Adv. Comp. Math. 2 (1994), J.C. Baltzer, pp. 143-154
This paper presents a short, simple, and general proof showing that the control polygons generated by subdivision and degree elevation converge to the underlying splines, box-splines, or multivariate Bézier polynomials, respectively. The proof is based only on a Taylor expansion. Then the results are carried over to rational curves and surfaces. Finally, an even shorter but as simple proof is presented for the fact that subdivided Bézier polygons converge to the corresponding curve.
We present a new algorithm which computes dot-products of arbitrary length with minimal rounding errors, independent of the number of addends. The algorithm has an O(n) time and O(1) memory complexity and does not need extensions of the arithmetic kernel, i.e., usual floating-point operations. A slight modification yields an algorithm which computes the dot-product in machine precision. Due to its simplicity, the algorithm can easily be implemented in hardware.
