 Solution

Choose a quadratic uniform B-spline segment on a unit spline knot interval. This choice is useful because adjacent pieces can later share de Boor points at simple, equally spaced spline knots. Its three active basis pieces on local $t\in [0,1]$ areThey follow from the Cox-de Boor recurrence for degree two and consecutive integer spline knots. If the three de Boor points are $D_0,D_1,D_2$, then the equivalent Bézier curve handles areOne can verify this conversion simply by substituting these expressions into the Bernstein basis formula and collecting each $D_i$. Solving givesThe desired matrix representation is thereforeThe left row is again the monomial row, the middle matrix contains the monomial coefficients of $b_0,b_1,b_2$, and the right matrix lists the de Boor points. Unlike the Bézier curve basis matrix, the constant row averages two handles. A clamped B-spline with repeated end spline knots could instead reuse $A,B,C$ directly; the simple uniform convention above specifies which representation is being used and supplies the shared handles required for the extension.