What is the maximum distance from an anchor point to a bezier curve?

Answer 1

Well, a bound is easy to compute.. just the maximum distance to the convex hull.

If you're looking for a precise answer, then you'll have to do a search.

Answer 2

The formula for a cubic Bezier curve is given in this wiki article. Using simple calculus, you can find the formula for the tangent B'(t) of the curve at parameter t.

The farthest point of the curve from P₃ is characterized by the tangent B'(t) being perpendicular to the vector B(t)-P₃.

So you need to find the value of t for which the dot product B'(t)⋅(B(t)-P₃)=0.

Offhand, you're solving a quintic in t, so prepare to do some root finding. I'd typically expect one root in the interval [0,1], but I suppose some configurations of the P_i could have more than one root (in which case you select one that gives the largest distance).