railways (I know nothing about rollercoasters). Real road and rail curves are circular arcs with (approximately) Euler spiral transitions.
I see you are a connoisseur of the fine arts x) I'm tempted to think that a game from 1999 or so decided against using the square of acceleration deliberatly for perfomance reasons. I know factorio also does take some liberty with physics for air_resistance that doesn't apply to braking trains. I had heard of that corkscrew magic too, but to my memory it was a trick relying on the fact that you could make a loop with pieces whose lengh didn't added up, something like you were going more "down" than "up" as an Escher painting. I've heard of the calculator Marcel Vos made in RCT2 and was able to find the video but it's unclear if that is gravity of lengh or position function that is 'problematic'.
I did that one quite some time ago without knowing if it would work for long time :quale wrote: ↑Wed Oct 25, 2023 2:11 pm There is no closed form for length. What matters for Factorio, though, is that you can set a length and it will behave as such. That length is defining. The position function is basically just flair. It will work fine as long as that function isn’t gravely off, such as by causing adjacent carriages to collide. Signal-less operation is likewise defined by those lengths that have been set. Long-term integration error is tiny in binary64 and affects different trains similarly. The visual requirements are much more stringent than that, yet are also easily met.
I've heard of No Limits but i think it looked like this FVD++ (Force Vector Design) thing : https://www.youtube.com/watch?v=o_cOiYa-0qM Where you can draw pieces of track from math functions as they say in first minutes of the video. Those track can then be imported in No Limits.quale wrote: ↑Wed Oct 25, 2023 2:11 pm Years ago I toyed a little with No Limits. I think it also used splines. I’d think that it would be much more stringent for that type of simulation, and it can afford to be. You can always subdivide a spline to get a more accurate length estimation or whatever you need. The smoothness can only be an asset to higher-order physics. The simulation doesn’t have to run hundreds of trains alongside a gigantic factory. If there are more steps to making it work the way it needs to, it can just take them, not to mention that there will be such steps regardless of the track’s representation at the level where I expect it to be.
Well nobody's perfect , rollercoasters i found are the main video game analog for adding the effect of gravity to trains in a fun way
They make up for fun curves like the Brachistochrone !https://www.ams.org/journals/mcom/1992- ... 4736-8.pdfAbstract. Highway and railway designers use clothoid splines (planar G1
curves consisting of straight line segments, circular arcs, and clothoid segments)
as center lines in route location. This paper considers the problem of finding a
clothoid spline transition spiral which joins two given points and matches given
curvatures and unit tangents at the two points. Conditions are given for the ex-
istence and uniqueness of the clothoid spline transition spirals, and algorithms
for finding them are outlined
They may perhaps be 'splines' in some generic sense of the term, but they are certainly not Bezier curves or anything closely related to them (e.g. Catmull-Rom), which is what is normally meant by the term in the context of computer graphics (and the way it has been used in this thread). The latter are polynomial curves (typically cubic), while the Euler/clothoid spiral involves transcendental functions (specifically, Fresnel integrals).
