Polar stroking: new theory and methods for stroking paths

Fig. 1. Polar stroking samples: A cubic Bézier segment with a cusp rendered properly with polar stroking while uniform parametric tessellation has no cusp, both using 134 triangles; B polar stroking improves the facet angle distribution compared to uniform tessellation, both using 126 triangles; C arc length texturing; D ellipse drawn as just 2 conic segments, one external; E complex cubic Bézier path (5,031 path commands, 29,058 scalar path coordinates) with cumulative arc length texturing; F centripetal Catmull-Rom spline.

Stroking and filling are the two basic rendering operations on paths in vector graphics. The theory of filling a path is well-understood in terms of contour integrals and winding numbers, but when path rendering standards specify stroking, they resort to the analogy of painting pixels with a brush that traces the outline of the path. This means important standards such as PDF, SVG, and PostScript lack a rigorous way to say what samples are inside or outside a stroked path. Our work fills this gap with a principled theory of stroking.

Guided by our theory, we develop a novel polar stroking method to render stroked paths robustly with an intuitive way to bound the tessellation error without needing recursion. Because polar stroking guarantees small uniform steps in tangent angle, it provides an efficient way to accumulate arc length along a path for texturing or dashing. While this paper focuses on developing the theory of our polar stroking method, we have successfully implemented our methods on modern programmable GPUs.