Finding all closed paths by overlap

[ErikSom]

I have a really hard time wording what I'm trying to do so I made an illustration:

Given 4 paths (the red circle, and the 3 line paths numbered 1-4) how could I find all the closed paths that are formed by their overlap?
I'm trying to find shape A, B and C.

Thanks in advance for any tips or advice.

[AngusJohnson]

There is a line-circle intersect function in the library somewhere that I suggest you use

[philstopford]

Untested, handwaving. You might be able to extract the intersection points of the open lines with each other by offsetting the open paths so that you can clip them (either by offsetting one of 2,3,4 to cut the other two open lines each pass, or offset two of 2,3,4 and trim the other open path). If you use some scaling tricks, you can probably make the offset vanish with rounding error (e.g. scale everything up 10000 times so that the minimal offset of 1 unit just disappears in the noise; scaling down would give you essentially the correct location). With the intersection points in hand, you can construct the triangle and use that with (1) to find the A,B,C outputs.

Again, hand-waving.

[ErikSom]

That's a very interesting suggestion! Thanks.

[AngusJohnson]

Phil, I don't understand why you'd offset to find line intersection points? Surely just solving simultaneous equations would be much faster.

[philstopford]

The library can't clip using open paths (The library supports open path clipping, though only subject paths may be open.) so I was using offset to generate closed clips that can be used to trim the open subject(s) to find the end result. It's not perhaps the best option, but it should work well enough, and also be fairly general (in case open paths end up non-linear or similar).

[AngusJohnson]

and also be fairly general (in case open paths end up non-linear or similar).

OK, that does make sense 😁.

[AngusJohnson]

There is a line-circle intersect function in the library

Oops, sorry, it's in my Image32 Delphi graphics library here, so Pascal only.