Knot Vector

A KnotVector of degree \(p\) and \(n\) number of points is described by

\[\mathbf{U} = \left[u_0, \ u_{1}, \ \cdots, \ u_{n+p}\right]\]

There are a total of \((n+p+1)\) elements in this vector which satisfies

\[\underbrace{u_0 = u_{1} = \cdots = u_{p}}_{p+1 \ \text{knots}} < u_{p+1} \le \cdots \le u_{n-1} < \underbrace{u_{n} = u_{n+1} = \cdots = u_{n+p}}_{p+1 \ \text{knots}}\]

The basis curves are piecewise polynomials. They are class \(C^{\infty}\) for each interval \(\left(u_{i}, \ u_{i+1}\right)\), but at \(u_{i}\) they are only class \(C^{p-m_i}\). Where \(m_{i}\) is the multiplicity of the knot \(u_i\) inside the KnotVector.