1-column matrix 1-row matrix almost block-diagonal almost block-diagonal almost block-diagonal almost block-diagonal appropriate knot sequence appropriate knot sequence aptknt augknt augknt augknt augknt augknt augknt augknt augknt augmented knot sequence aveknt aveknt aveknt aveknt B in B-spline banded banded basic interval basic interval basic interval basic interval basic interval basic interval basic interval basic interval basic interval basic interval basic interval basic interval basic interval for the B-form for the pp-form for the pp-form of a pp of a pp of a spline of a spline basis bell-shaped best interpolant B-form B-form B-form B-form B-form bias bicubic spline bicubic spline bivariate bkbrk bkbrk boundary layer break break break break interior break sequence break sequence break sequence breaks breaks breaks vs knots breaks vs knots B-representation brk2knt bspligui B-spline B-spline B-spline B-spline B-spline B-spline coefficients in CAGD normalized of order k some sample figures support of bspline bspline CAGD center of a shifted power form centripetal chbpnt chebdem Chebyshev polynomial Chebyshev spline circle, spline approximation to clamped end condition collocation collocation matrix matrix matrix column-vector composing function with a matrix constructive approach to splines control point control point control polygon control polygon control polygon conventions in our documentation (table) conversion conversion coordinates with respect to a basis csape 2-18 csapi csapi csaps cscvn cscvn cubic cubic cubic cubic cubic cubic cubic cubic cubic cubic cubic cubic Hermite smoothing spline spline Curry-Schoenberg Theorem curvature curve curve curve curve curve curve curve curve curve curve data point multiplicity multiplicity data site data value degrees of freedom derivative of a rational spline differential equation differential equation differentiation differentiation discrete in the pp sense dimension discrete differentiation least-squares approximation domain of a function draftsman's spline dual functional dual functional d-vector end conditions clamped complete curved Lagrange natural not-a-knot other variational variational equidistribute error measure error measure error measure error measure error weight evaluation evaluation evaluation of tensor product spline extension beyond basic interval extension beyond basic interval fn2fm fn2fm fnbrk fnbrk fnbrk fncmb fncmb fncmb fncmb fnder fnder fnder fnder fndir fnint fnint fnint fnint fnjmp fnplt fnplt fnplt fnplt fnplt fnplt fnplt fnplt fnrfn fnrfn fntlr fnval fnval fnval fnval fnval fnval franke Franke function Franke function function functional dual Gauss points getcurve good interpolation sites good interpolation sites graphic accuracy Greville site gridded data gridded data gridded data Hermite cubics Hermite interpolation implicit integral definite indefinite integral equation integration interior break interior break interior knot interpolate interpolation interpolation interpolation interpolation interpolation interpolation Hermite Hermite interpolation points, good jump jump jump jump in derivative knot average average insertion insertion insertion interior multiplicity at endpoints at endpoints sequence sequence sequence sequence appropriate improved simple simple simple knots vs breaks knots vs breaks knt2brk knt2mlt Lagrange end condition least-squares least-squares least-squares approximation least-squares approximation discrete discrete limit from the left limit from the left limit from the left limit from the right linear combination of functions linear dependence linear operations linear space local polynomial coefficients local power form local power form local power form matrix banded mesh meshgrid minimize Moebius multiplicity multiplicity multiplicity multiplicity multiplicity of a data point of a knot of a knot smoothness conditions multivariate multivariate multivariate multivariate multivariate multivariate m-variate Naming conventions natural natural natural nested multiplication newknt newknt Newton's method Newton's method noise noisy nonlinear system nonlinear system normalized B-spline not-a-knot not-a-knot not-a-knot end condition not-a-knot end condition not-a-knot end condition not-a-knot end condition NURBS of a pp-form of the pp-form optimal interpolation optimal interpolation optknt order of a polynomial of a pp of a spline osculatory parabolic parabolic spline parametric parametric parametrization parametrization, chord-length parametrized parametrized perfect spline periodic PGS piecewise-polynomial piecewise-polynomial placeholder notation plotting plotting polygon polyval power form pp ppbrk ppbrk ppform ppform ppform ppform ppform ppform ppform of a B-spline ppmak ppmak ppmak pp-representation QR factorization QR factorization QR factorization quadratic convergence quartic range of a function rational spline rational spline rational spline rBform recovery scheme recurrence relation recurrence relation recurrence relation Remez algorithm restriction to an interval roughness measure roughness measure roughness measure roughness measure roughness weight row-vector rpform rpmak rsmak scalar-valued scaling of a function Schoenberg Schoenberg-Whitney conditions conditions conditions conditions theorem theorem secant method shifted power form side conditions simple knot simple knot simple knot simple knot site slvblk slvblk slvblk slvblk slvblk slvblk slvblk smoothing smoothing parameter smoothing parameter smoothing parameter smoothing parameter smoothing spline smoothness across breaks across knot condition condition condition multiplicity of sort sorted sp2pp sp2pp sp2pp spap2 spap2 spap2 spap2 spap2 spap2 spapi spapi spapi spapi spaps sparse sparse matrix spbrk spbrk spbrk spbrk spbrk spbrk spbrk spbrk spcol spcol spcol spcol spcol spcol spcol spcol spcol spcrv spcrv sphere sphere sphere spline spline spline draftsman's spline approximation to a circle splinetool splpp spmak spmak spmak spmak spmak spmak spmak spmak spmak sprpp staircase shape subdivision support of a B-spline surface surface target Taylor series tensor product tensor product tensor product tensor product tensor product tensor product tensor product trivariate truncated tspdem uniform knot sequence uniform knot sequence uniform mesh unimodal unique spline uniqueness of B-form unit circle univariate value outside basic interval variational approach to splines vector vector vector curve is always a column matrix scaling -valued -valued -valued -valued splines weight