How to calculate normal vector of a Bézier curve?

Question

How I can calculate the normal vector for two dimensional Bézier curve?

(I have the control points of the curve in the x y plane)

Answer 1

If vector (x,y) is a tangent vector of your curve, then the normal vector is simply (y,-x). So you just need to find a tangent vector, and it all depends on how exactly you define the curve.

If your curve comes from a parametric representation p(t) = (x(t),y(t)) which is sufficiently continuous, then a good approximation of the tangent vector at t is, given a very small ɛ:

       p(t+ɛ)-p(t)
v(t) = ———————————
             ɛ

         x(t+ɛ)-x(t)   y(t+ɛ)-y(t)
v(t) = ( ——————————— , ——————————— )
              ɛ             ɛ

This applies to e.g. Bézier curves.

If your curve comes from a dense list of points then a good approximation comes from taking the two closest points on the curve, p0 and p1, and then your tangent is:

v(t) = (p1 - p0) / distance(p1, p0)

If your curve comes from a scattered list of points then the above method will probably exhibit too many discontinuities, but so will your curve unless you use some kind of interpolation. What I suggest is consider the interpolation function (for instance, cubic interpoltion) as a parametric function, and apply the first method.