witchy

1:64 (or, to be precise, I usually scale to 1:62.5 where 1 plate is 20cm/8" and 1 stud is 50cm/20") is a very convenient scale for making people because you have 2 bricks' height for the legs and body, and 2 plates (or more with tall headgear) for the head. The width of such people is roughly 0.5m which is reasonably close to a train seat, so depending on how thick you make the walls of a 6wide coach you can easily get a realistic number of passengers side by side (on the other hand, trying to get 4 minifigs side by side would require the train to be at least 13 studs wide). This isn't a train, but it's another example of what you can do with the scale; a 1830ish Royal Mail stagecoach, with the guard having a bugle and a shotgun. I haven't figured out how exactly to do the horse harness yet (I need to study how actual coach harnesses worked in the period, and the build would probably end up relying on flex hose a lot). The Houses of the World sets are, incidentally, at a scale that matches these trains and people well enough to be compatible on a layout.

These are not true 4wide, using narrow gauge track to represent standard gauge in 1:64 scale (modern rolling stock would be 5-6 wide), but they are 4 studs wide due to being old and small: The Rainhill trials contestants, from left to right: Cycloped, Novelty, Sans Pareil, and Rocket. The gears on Novelty and Sans Pareil are acting as stand-ins for third-party #4 wheels. Stephenson's Rocket and its evolution as an engineering testbed; from its Rainhill trials condition, to getting a more utilitarian paintjob and lowered cylinders, and finally having the steam pipes moved inside the boiler Liverpool & Manchester Railway Northumbrian with a 2nd class consist L&MR Planet with a 1st class consist. This one can actually be motorised with a Studly Trains micromotor and is capable of pulling 2-3 coaches with some tuning (ballasting all open cavities, using higher friction o-rings for traction, lubricating the coaches). The tender is designed to fit a Deltang or similar receiver, and a small battery. And on the American side we have DeWitt Clinton of the Mohawk & Hudson Railroad with Goold cars. The cars have passengers both inside and outside (the inside passengers are actually load-bearing) but it isn't very visible in the renders. I also have a motorised John Bull in the works, with its weird 4-2-0 layout that looks like a 2-4-0 being authentically represented (only the rear axle is powered) but the tender needs a bit more work first. I've built Planet, Rocket and the L&MR coaches, and the rest should also be physically buildable without severely illegal connections (although some are fiddly to assemble) or third-party parts (except for the motor and associated electronics in Planet, which are optional, and the #4 wheels of Novelty and Sans Pareil). I've also generally avoided excessively rare or expensive parts, and where they have been used (such as the medium blue 4081b in the light blue car) an acceptable substitute of a different colour can usually be found.

To be precise, as the length of this track is measured near the inner rail, to have a R50 circle at the centre of the track you need a radius of 47 studs at the hinge position, giving a hinge circumference of 295.16 studs or ~148 segments (round up to 150 segments for convenient round numbers). The pictured curve has a 2 degree angle between each segment, which would give a hinge circumference of 180 segments, so the desired curve is only slightly tighter than pictured (2.4 degrees per segment) and appears viable to build in the manner pictured.

Like this. Cost (in efficient colours) is around £2/2studs, or around £300 for a circle of R50 (80 cm diameter). I've tested the narrow gauge equivalent and it works, at least in large enough radii. In tighter radii it may be necessary to build each rail independently so that the outer rail has more segments, to make the gaps smaller. This is left as an exercise to the reader. The plates with the rail need to have 4 studs and 2 plates between them, otherwise they will interfere with wheel flanges as the regular rails are slightly narrower than the thickess of a plate. The flex hose sections should be lapped so that the joint of one is in the middle of another. The print on the studs of the wedge doesn't seem to foul the snot pieces above because the hole in technic pieces is slightly higher than would be in system, so the rest of the light grey plates could also be replaced by a 2x6.

Would narrow gauge track work? R48 would be quite close to the size you're looking for. Otherwise the most reasonable option is to accept a slightly different size from standard R40 or R56 track. If you like spending a lot of money it is also possible to brick-build rails to the exact specification using 32028 but that is a lot of effort and expense.

Using third-party R56 track would get you a diameter of roughly 90cm. At least Trixbrix and HA Bricks have R56 track available in Europe. Other than that, another option is using straight sections between some of the curves. Alternatively, mixing R40 and R56 curves (specifically, every other piece R40 and every other R56) would net you a diameter very close to what you're looking for. In the latter two options the frequent curvature changes may make the tracks and the trains running on them look a bit weird.

In addition to the calculator and its visualiser, I developed a visual/tactile tool for demonstrating and working out curvatures, which I call "Euler blocks" or "curvature blocks". These are colour-coded blocks built from LEGO, each corresponding to the shortest curved track piece (that conforms to the 5.625 degree division) available for a given radius. The width of the block equals the length of the corresponding track piece, the height of the block equals the curvature, and the number of vertical dashed stripes denotes the number of 5.625 degree segments the block represents To measure the curvature changes of a curve with these blocks, you set them side by side against a straight bottom edge, in whatever configuration you wish. The resulting shape is what the spreadsheet's visualiser would give you. The widths of the blocks are exact, and the heights are accurate to less than 0.25 plates. You can count the number of coloured stripes to calculate the total angle of the curve; 16 stripes for 90 degrees, 8 stripes for 45 degrees, etc. The curve shown in the above image is 72sa2, a curve with the net radius of Rn72, an apex of R40 (tightness value 2), and an error of 0.2 studs compared to a standard R72 curve. Unfortunately the curvature blocks cannot determine the net radius of a given curve, as there is no simple way to translate length to X-axis and Y-axis distances for curves of varying radius, but if you have a curve in front of you that works, you can use the blocks to see how abrupt its curvature changes are without needing a computer. Curvature blocks studio file

Real railroads don't use curves of constant radius from beginning to end because it creates abrupt changes of curvature, leading to theoretically infinite jerk. Instead, track transition curves are preferred, with the Euler spiral being the classic "ideal" example (although technically speaking it is not actually ideal for reasons I won't go into in this post). However, it appears that this practice is almost unheard-of in LEGO trains, with even clubs with high resources building large layouts seemingly relying on constant-radius curves everywhere. I find this dissatisfactory and decided to do something about it. I developed a calculator for laying out custom combinations of curved track in increments of 5.625 degrees (16 segments per 90 degrees) and determining where the resulting curve ends up, with the end goal of producing curves that match the existing standard curves in net radius, even though the individual components have differing radii. This makes it easy to develop and verify custom curves with proper transitions. Curve calculator (.ods format; it might be compatible with Microsoft Excel but if you're having problems, LibreOffice is free, requires no account, and doesn't spy on you) The calculator is operated by inputting the radius of each 5.625 degree segment in the corresponding field F2 to F17. When working with smaller-radius tracks you need to fill in a number of fields that matches the length of the curve piece you're using; for pieces of 11.25 degrees you need to fill in 2 fields (for example, the "56" in F4-F5), and for 22.5 degree pieces fill in 4 sequential fields (for example, the "40" in F6-F9). By default fields F10-F17 will mirror the contents of F9-F2, producing symmetric 90-degree curves. To calculate curves other than 90 degrees, you can override these values with your own input. The "Total" fields I19 and J19 show the total length of the curve on the X-axis and the Y-axis respectively, measured in studs. If the curve is symmetric, these values should be the same. If this value is very close to a standard curve radius (e.g. 72.01) you should be able to substitute the curve specified in F2-F17 for a curve of that radius with no problems. If the value differs from a standard radius it requires an offset, which equals the amount of straight track that needs to be added to the ends of the curve to line it up with the matching standard radius (in the example picture, the curve has a net radius of 64.07 meaning that a 8 studs long straight track at each end lines it up with a standard R72 curve). Below the calculator is the visualiser graph. This plots the length of the used segments on the X-axis, and the curvature (the inverse of the radius; the smaller the radius the higher the curvature i.e. the tighter the curve) on the Y-axis. An ideal Euler spiral would look like a triangle or a trapezoid. As we are dealing with track pieces of specific lengths and radiuses, we always get a stepped graph instead, but if the stepped line on the visualiser is close to a consistent diagonal the curve is a decent approximation of the Euler spiral. As we can see in the example, the graph is pretty close to a trapezoid with a short top, meaning that the curve achieves close to ideal transitions. A standard curve of constant radius looks like a rectangle. If you want to use the calculator for curves that aren't 90 degrees, you can replace the segments you aren't using with segments of the net radius you're aiming for. For example, to calculate a Rn104 curve of 67.5 degrees, you can set F14-F17 at 104 and build your curve in the remaining fields F2-F13. This makes the offsets slightly harder to calculate, but if you manage to get close enough to the target net radius it won't matter. At the right side of the sheet I have recorded a number of curves I've found using the tool, with the example curve being called 72sb2o8. The "72" stands for the net radius, meaning that this curve is what I call an Rn72 curve, i.e. it can substitute a R72 standard curve. The "sb" is the identifier of the curve's family and variant; I have a number of different families of curves with identical or related profiles, each with a letter code. This curve belongs to the S family, which is notable for being close to Euler spirals, and is specifically the variant Sb. Curves with the same identifier have an equal profile, meaning that their segments have the same relation to the net radius. For example, Sb starts with one segment of 7 steps (112 studs) greater radius than the net, then one segment of 1 step (16 studs) greater than Rn, then 2 segments of 1 step smaller, and finally 4 segments of 2 steps smaller than Rn before being mirrored on the other end. A hypothetical curve 88sb2o8 would start with 1x R200, then 1x R104, then 2x R72, then 4x R56 and mirrored, with each segment having a 16 stud larger radius than in 72sb2o8. The "2" stands for the apex tightness; the curve is R40 at its tightest, which is 2 steps smaller than Rn72. If the apex was R56 instead, this number would be 1, and so on. Finally, the "o8" stands for the required straight offset. Adding a straight track piece of 8 studs brings the curve to the net radius value of 72. If the offset code is o1 it means that 1 stud of straight track is required. On some curves the offset is denoted with an X instead of an O. This means the offset is negative; instead of adding a straight track section the adjacent straight has to be shorter by the corresponding value. For example a curve with x2 needs to be paired with 14 studs long straight sections at each end to fit in its net radius. This is a lot more inconvenient, so I've generally avoided those curves. If no offset is needed, this code is absent. Thus, we get the full code 72 (net radius) sb (family and variant) 2 (tightness) o8 (offset). Other example codes are 120b1 (net radius = Rn120, curve family B main profile, tightness of 1 i.e. the apex of the curve is R104, no offset required) and 168dc1x1 (net radius = Rn168, family D variant Dc, tightness 1 = apex R152, and negative offset of 1). In addition to 90-degree curves, I've developed some 45-degree, 22.5-degree, and 180-degree curves. I have also created a BlueBrick file visualising a number of the curve families I've created, with the different radius versions laid out next to each other with some comments about the family: (example image, the actual file contains significantly more curve families) BlueBrick file Because BlueBrick doesn't have the full portfolio of available track pieces some families do not have all of their curves represented, but most curves of most families are. The weird code under the profile family name is the profile specification; each number of letter denotes one 1/16th of a 90-degree curve segment, with 0 meaning a segment of the same radius as the net radius of the curve, positive numbers meaning a larger radius, and letters meaning a smaller radius; for example on a Rn72 curve 4-11-aaaaa+ means 1x R136, 2x R88, 5x R56, then mirrored (the "+" sign at the end). The value after the slash is the error in studs; a single slash like "/0.4" means the error is in the positive offset direction and a double slash "//0.4" means it is in the negative offset direction. I haven't tested these curves physically yet, and I don't have the space for a large layout, but at least visually and in theory they seem significantly improved over the constant-radius standard curves, in both aesthetics and running qualities. I developed the curves and the theory on my own, after finding no existing information on LEGO track transition curves. After hours of looking if anyone else has tried this, I managed to dig up a Flickr post from 8 years ago and the 2021 Fx Bricks track reference with some prior art, but Michael Gale's curves mix straight pieces with curves, creating exactly the abrupt changes in curvature that I wanted to avoid, and FX bricks only has one curve which I consider suitable for 180 degrees when doubled (the R56 end has an abrupt curvature change when used as a 90-degree curve), which has a slightly different profile to my U and V curve families.