How to get the outline of a stroke?

Question

I want to convert a stroked path to a filled object. (Programmatically, in JavaScript.)

The line is just a simple curved line, a sequence of coordinates. I can render this line as a path, and give it a stroke of a certain thickness... but I'm trying to get a filled shape rather than a stroked line, so that I can do further modifications on it, such as warping it, so the resulting 'stroke' might vary in thickness or have custom bits cut out of it (neither of these things are possible with a real SVG stroke, as far as I can tell).

So I'm trying to manually 'thicken' a line into a solid shape. I can't find any function that does this – I've looked through the docs of D3.js and Raphaël, but no luck. Does anyone know of a library/function that would do this?

Or, even better: if someone could explain to me the geometry theory about how I would do this task manually, by taking the list of line coordinates I have and working out a new path that effectively 'strokes' it, that would be amazing. To put it another way, what does the browser do when you tell it to stroke a path – how does it work out what shape the stroke should be?

Answer 1

There has been a similar question recently: svg: generate 'outline path'

All in all, this is a non-trivial task. As mentioned in my answer to the linked question, PostScript has a command for generating paths that produce basically the same output as a stroke, called strokepath. If you look at what Ghostscript spits out when you run the code I posted at the linked question, it's pretty ugly. And even Inkscape doesn't really do a good job. I just tried Path => Outline stroke in Inkscape (I think that's what the English captions should say), and what came out didn't really look the same as the stroked path.

The "simplest" case would be if you only have non-self-intersecting polylines, polygons or paths that don't contain curves because in general, you can't draw exact "parallel" Bézier curves to the right and the left of a non-trivial Bézier curve that would delimit the stroked area - it's mathematically non-existent. So you would have to approximate it one way or the other. For straight line segments, the exact solution can be found comparatively easily.

The classic way of rendering vector paths with curves/arcs in them is to approximate everything with a polyline that is sufficiently smooth. De Casteljau's Algorithm is typically used for turning Bézier curves into line segments. (That's also basically what comes out when you use the strokepath command in Ghostscript.) You can then find delimiting parallel line segments, but have to join them correctly, using the appropriate linejoin and miterlimit rules. Of course, don't forget the linecaps.

I thought that self-intersecting paths might be tricky because you might get hollow areas inside the path, i.e. the "crossing area" of a black path might become white. This might not be an issue for open paths when using nonzero winding rule, but I'd be cautious about this. For closed paths, you probably need the two "delimiting" paths to run in opposite orientation. But I'm not sure right now whether this really covers all the potential pitfalls.

Sorry if I cause a lot of confusion with this and maybe am not of much help.

Answer 2

This page has a fairly good tutorial on bezier curves in general with a nice section on offset curves.

http://pomax.github.io/bezierinfo/

A less precise but possibly faster method can be found here.

http://seant23.wordpress.com/2010/11/12/offset-bezier-curves/

There is no mathematical answer, because the curve parallel to a bezier curve is not generally a bezier curve. Most methods have degenerate cases, especially when dealing with a series of curves.

Think of a simple curve as one with no trouble spots. No cusps, no loops, no inflections, and ideally a strictly increasing curvature. Chop up all the starting curves into these simple curves. Find all the offset curves of these simple curves. Put all the offset curves back together dealing with gaps and intersections. Quadratic curves are much more tractable if you have the option to work with them.

I think most browsers do something similar to processingjs, as they have degenerate cases even with quadratic curves. For example, look at the curve 200,300 719,301 500,300 with a thickness of 100 or more.

Answer 3

The standard method is the Tiller-Hanson algorithm (Offsets of Two-Dimensional Profiles, 1984, which irritatingly is not on line for free) which creates a good approximation. The idea is that because the control points of each Bezier curve lie on lines tangent to the start and end of the curve, a parallel curve will have the same property. So we offset the start and the end of the curve, then find new control points using these intersections. However, that gives very bad results for sharp curves, so the first step is to bisect the original curve, which is very easy to do to Bezier curves, until it turns through a sufficiently small angle.

Other refinements are needed to deal with (i) intersections between the parallels, on the inside of each vertex; (ii) inserting an arc of a circle to fill the gap on the outside of each vertex; and (iii) adding end-caps - square, butt or circular.

Tiller-Hanson is difficult to implement, but there's a good open-source implementation in the FreeType library, in ftstroke.c (http://git.savannah.gnu.org/cgit/freetype/freetype2.git/tree/src/base/ftstroke.c).

I'm sorry to say that it can be quite difficult to integrate this code, but I have used it successfully, and it works well.