Bezier clipping

Question

I\'m trying to find/make an algorithm to compute the intersection (a new filled object) of two arbitrary filled 2D objects. The objects are defined using either lines or cub

Answer 1

I know I'm at risk of being redundant, but I was investigating the same issue and found a solution that I'd read in academic papers but hadn't found a working solution for.

You can rewrite the bezier curves as a set of two bi-variate cubic equations like this:

• ∆x = ax₁*t₁^3 + bx₁*t₁^2 + cx₁*t₁ + dx₁ - ax₂*t₂^3 - bx₂*t₂^2 - cx₂*t₂ - dx₂
• ∆y = ay₁*t₁^3 + by₁*t₁^2 + cy₁*t₁ + dy₁ - ay₂*t₂^3 - by₂*t₂^2 - cy₂*t₂ - dy₂

Obviously, the curves intersect for values of (t₁,t₂) where ∆x = ∆y = 0. Unfortunately, it's complicated by the fact that it is difficult to find roots in two dimensions, and approximate approaches tend to (as another writer put it) blow up.

But if you're using integers or rational numbers for your control points, then you can use Groebner bases to rewrite your bi-variate order-3 polynomials into a (possibly-up-to-order-9-thus-your-nine-possible-intersections) monovariate polynomial. After that you just need to find your roots (for, say t₂) in one dimension, and plug your results back into one of your original equations to find the other dimension.

Burchburger has a layman-friendly introduction to Groebner Bases called "Gröbner Bases: A Short Introduction for Systems Theorists" that was very helpful for me. Google it. The other paper that was helpful was one called "Fast, precise flattening of cubic Bézier path and offset curves" by TF Hain, which has lots of utility equations for bezier curves, including how to find the polynomial coefficients for the x and y equations.

As for whether the Bezier clipping will help with this particular method, I doubt it, but bezier clipping is a method for narrowing down where intersections might be, not for finding a final (though possibly approximate) answer of where it is. A lot of time with this method will be spent in finding the mono-variate equation, and that task doesn't get any easier with clipping. Finding the roots is by comparison trivial.

However, one of the advantages of this method is that it doesn't depend on recursively subdividing the curve, and the problem becomes a simple one-dimensional root-finding problem, which is not easy, but well documented. The major disadvantage is that computing Groebner bases is costly and becomes very unwieldy if you're dealing with floating point polynomials or using higher order Bezier curves.

If you want some finished code in Haskell to find the intersections, let me know.