How to prove if all points of Bezier-Bernstein belong to the same plane?

Question

I have a deggree $\ n=3 $ Bezier - Bernstein curve with the following control points:

$\ p_{o} = [0, 0, 1]^T $

$\ p_{1} = [1,2, 2]^T $

$\ p_{2} = [2,2,3]^T $

$\ p_{3}=[3,4,4]^T $

I have found the $\ 4 $ polynomials that describe the curve using the mathematical equation for Bernstein polynomials. How can I now prove that all points of this curve belong to the same plane ?

Answer 1

You can check if all control points are co-planar. If yes, the Bezier curve is a planar curve (deduced from Bezier curve's convex hull property) and therefore all points on the curve belong to the same plane.