4 ms·

What about SDFs of cubic bezier curves and rational bezier curves? Because these appear in vector graphics and I think there is no analytic solution for them (y

What about SDFs of cubic bezier curves and rational bezier curves? Because these appear in vector graphics and I think there is no analytic solution for them (yet?).

moonchild 4y ago
> yet? SDF of a cubic bezier involves solving a quintic, so it's not analytic. There are approximations, of course, but for an outline, using an sdf is just silly. (For a stroke, you don't really have much choice--though it's common to approximate strokes using outlines.) I'll add that sdf aa is not as good as analytic aa.
- pcwalton 4y ago
  IIRC Loop-Blinn gives you a pretty good distance approximation using the 2D generalization of Newton's method to avoid having to solve the quintic. (Though I've never actually seen anybody implement Loop-Blinn for cubics because the edge cases around cusps/loops are annoying. Every Loop-Blinn implementation I've actually seen just approximates cubics with quadratics.) (Fun fact: Firefox uses the same 2D Newton's method trick that Loop-Blinn does for antialiasing of elliptical border radii in WebRender—I implemented it a few years back.) :)
  - Lichtso 4y ago
    > Though I've never actually seen anybody implement Loop-Blinn for cubics I implemented them (implicit formulation of rational cubic bezier curves) in my renderer (see one of my other replies in this thread). Here is an extract of the relevant code in a shader toy: https://www.shadertoy.com/view/WlcBRn https://www.shadertoy.com/view/WlcBRn Even in Jim Blinn's book "Notation, Notation, Notation" he leaves some of the crucial equations as an exercise to the reader. I remember spending 2 or 3 weeks reading and trying to understand everything he wrote to derive these equations he hinted at myself.
  - moonchild 4y ago
    Out of curiosity, where have you seen loop-blinn implemented? I was under the impression that it was generally a no-go due to the patent (which, incidentally, expires in 2024).
johndough 4y ago
You can approximate the Bézier curves with circular arcs and store those in the "SDF" instead. https://github.com/behdad/glyphy https://github.com/behdad/glyphy

What about SDFs of cubic bezier curves and rational bezier curves? Because these appear in vector graphics and I think there is no analytic solution for them (yet?).

moonchild 4y ago

> yet? SDF of a cubic bezier involves solving a quintic, so it's not analytic. There are approximations, of course, but for an outline, using an sdf is just silly. (For a stroke, you don't really have much choice--though it's common to approximate strokes using outlines.) I'll add that sdf aa is not as good as analytic aa.

pcwalton 4y ago
IIRC Loop-Blinn gives you a pretty good distance approximation using the 2D generalization of Newton's method to avoid having to solve the quintic. (Though I've never actually seen anybody implement Loop-Blinn for cubics because the edge cases around cusps/loops are annoying. Every Loop-Blinn implementation I've actually seen just approximates cubics with quadratics.) (Fun fact: Firefox uses the same 2D Newton's method trick that Loop-Blinn does for antialiasing of elliptical border radii in WebRender—I implemented it a few years back.) :)
- Lichtso 4y ago
  > Though I've never actually seen anybody implement Loop-Blinn for cubics I implemented them (implicit formulation of rational cubic bezier curves) in my renderer (see one of my other replies in this thread). Here is an extract of the relevant code in a shader toy: https://www.shadertoy.com/view/WlcBRn https://www.shadertoy.com/view/WlcBRn Even in Jim Blinn's book "Notation, Notation, Notation" he leaves some of the crucial equations as an exercise to the reader. I remember spending 2 or 3 weeks reading and trying to understand everything he wrote to derive these equations he hinted at myself.
- moonchild 4y ago
  Out of curiosity, where have you seen loop-blinn implemented? I was under the impression that it was generally a no-go due to the patent (which, incidentally, expires in 2024).

johndough 4y ago

You can approximate the Bézier curves with circular arcs and store those in the "SDF" instead. https://github.com/behdad/glyphy https://github.com/behdad/glyphy