Railroad curves are usually splines. That was the original physical application. You need a curve that minimizes jerk as the radius changes.

Now I'm wondering how the steel tracks get bent- I figure it's done on site rather than at the factory. Dunno what sort of equipment they use to bend them though. Maybe they can just use bigger "spline weights" and pin the bend points to scale from the drawing! Hmmmmmmmmm probably not but I want it to be true

I would guess in a ring roller.

Basically a roller on top pressing down and 2 rollers on the bottom and you wind the metal through tightening the top roller to increase the curve.

Sorry to disappoint you.

If it makes you feel better though, the whales used to set the curves on boats used to be real whales.

But a ring roller produces curves of constant radius, and for the railway application it would have to be absolutely enormous.

From a quick DuckDuckGo search, it looks like they basically are just laid out the same way the "splines" from the article are: the curves are gentle enough that the steel is not very hard to bend.

Talking of ducks, some think that the spline weight 'ducks' could be the origin of the expression 'have your ducks in a row'.

Railroad tracks are pretty flexible, there's no need to bend them separately, you just put them in the sleepers and can fine tune them in there.

You can see how much they bend in this video: https://www.youtube.com/watch?v=CpP6ar63Qxw (1:40). In this case they get delivered as 200m sections on normal rail cars, so they hava to withstand normal curves during transport.

before ships ?

Interesting, do you have a source where I can get deeper into the application of splines to reilroads? Specifically, how to arrive at the jerk minimization problem. All things railroads are too descriptive with little engineering/physics formulas.

chapter 3.2 has a good explanation of it all. https://nap.nationalacademies.org/read/22800/chapter/3

There are tons of papers and research on this topic. Here's just one review of some of the various approaches that have been used.

https://www.researchgate.net/publication/341476435_Railway_T...

i think - but am not sure - they use clothoid definitions to characterize jerk, as it is more restrictive regarding the maximum rate of turn change. you can make quite a sharp turn with a mathematical spline. but i can see how they would use the physical spline to solve this…

Googling “railroad spline clothoid” got some intriguing hits — and this HN comment. https://news.ycombinator.com/item?id=10446154

semi-relatedly, this is a fun watch. https://youtu.be/aVwxzDHniEw

even less relatedly, here is a drawing i did by torturing a clothoid optimizer https://www.instagram.com/p/CSe40DTp-WW/?igshid=YmMyMTA2M2Y=

See also French curves: https://en.m.wikipedia.org/wiki/French_curve

And, more like that in the post, the "flexible curve ruler": https://www.ryman.co.uk/helix-flexi-curve-ruler

I used both during my Industrial Design degree in the mid 00s, but never in practice after. I suspect some people still do though.

My dad used to have a set of French curves - he was a sign writer and used them for cutting the templates for silk screen printing, probably in the 60's or there abouts.

That “flexible curve ruler” seems to be called a Lesbian rule (after the island; https://en.wikipedia.org/wiki/Lesbian_rule), and can also be used to measure curve length (bend the rule so that is matches the curve to be measured, mark begin and end of the to-be measured length, straighten the rule, and use a straight ruler to measure the length between the marked points)

He didn’t answer the question the first come to my mind: Is the curve from the physical spline a bezier?

You should be able to describe (within any reasonable tolerance) the same curve with a bezier, however the control points will obviously be different.

Any imperfections in the physical spline, say a knot of it's wooden, would translate to a slight curvature difference.

Sounds like the physical curves are a subset of bezier? So there might be some bezier curves that cannot be replicated physically

Not really, you can just add more and more control points to a Bézier and it will be possible to describe a curve to a greater accuracy.

However, in reality when using Bézier curves you want to really want to use the minimum number of control point neccesery to ensure a nice smooth curve.

I mean vice versa. Can the physical thing do any CAD Bezier?

A Bézier curve can have a cusp, which you can’t do with a physical spine.

> Not really, you can just add more and more control points to a Bézier and it will be possible to describe a curve to a greater accuracy.

The more control points you add to your curve the more computationally expensive it becomes, though.

This is just the law of spline demand.

Excellently integrated pun!

IIRC, generally speaking a beam with a transverse point load (think like a diver on the tip of a diving board) deforms into a parabolic curve, and beams with more point loads end up taking higher order polynomial curves. Engineering beam math usually much assumes that the beam is acting under small deflections though.

I don't remember as much about beziers but I think they are also often essentially a series of cubic curves. There are a couple of different types of splines- one of them is controlled locally only by it's local control points (I think that one's beziers) and one requires all control points to control the whole curve.

That's incorrect. That curve is Bernoulli's elastica, which is different from a parabola (though for small deformations they can be a reasonably close approximation).

There is a neat connection between a rolling hyperbola and an elastica though: https://twitter.com/KangarooPhysics/status/14337538508177858...

No - it's an elastica, which cannot be represented exactly as a bezier. Raph Levien's thesis is a great read on this https://www.levien.com/phd/thesis.pdf

So if you put a few of these together in a lattice, would you be reticulating splines?

I was looking for this reference.

Website is completely broken in firefox when javascript is enabled.

no it isn't. Running 107.0 here and it looks fine

Broken in Firefox for Android 107.1 as well.

Does seem a bit odd at second glance, seems like hitting the back button undos scrolling instead of navigating back to HN, depending on how much I scrolled. Maybe it's a feature though. Firefox for Android 107.1.0 with uBO.

Fine here. Firefox 107.0 w/ uBlock Origin

In Brave I see "Enter a caption (optional)" on every picture. <span class="caption public_hide" contenteditable="false" id="78a560_1776">Enter a caption (optional)</span>.

They still are in wooden boat building. See Sampson Boat Company on YouTube from a few years ago.

Came here to say exactly that. In particular, the episodes that deal with lofting (24 & 25): https://www.youtube.com/watch?v=oKJagvumvCI&list=PLB00JHoTw1...

Intriguing videos! There's a good section of him laying down splines at 10:58 in episode 25:

https://youtu.be/gfuPTLSBQfY?t=658

Is this where they would be used with mitres?

Now bronze spline weights go $66, not $50. +8% in 6 years.

It's +32%

Ooops, divide instead of multiply :-(

Aside, there is no reason for them to be bronze, they look nice, but you could 3d print them and fill them, cast out of ceramic in plaster mold, make them from plaster, etc.

If I were going to do a one-off set, I'd 3d print them and fill with sand.

My father used to have these curves that were segmented rulers. You could bend them, and they would retain the shape.

He also had a couple of slide rules.

I have a certain nerdy penchant for the "stationary" objects we used to pack in our school bags; slide rules, protractors, set squares, compasses, stencils sets for things like chemistry and flow charts, hard pencils and sharpeners. They still feel like instruments of ritual and magic.

Is it possible to reticulate physical splines?

Yes, but this requires specialized equipment and a licensed technician.

My uncle used to work at a tool maker shop for the stamping industry in the 70’s. For the round molds for the wheel wells, they did grind the piece of steel by hand, and the curves were checked by a soft bending ruler.

I'm so curious but the article doesn't mention: how were these splines "defined"? In other words, if you have to reproduce your work for a new draft, how do you make sure you draw the exact same curve? (Without tracing the prior draft.)

Are there "control points" like Bézier curves have? Do those have anything to do with the placement of the whales? Or was this all just done "by eye" so that these splines assisted in drawing a perfectly smooth curve, but the result was just trial-and-error to find a curve that looks good?

An interesting question, since you could record the positions of the whales, but that alone doesn't fully determine the curve. I guess you'd have to also record the distances along the spline between those pinned points.

I think the pins aren’t ‘pinning’ specific points on the wood strips to specific locations, but just prevent the strip of wood from becoming a straight-lined strip of wood. The tendency of the strip of wood to straighten itself would provide enough force to make the wood slide along the pins to find the minimum energy solution.

(Edit: https://en.wikipedia.org/wiki/Flat_spline seems to confirm that)

Both types are used - pinning and sliding. See figure 3.19 in https://www.levien.com/phd/thesis.pdf

Is it a requirement, to be able to copy a design without tracing? Blueprints seem to have been the main way of storing designs for quite a while. Maybe the drawings are just as good as it gets until you have computers that really want numbers.

Think about buying a ship for example — I guess the person buying it would just specify a type of hull (or for high end stuff, go to a shipwright they specializes in a type of hull), and a couple dimensions.

As to actually doing the design, must be their experience plus conventions built on outcomes from the field — this part of the hull has to be convex, etc etc. So, trial an error, but built up over generations.

My dad was a naval architect, so I can answer that one. Drawing was done on translucent paper (tracing paper). If you had to copy another drawing, you could just slide it under your current drawing and position a spline to match the line.

Once the drawing was done, it was brought to a copy shop that used a special machine to copy it to a paper that turned blue when exposed to light (hence the term blueprint - https://en.m.wikipedia.org/wiki/Blueprint).

The original drawing on tracing paper was then carefully archived, and more blueprints could be made out of it as needed.

I guess measure where the whales are, and hope that the flexibility of the thing you're using to re-create the line is close enough to what was used when the curve was initially created.

The order's different (the curve defining the points, as opposed to points defining the curve), but it might look like what Apple provides in their CAD files for accessory makers. Take a look at p. 362 for an example: https://developer.apple.com/accessories/Accessory-Design-Gui...

An elastica curve doesn't depend on the stiffness of the material - as long as it is linearly elastic.

Boat builders do talk about “fair battens”, where batten = spline, and “fair” refers to being linearly elastic. Boat building battens can be really long, so fairness is not guaranteed. Long battens are scarfed together out of pieces cut from the same timber plank to try to maximise uniformity.

It seems that the software interaction is not as nice as in the real world. I often end up fussing with the spline tools in programs to get what I want but this looks much more straigtforward.

I loved playing with those weights as a kid. Their funny shape and hefty weight made them into interesting characters. The spline themselves were definitely off limits to my little hands, though.

One thing the article doesn't mention is that splines were also made out of clear plastic, not just wood. This made them more bendable and durable, I believe, and it was nice to be able to see through them when positionning them.

I have some stuff on real splines (their similarity and difference from the mathematical idealization) in section 3.12 of my thesis (page 39 of https://www.levien.com/phd/thesis.pdf).

There's also a bunch of stuff in there about Euler spirals (aka clothoids), including application to railways - see section 6.6, page 106. Fun fact, I'm going to get my first tattoo soon, and it will be an Euler spiral.