= Solution
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 sequence>
$$
\tau_{3i}=\tau_{3i+1}=\tau_{3i+2}=i.
$$
With the standard <B-spline> convention that control $Q_j$ multiplies $N_{j,4}$ supported on $[\tau_j,\tau_{j+5}]$, the answer is
$$
\boxed{Q_{3i-1}=\frac{P_{i-1}+3P_i}{4},\qquad Q_{3i}=\frac{P_{i-1}+10P_i+P_{i+1}}{12},\qquad Q_{3i+1}=\frac{3P_i+P_{i+1}}4.}
$$
In 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,
$$
\frac{E_{3,i-1}+E_{1,i}}2=\frac{(3P_{i-1}+P_i)+(P_{i-1}+3P_i)}8=\frac{P_{i-1}+P_i}2=E_{0,i}.
$$
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:
$$
4(E_{4,i}-E_{3,i})=P_{i+1}-P_i=4(E_{1,i+1}-E_{0,i+1}).
$$
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.
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