Shape generator help (Armin Hofmann's 'rubber band' shape generator)

Hi, I actually gave it quite some thoughts but gave up after some times. I’m at the same point as in my 07/11/21’s answer.

I agree that this question could be a problem on its own.

To sum up my thoughts:

  1. Keep generating shapes until one is not self-intersecting. I wrote that the algorithm is not that trivial but it is just some line/line intersections and line/arcs intersection to be done. But as I wrote, it seems quite rare to find a non intersecting shape randomly and the method is quite inefficient.

  2. Find a more systemic approach. This is where I spent most of my time thinking. My idea was the following: start with just 3 circles (it always works) and find a way to identify in which area of the plane we could pick a new pin without creating any self intersections and then pick one randomly from that zone. Repeat the same process over and over until the desire number of pins is added. Sadly I could not figure any way to compute the “OK” zone easily.
    To define that safe zone, I started to think about the simple shape created by connecting the center of each selected pin instead of the more complicated one with the arcs. Then on a piece of paper I was mapping the safe zone for different scenarii and trying to find common rules but no luck so far.

  3. Instead of generating random shapes, the pins could be manually selected by the user. In this case no need to check for self intersection, the user would do it itself.

Anyway, if you find some insight on this problem, I would like to read it. Maybe I’ll go back to think about it.

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Instead of thinking about the discs, one could think of the rubber-band closed path that is also the perimeter of the filled shape.

This path is made from a number of shorter paths. It’s a bit like a child’s railway track.

These paths can be in one of three states, available to use but unused, unavailable to use and unused, or used.

As soon as some paths have been picked and used, this makes some of the yet unused paths unavailable.

So, when you switch a path on, the code also switches some paths to unavailable.

Light blue: available and unused.
Green: used.
White: unavailable and unused.

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Could you please elaborate?
I fail to see how it can produce the shape at the end.
What are the rules to deactivate the connections when one is turned on?

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Apparently you can make quite a lot of money with this kinf of generative art :sweat_smile:

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Today I came across Dubins paths and remembered this thread. They seem describe exactly the same problem, and Dubins path are a well studied topic in the field of (robotic) motion planning.

In geometry, the term Dubins path typically refers to the shortest curve that connects two points in the two-dimensional Euclidean plane (i.e. x-y plane) with a constraint on the curvature of the path and with prescribed initial and terminal tangents to the path, and an assumption that the vehicle traveling the path can only travel forward.

In 1957, Lester Eli Dubins proved that any such path will consist of maximum curvature and/or straight line segments. In other words, the shortest path will be made by joining circular arcs of maximum curvature and straight lines.


On github, there just was a single well-stared (C++) library for Dubins curve generation (all Java repositories were hobby projects at best). So I’ve just ported it that library to Java and it’s available here:

@jb4x Perhaps this would make your code simpler? To find the rubber band between two pins all you’d have to supply is the pin coordinates and the heading angles (one of N,S,E,W I believe).

In time I’d like to have a go myself at applying this library to the grid of circles to generate the shapes.


The shortest dubins path between two points (fixed terminating heading angle).
dubins2

All possible (6) dubins paths between two points.
dubins

Dubins curves are similar to Reeds-Shepp curves, only that the Reeds-Shepp “car” can move reverse, so some paths are shorter:

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Reading this back again I realize there’s an obvious approach regarding the arrangement of the pins, and it happens that some fine gentlemen of this forum already discussed it here.

If you consider the set of selected pins as a collection of nodes in a graph then generating a “valid” shape amounts to finding a possible non-self-intersecting Hamiltonian path that is cycling through them.

While not perfect this method has 2 benefits:

  • deterministic aspect: it mitigates randomness, no need to generate different orders/arrangements of pins by trial and error as the procedure will find a solution for a given set.

  • combinatorial aspect: because it will traverse a graph and explore its different branchings the procedure will output, not one, but all the solutions to a given set.

Steps:

  • Randomly select a specific amount of circles/pins from a grid
  • Find all non-intersecting lines between all pairs of circles (center to center). Consider those selected lines as the edges of an undirected graph.
  • Create an adjacency matrix representing the topology of this graph.
  • Traverse the graph and find all Hamiltonian cycles that aren’t self-intersecting on a 2D plane. A cycle is an ordered list of indices that will pave the way for a possible Dubins paths.

Taking the example set of pins (in any order) chosen by @jb4x earlier, this algorithm will output the following:

The shape made by jb4x can be spot at position B10. All the rest are the other possible path combinations given the same set of nodes.

Now for fun, displaying just a portion of the solution space for a set of 16 nodes (whole 4x4 grid):

A few notes:

  • I’m creating a graph whose edges are the center lines between pairs of circles. Normally one would choose the internal and external tangents instead. However doing so would create an adjacency matrix too large to be processed in reasonable time (not because of the number of tangents but because of the number of tiny small arcs that would create around each pin as a result) . As a consequence some edges create connections between pairs of circles that normally couldn’t be joined by tangents. This will sometimes result in Dubins paths slightly intersecting themselves. In this case, either discard the solution or reduce the radii of the pins.

  • It’s often said that one of Hofmann’s rule is to wrap around each circle once and once only. I believe this is not true. If you look at the examples provided in his original publication (sample image at the top of this thread) you’ll see that one of the shapes (top right, middle row) has a pin that is wrapped around twice. This is not trivial as it means one would have to come up with a different / more specific set of rules. In that case, using a grid system with pins of different sizes may be useful.
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Hello!
Could you please share the code? I’m too dumb to understand how to do it myself :smiling_face_with_tear:

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Hi @JoWood,

Sorry for the late reply, I’m swamped with work lately and don’t have much time to check to forum.

I won’t be posting the full code here but if you share some of your attempts and clarify which part is giving you a hard time we can sure try to help you achieve your goals.

Also no one is “dumb”, just:

  • don’t be afraid to ask questions (I do ask a lot of questions myself)
  • and give you some time to digest the answer

Hope we can help.

PS: I see what you did with your PP :wink:

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How do you offset the Start [x,y], End [x,y], and tangents so they wrap around the circles?

With all the versions I’ve read about and seen examples of online for Dubins Path there is a Start Point and End Point with their respective tangent directions. However, I haven’t been able to work out a consistent way to get the path to wrap around the circles instead of the center of the circles. Any tips from much smarter people than myself?

This is a simplified version but I’d like to expand it to a grid.

Rather than connecting circles think about the problem in terms of connecting pins on the circles. Each circle has 4 pins located at N,S,E,W on its perimeter. These are the coordinates dubin’s paths expect; the paths will wrap the circles automatically once these are given.

So the problem is no longer which circles to connect, but which pins between successive circles on a chosen circle sequence to connect.

To illustrate, your example would connect the W pin of the leftmost circle to the S pin of the other circle with a RSR dubin’s path.

It also looks like your path’s radius is twice as large as it should be (probably using circle diameter rather than radius for the path radius).

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