B-Splines

General B-Splines

• Nonuniform B-splines (NUBS) generalize this construction
• A B-spline, , is a piecewise polynomial:
  • each of its segments is of degree
  • it is defined for all t
  • its segmentation is give by knots
  • it is zero for and
  • it may have a discontinuity in its d-k+1 derivative at ,
    if has multiplicity k
  • it is nonnegative for
  • for ,
    and all other are zero on this interval
  • Bézier blending functions are the special case where
    all knots have multiplicity d+1

Example (Quadratic):

Evaluation

• There is an efficient, recursive evaluation scheme for any curve point
• It generalizes the triangle scheme (de Casteljau) for Bézier curves
• Example (for cubics and ):