Non-uniform rational B-spline

Remarks

The denominator is positive on $[t_k, t_{n+1})$, because the B-splines are nonnegative and do not all vanish there; so $C(u)$ is a convex combination of the control points. At $u = t_{n+1}$, where the right-continuous B-splines may all vanish, $C$ is defined by its limit from the left, as usual. In homogeneous coordinates, $C$ is the central projection of the polynomial B-spline curve with control points $(w_i P_i, w_i)$ in $\R^{d+1}$. If all weights are equal, $C$ is the ordinary B-spline curve $\sum_i B_{i,k}(u) P_i$. Example: take $k = 3$, knots $0, 0, 0, 1, 1, 1$, points $(1, 0)$, $(1, 1)$, $(0, 1)$ and weights $1$, $1/\sqrt{2}$, $1$. Then $C$ is a rational quadratic Bézier curve, and $C(1/2) = (1/\sqrt{2}, 1/\sqrt{2})$. In fact $C$ traces the quarter of the unit circle in the first quadrant. Non-example: the same data with weight $w_2 = -1$ fail the positivity clause, and then the denominator $(1-u)^2 - 2u(1-u) + u^2 = (2u - 1)^2$ vanishes at $u = 1/2$, so $C$ is not defined on all of $[0, 1]$.