Blossoms and Splines

A Summary of the Key Ideas

• Each coordinate function of a spline curve is a part of a (different) N-degree polynomial P(u).

• Each such polynomial can be written as an N-degree Blossom: Blm( P(u) ) = p(u₁, ..., u_i, ... u_N).

• Each such blossom is a collection of Nth-degree polynomial expressions, symmetrical in N new variables u_i.

• If all u_i of a blossom are the same (= u_t), then the value of p(u_t, ..., u_t, ... u_t) = P(u_t).

• If this is true simultaneous for all coordinate polynomials, then the blossom point lies on the spline.

• Assume that for u_i=a, we get point A on the curve, and for u_i = b we get point B,
  then for a < u_i < b we trace out the curve between points A and B.

• Thus if p(a, a, ... a, a) is the start point, and p(b, b, ... b, b) is the endpoint of the curve segment,
  what is p(a, a, ... b, b) , { with i "b"s and (N-i) "a"s } ?
  ===> it is the i-th Bezier control point !

• And what is p(1, 2, 3) or p(0, 1, 2) or p(-1, 0, 1) or p(-2, -1, 0) ?
  ===> These are de Boor points, or B-spline control points !