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I have an original figure in the SVG file. I need to break it randomly on parts, an example of which is specified below:
sample image
I have an idea, to use for partitioning Voronoi diagram (Fortune's algorithm). Can I then change the line of intersection of figures to give them a curvature? Can be obtained to identify each shape as a path? To create a game I'll be using cocos2d-x.

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I think you've got the right idea, but the execution is going to be immensely challenging. I'm presuming when you say 'in an SVG file' you mean that the shape is defined by one (or more — your sample figure has an internal hole!) stroke paths. Unfortunately, SVG paths can be remarkably complicated, with both quadratic and cubic Bezier curves and elliptical arc segments. The intersection of a line with any of these is a solved problem, but a highly non-trivial one, and even the matter of re-parametrizing Bezier curves to split them in two at the point of intersection is a little tricky — basically, what you're looking for is complex enough to take a small book to answer, not a forum post! That said, there are several excellent references out there; the short short version of the answer is that you want to use recursive subdivision along with the so-called 'variation diminishing' property of the Bezier curve (essentially, the property that a Bezier curve is contained within the convex hull of its control points) to narrow down the possible locations of an intersection (and note that a line can have multiple intersection points, and that's another special-case that you may have to take into account). And once you solve the intersection problem, you've still got a long road ahead — you're going to have a ton of bookkeeping to do to keep track of curve and line segments and partition them correctly into pieces. This isn't to discourage you at all; just be aware of the magnitude of what you're trying to do!

Fortunately, your other question — 'can I change the line of intersection of figures to give them curvature?' — has an easier answer: absolutely. Since you're already in SVG-land, I'd use Bezier curves here too: first, choose five points p1..p5 along the line segment you want to 'curve out', probably somewhat at random in roughly equally-spaced intervals; then displace them a small random distance off the line, and use them as the control points of two separate Bezier curves, one using points p0, p1, p2, p3 and the other using points p3, p4, p5, and p6 (where p0 and p6 here are the two endpoints of the original line segment). You can take advantage of the convex-hull property of the Bezier curve here too, making sure that your displaced points don't displace too far so that the two regions to either side of the curve(s) are still simple regions. This should give you the sort of curved boundary that you're after.

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Thank you for answer. I decided to use the SVG file because SVG images and their behaviors are defined in XML text files, and maybe I could break the figure using the internal information of the figure. I suspect that this is not the best solution, but this area of knowledge is new to me. How do you think if I replace the SVG file to something else, it will simplify the solution? Maybe you know other ways to partition non-convex shape with curves on the side? – Crazy D0G May 25 '12 at 19:42
I think SVG is fine as a format - as you note, it's a well-defined text format that can be straightforwardly parsed. SVG isn't what makes the problem difficult - it's the core issues of working with splines and such. But I don't think any other format can express figures like the one in your example concisely enough. It's just an inherently challenging problem. – Steven Stadnicki May 25 '12 at 22:13
...that said, it's not insurmountable; it's even relatively straightforward (although there are a lot of little details to worry about). You just need to build (or find) a good Bezier library. The only issue that didn't get raised, that might be a little tricky, is how to choose the points within your figure for building the Vornoi diagram from. – Steven Stadnicki May 25 '12 at 22:16
Placement of points can be made at random, so that they fall into the area of ​​the figure, or specify multiple lines and further take a random point on them. – Crazy D0G May 28 '12 at 10:54

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