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docs/chapters/shapes/content.en-GB.md
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@@ -40,9 +40,9 @@ Once we have all the new poly-Bézier curves, we run the first step of the desir
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The main complication in the outlined procedure here is determining how sections qualify in terms of being "inside" and "outside" of our shapes. For this, we need to be able to perform point-in-shape detection, for which we'll use a classic algorithm: getting the "crossing number" by using ray casting, and then testing for "insidedness" by applying the [even-odd rule](http://folk.uio.no/bjornw/doc/bifrost-ref/bifrost-ref-12.html): For any point and any shape, we can cast a ray from our point, to some point that we know lies outside of the shape (such as a corner of our drawing surface). We then count how many times that line crosses our shape (remember that we can perform line/curve intersection detection quite easily). If the number of times it crosses the shape's outline is even, the point did not actually lie inside our shape. If the number of intersections is odd, our point did lie inside out shape. With that knowledge, we can decide whether to treat a section that such a point lies on "needs removal" (under union rules), "needs preserving" (under intersection rules), or "needs flipping" (under exclusion rules).
The main complication in the outlined procedure here is determining how sections qualify in terms of being "inside" and "outside" of our shapes. For this, we need to be able to perform point-in-shape detection, for which we'll use a classic algorithm: getting the "crossing number" by using ray casting, and then testing for "insidedness" by applying the [even-odd rule](https://folk.uio.no/bjornw/doc/bifrost-ref/bifrost-ref-12.html): For any point and any shape, we can cast a ray from our point, to some point that we know lies outside of the shape (such as a corner of our drawing surface). We then count how many times that line crosses our shape (remember that we can perform line/curve intersection detection quite easily). If the number of times it crosses the shape's outline is even, the point did not actually lie inside our shape. If the number of intersections is odd, our point did lie inside out shape. With that knowledge, we can decide whether to treat a section that such a point lies on "needs removal" (under union rules), "needs preserving" (under intersection rules), or "needs flipping" (under exclusion rules).
These operations are expensive, and implementing your own code for this is generally a bad idea if there is already a geometry package available for your language of choice. In this case, for JavaScript the most excellent [Paper.js](http://paperjs.org) already comes with all the code in place to perform efficient boolean shape operations, so rather that implement an inferior version here, I can strongly recommend the Paper.js library if you intend to do any boolean shape work.
These operations are expensive, and implementing your own code for this is generally a bad idea if there is already a geometry package available for your language of choice. In this case, for JavaScript the most excellent [Paper.js](https://paperjs.org) already comes with all the code in place to perform efficient boolean shape operations, so rather that implement an inferior version here, I can strongly recommend the Paper.js library if you intend to do any boolean shape work.
(Of course, as a general geometry library, Paper.js is also roughly the size of this entire primer, so for illustrative purposes the following graphic implements its own boolean operations, and may not do quite the right thing on all edge cases!)