You're missing one very important type of curve: a clothoid (or "Euler spiral") is a curve of continuously-varying radius, these are encountered on roads very frequently. And especially on race circuits.
A clothoid is used to connect two lines the same way your fillet is, except instead of just 1 radius it has a radius configured for each end and smoothly changes in between.
https://en.wikipedia.org/wiki/Euler_spiral
They are also used in railways, because on a railway you don't have the freedom of moving the car's position across the road, so a transition from a straight track to a constant radius would imply an instantaneous step change in centrifugal force, or infinite jerk. Using a clothoid to smooth the change between the straight track and the constant-radius turn means the lateral acceleration increases smoothly instead of instantaneously.
Author here. Engineers use clothoids as the primary geometry for high-speed roads by offsetting a centerline clothoid. However, the offset of a clothoid is not itself a clothoid, so the left and right edges are not mathematically clothoids.
A clothoid is simply a mathematically ideal way to achieve continuously increasing curvature along a path. In practice, it can be approximated by chaining multiple circular arcs with decreasing radii.
I was also confused about that because they did mention clothoids in their first post: https://sandboxspirit.com/blog/art-of-roads-in-games
Although re-reading that it seems they just don't want to deal with the math involved
Here's a nice article on it:
https://www.dgp.toronto.edu/~mccrae/projects/clothoid/sbim20...
I think this comment section would be a good place to ask for help with a related problem.
I want to design a "smooth" closed path that fits in a square, as long as I can make it. It needs to smooth in such a way that a constant velocity can be maintained subject to limits on acceleration and jerk. The point is to develop a test for maximum flow rate on a filament 3D printer without the motion system ever slowing the tool-head down. (In reality, the standard smoothed "E" shape is good enough. This is more of an exercise.)
I know roads are designed to limit acceleration and jerk, and someone knowledgeable about road design would know about finding curves with constraints on acceleration and jerk.
Do you know any free tools or resources on theory that I could use?
I recall reading that Markov (of Markov chain fame) had also worked on transition curves for railway tracks. I have tried and failed to track down any references, or even recall where exactly I read this.
If any HN folks have pointers, would love that.
The care taken here to achieve aesthetically pleasing results reminds me of this post about creating nice transit maps in the Transit App: https://blog.transitapp.com/how-we-built-the-worlds-pretties...
The wikipedia page on some of the considerations is a good introduction on where the real world goes beyond this:
https://en.wikipedia.org/wiki/Geometric_design_of_roads
It's tempting to think that the cross-section geometry of a road only applies in three dimensions and can be elided for 2D overheads, but the parabolas that smoothly connect straights and curves in the overhead perspective are often subtly warped to permit any requirements of cant and superelevation.
And then you have various types of hairpin bend where you actually vary the width of the lanes with the radius: https://www.google.com/maps/@46.8360535,9.6369913,68m
vehicular speed is very important as a consideration in any road curvature, as well as “pitch and yaw” when changing slope and direction at speed, so… simple it is not, and if we are mostly “offsetting” straight lines and arcs, we are doing it wrong
Expected it to at least mention the slant imposed on any road surface so water does not pool. Disappointed to tears and thus salt-water-aquaplaning in all games build upon this.
many are yet to catchup