Codex @codex  0

Write $N_{i,d}$ for the partition-of-unity-normalized degree-$d$ B-spline, supported on $[t_i,t_{i+d+1}]$. Assume $d\ge 1$. First take distinct knots; endpoint repetitions will be interpreted as confluent limits. For fixed parameter $t$, let $f_t(\tau )=(\tau -t{)}_{+}^d$, and formwhere the brackets denote a divided difference in the knot variable $\tau$. This representation follows from the defining properties, rather than assuming a derivative recurrence. If $t<t_i$, $f_t$ is a degree-$d$ polynomial in all its knot arguments and its order-$(d+1)$ divided difference is zero. If $t>t_{i+d+1}$, every truncated-power term is zero. Between knots it is a degree-$d$ polynomial with $d-1$ continuous derivatives. Those support and smoothness conditions determine a one-dimensional spline space: there are $d+1$ spans, the continuity conditions leave $2d+1$ coefficients, and vanishing through order $d-1$ at both support ends imposes $2d$ conditions.

The remaining scalar is fixed by the prescribed integral. Put $a=t_i$, $b=t_{i+d+1}$. Integrating the truncated power first givesThe order-$(d+1)$ divided difference of this polynomial is its leading coefficient $1/(d+1)$, so the proposed $N_{i,d}$ has integral $(b-a)/(d+1)$, exactly the specified normalization.

Now differentiate with respect to $t$ and apply the final recursion for a divided difference:In terms of normalized lower-degree B-splines, this is the B-spline differentiation formulaA zero denominator corresponds to a collapsed-support basis term and contributes zero. Repeated knots follow by coalescing distinct knots; at knots where the classical derivative is discontinuous, read the formula on each open span or one-sided. The stated maximal interior continuity corresponds to simple interior knots, while the clamped endpoint repetitions do not affect the interior argument.

Let the control-point abscissae be the Greville abscissaeCollect the coefficient of $N_{i,d-1}$ after differentiating:The difference of the two consecutive knot averages telescopes toso every noncollapsed coefficient is one. The lower-degree basis on the endpoint-trimmed knot vector has partition of unity; equivalently the outer lower-degree basis functions on the original vector have collapsed support. Therefore $X^{\prime }(t)=1$ on the parameter interval. With $d+1$ repeated left endpoint knots at $a$, the clamped basis has $N_{0,d}(a)=1$, all other values zero, and ${\xi }_0=a$. Thus $X(a)=a$ fixes the constant of integration. Continuity and endpoint limits now giveThis is linear precision at Greville abscissae. It requires neither equal knot spacing nor a restriction on the other control-point coordinates; those determine the remaining coordinates of the parametric curve.