Fascinating and depressing at the same time. Posts like this make me think I'm in the wrong job as if asked to write a similar program, I'd have no idea where to start. JGC labels this as a "small program", like it was knocked up on a whim :(
It was knocked up on a whim, and it is small, but don't be depressed: it wasn't easy...
I had to bang my head against the trigonometry to get it all correct. Although I know the rules of trig, I always have a hard time applying them in the real world because I have a hard time imagining the shapes involved. Ended up drawing little pictures of what was doing on.
I think it would be quite interesting to know more about how you approached this problem for solution, than the solutions presented by the program itself .. I didn't really get much of the details about how you'd sorted through this from your post .. perhaps a followup topic?
Its really great, btw .. I play with this toy with my son on a regular basis, and found myself just a few days ago wondering what is the maximum number of shapes we can build with the set .. okay, this got a bit screwed up by the addition of another mini-set gifted to my son by a friend, so there is room for improvement in my thinking now that I've learned what you have discovered. Nice job!
I tried to explain how it works in the comments in the code. Probably worth reading them in detail as they do explain the algorithm used.
My thought process was something like this (over a number of days grabbing time when I could):
Emulate a human selecting pieces and clicking them together. Model pieces as straight lines of the right length. There are two important things: how long the line is and how much it modifies the angle at which the next piece will join. Thus all I need is to keep track of the position of the end of the last piece added and the angle.
Spend a while figuring out the trigonometry and getting that code working correctly. To generate the permutations I realized that the process could be split into three parts:
1. Always place the bridge. Since there's one just add it at the end.
2. Separate the straights and curves. Generate all the permutations of clockwise and anticlockwise curved pieces. To do that I used the trick of all 12-bit binary numbers between 0 and 2^12-1.
3. Then insert the straight pieces between the curved pieces in all places possible.
Feed each possible track layout into code that builds the track and check that the end pieces match up. Add code to eliminate end pieces that join impossibly (come from the wrong angle), add code to remove duplicates caused by rotation and reflection.
Once I got that working I then wrote code in Processing to draw a pretty picture.
You might be able to speed it up considerably by adding
2a. Continue with next iteration unless the turning number of the sequence of curves is a whole number.
You add 1/8 to the turning number for each clockwise curve and subtract 1/8 for each CCW (for numerical stability you would rather add of subtract 1 and check for multiples of 8).
I am genuinely interested: how could you have no idea how to start? Isn't it obvious to check all possible permutations of the 9 pieces (and orientations I suppose, also accounting for duplicate pieces)? I suppose that is what you would be doing even if you didn't have a computer?
That shouldn't be too hard. A little harder perhaps to then determine if a given permutation is a closed loop or not, but essentially it should be basic geometry (didn't look at the code). Even if that naive approach would not work, it would at least be a way to start. The remaining sub problems should be less scary to tackle.
I did a similar thing with our kids' LILLABO set. Except that rather than build from zero, my algorithm was to start with the smallest possible loop (8 curves - a circle) and add similar parts to both sides until the set was used up. The main criteria for the additions was that they had to translate but not rotate, ie. AC or SS and B, although more complex ones are possible, like BCA vs. SCAS. It doesn't seem to make any difference as long as they translate to the same thing.
You should be able to generate all possible loops relatively cheaply, since you're just moving characters around. You might need a separate character for the original circle/curve parts though, to make it possible to find the originals.
I never got around to implementing this - it was just an observation while building interesting tracks for my children, but I might have a go...
I just noticed that the bridge can be divided into two parts. Have you thought about how many new track patterns you can create if you allow some of the pieces to be elevated and use the two separate parts of the bridge as slopes onto and off of the elevated track?
There are special ramp pieces you are supposed to use for that purpose and bridging blocks that allow you to elevate regular track. I've never heard of this brand but it used to be called "Brio" when I was a kid, http://brio.knex.com/
I have a box full of brio from my childhood at home - and, yeh, they did have blocks to let you make raised track.
(it was a pain though because it was easiest to build the bridges first - and then just as you laid the last piece of track a trip/wobble/stumble collapsed the raised bits :P)
My son got a similar train track when he was little but with a lot more pieces. I wanted to write something similar but never got around to it. One twist is that his set has several sets of forks and some pieces to make an elevated track that can double as a bridge.
I'm going to look through the program and see if I can add these types of pieces. We always just made up the tracks as we went and quickly learned to follow some rules so the track would loop but not reverse the train because we had 2 sets of trains. It became a contest between my wife and I to see who could come up with the coolest layout. It was lots of fun.
I've got a whole crate full of Brio wooden trains from when I was a kid, including up/down pieces. I bet if I modified this software to work in three dimensions it could build millions of configurations with the parts I've got. Could be fun.
My brothers and I have split our massive compiled collection of Brio for our kids. It makes me sad that the company is no more. I've heard bad knockoffs are now available in discount stores, but they don't fit the original track.
I've got the same train set at home. I also bought the extra parts with the fork an extra brigde, trestles, and tunnel. Sounds like I've got a new project to help me learn processing.
Awesome. I'd bet most parents reading this have two sets. You should update your post with all the other possible configurations. How many 3 and 4 leaf clovers can you make?
