Let \(F(u)=\sum_{i=0}^{m-1}\,N_i^n (u)\,\mathbf{d}_i\) be a B-spline curve of degree \(n\) over the
knot vector \(T\). Then the following properties hold true on \(\left[t_n,t_m\right]\):
In general there is no endpoint interpolation.
For \(t_i\leq u < t_{i+1}\), \(F(u)\) lies in the convex hull of the \(n+1\) control points
\(\mathbf{d}_{i-n}\ldots,\mathbf{d}_i\), the local convex hull.
Local control: For \(u\in [t_i,t_{i+1})\) the curve is independent of \(\mathbf{d}_j\) for
\(j < i-n\) and \(j > i\).
If \(n\) control points coincide, then the curve goes through this point and is tangent
to the control polygon.
If \(n\) control points are on a line, then the curve touches this line.
If \(n+1\) control points are on a line \(L\), then \(F(u)\in L\) for \(t_i\leq u < t_{i+1}\),
i.e. a whole segment of the curve \(F(u)\) coincides with \(L.\)
If \(n\) knots \(t_i=t_{i+1}=\cdots=t_{i+n-1}\) coincide, then \(F(t_{i})=\mathbf{d}_{i-1}\),
i.e. the curve goes through a control point and is tangent to the control polygon.
If \(\phi\) is an affine map, then (affine invariance):
\[
\phi\left(\sum_{i=0}^{m-1}\,N_i^n (u)\,\mathbf{d}_i\right)=\sum_{i=0}^{m-1}\,N_i^n (u)\,\phi(\mathbf{d}_i)
\]
Variation diminishing property: If \(H\) is a hyper plane in \(\mathbb{R}^d\), then
\[
\#(F\cap H)\leq{}\#(H\cap\text{control polygon})
\]