Intersection points of a quadratic Bezier curve and a plane

quadratic_bezier_plane_intersections.txt

Find the values of the parameter t (in the range 0..1 inclusive) at the intersection points of a quadratic Bezier curve 
and a plane in three dimensions.

Let P₀, P₁, and P₂ be the control points of a quadratic Bezier curve B such that

B{t} = (1-t)²P₀ + 2(1-t)(t)P₁ + (t²)P₂

Let A be a unit normal vector to the plane where Q is any position vector on the plane such that

A⋅Q = d
  0 = A⋅Q - d

Now, at the intersection points we have B{t} = Q, thus

0 = A⋅B{t} - d
  = A⋅[(1-t)²P₀ + 2(1-t)(t)P₁ + (t²)P₂] - d
  = t²[A⋅(P₀-2P₁+P₂)] + t[2A⋅(P₁-P₀)] + A⋅(P₀-Q)
      
We now have a quadratic equation in terms of t which can be solved for t. The value(s) of t can be plugged back into
B{t} to find the intersection position vector(s).

If there are no real solutions for t in the range 0..1 inclusive then either the plane does not intersect the Bezier 
curve or the plane coincides with the Bezier curve's intrinsic plane meaning there is an infinite number of intersection
points.