CS 284: CAGD (SPLINES)

KNOT - FACTS

• "Knots" are located where distinct B-spline segments join together.

• For knots with multiplicity 1, the continuity is d-1 (d=degree of B-spline); in general, the continuity at a knot is d-k (k=knot-multiplicity).

• A knot multiplicity of k will result in basis functions that are missing k-1 segments of their regular d+1 segments; the continuity at that knot is d-k.

• The parameter value "t" assumed at a knot in the curve is the "knot value."

• Knot values will show up as one of the d numbers making up a Blossom label assigned to the deBoor control points; if there are knots with multiplicity k, the corresponding label will show up k times in sequence.

• If the same value shows up d times in the same Blossom label, then the curve will interpolate that deBoor point.

• If the same value shows up d-1 times on two subsequent Blossom labels, then the curve will have an intermediate point on the control polygon segment between those two deBoor points.

• The difference in knot values on subsequent knots controls the length of the curve segment between the two knots. When that difference goes to zero, a knot with higher multiplicity results, and correspondingly, a curve segment is "lost."

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