How to Determine a clamped B-spline curve passes through a given point q Let P be a clamped B-spline curve of degree two defined by the control
points,
The control points are :    $\binom{-2}{-2},\binom{-2}{0},\binom{0}{2},\binom{2}{2},\binom{2}{0},\binom{0}{-2} $
and over the knot vector of $\tau$ := $ (0, 0, 0, \frac{1}{4},\frac{1}{2} ,\frac{3}{4}, 1, 1, 1)$. 


*

*Does P pass through the point  $q := \Biggl(\begin{smallmatrix}
    \frac{-1}{2} \\ \frac{1}{2} \end{smallmatrix} \Biggr)$  ? 

*Could we get different results for other clamped (possibly non-uniform) knot vectors instead of $\tau $?
 A: Forget about the knot sequence and just draw the control points:

Any B-spline of degree 1 with those control points is contained in the union of convex hulls of pairs of successive points (left figure).
Any B-spline of degree 2 is contained in the union of convex hulls of triples of points (right figure). It is obvious that $q$ isn't contained not even in the first 2 convex hulls. To prove it you need to show that if $\ c_1P_1+c_2P_2+c_3P_3 =q\ $ where the coefficients sum to $1$, then one of the coefficients must be negative.
A: The question doesn't really make sense. We have 6 control points and 9 knot values, and these completely determine a quadratic b-spline curve of degree 2.
The word "clamped" refers to an end condition that you use when constructing a spline curve. It serves to add two more constraints so that you end up with a linear system that has the same number of equations as unknowns. Since our b-spline curve is completely determined by the given data, it doesn't matter whether it was constructed with "clamped" ends or not.
I'd say that your instructor is trying to make you think by asking you a nonsensical question, or you copied the question incorrectly.
