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Use the quartic treble-knot representation of a quadratic B-spline. Specify the indexing as well as the controls: for the interior of the long curve, take the quartic spline knot sequenceWith the standard B-spline convention that control $Q_j$ multiplies $N_{j,4}$ supported on $[{\tau }_j,{\tau }_{j+5}]$, the answer isIn other words, keep the three interior quartic Bézier curve controls $E_{1,i},E_{2,i},E_{3,i}$ for each span in that order. The offset $3i-1$ matters for the stated knot indexing; shifting all indices together gives an equivalent convention.

To prove the representation, insert each integer interior knot once, changing multiplicity three to four. A degree-four knot of multiplicity four separates Bézier curve spans while leaving their shared endpoint common. The three interior controls $E_{1,i},E_{2,i},E_{3,i}$ remain, and knot insertion constructs the shared endpoint from its neighboring controls with weights $1/2,1/2$, since adjacent knot intervals have equal length. In fact,The right endpoint is obtained in the same way. Hence every extracted quartic span has exactly the five controls proved in part (b). Knot insertion leaves the curve unchanged, so the assembled quartic B-spline is identical to the original quadratic one on every span.

There is also a derivative check at each join:This is why treble knots are appropriate: degree four with interior knot multiplicity three permits $C^{4-3}=C^1$ continuity, exactly the generic continuity of the uniform quadratic B-spline. For a finite clamped curve, include its endpoint controls and endpoint knot multiplicities using the same degree elevation of Bernstein coefficients and knot insertion; the displayed formulas describe the long interior requested here, without imposing an unstated end condition.