Knot vector tutorial

Albert Technical Memo#5: How to use Knot VectorsWritten by: John Peterson June 1990This gives a basic explanation of how to use the knot vector of aNurb curve (or surface) to manipulate its shapeIntroductionA Non-Uniform Rational B-Spline curve is defined by three things:• Control points• The curve's order.• A knot vectorControl points are familiar to people who have used Bézier curvesin applications such as Illustrator and MacDraw. The knot vectoris an additional feature of the Nurb representation. It is easy to usethe curves without ever accessing knot vectors, but learning theinformation in this note extends what you can do with them.Curves like Nurbs and Béziers are parametric curves. The curve isa function P(u) that returns a point P on the curve for a particularvalue of the parameter u. As u varies from initial value umin -towards u the curve is drawn. Knots are defined in themax parameter space of a curve. We’ll use the notation u to refer to aiparticular knot. The collection of knots for a particular curve iscalled the knot vector.1A curve is defined with a series of polynomials. The number ofpolynomials needed to define a curve depends on the number ofcontrol points and the order of the curve. The knot vectordetermines where in the parameter range these polynomials startand stop as the curve is drawn. (They’re called “knots” because thevalues “tie together” the polynomials.) Fortunately, you don’t needto understand the ...